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BREIT, D., L. DIENING, and S. SCHWARZACHER. "SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs." Mathematical Models and Methods in Applied Sciences 23, no. 14 (October 10, 2013): 2671–700. http://dx.doi.org/10.1142/s0218202513500437.
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We consider functions u ∈ L∞(L2)∩Lp(W1, p) with 1 < p < ∞ on a time–space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation uλ of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, Ruzicka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1–46. Since div uλ = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics, J. Differential Equations253 (2012) 1910–1942.
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Bilige, Sudao, and Yanqing Han. "Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 3 (March 5, 2018): 518–31. http://dx.doi.org/10.1108/hff-08-2016-0304.
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Purpose The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics. Design/methodology/approach The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods. Findings First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions. Originality/value The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.
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Swapna, Y. "Applications of Partial Differential Equations in Fluid Physics." Communications on Applied Nonlinear Analysis 31, no. 1 (March 1, 2024): 207–20. http://dx.doi.org/10.52783/cana.v31.396.
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Partial differential equations, or PDEs, assume a critical part in grasping and outlining different fluid physics peculiarities. They have an expansive scope of utilizations, from expecting weather patterns to consolidating ocean streams, fire cycles, and fluid streams into system plan. These equations oversee the way of behaving of fluid amounts like as speed, stress, temperature, and consistency. They portray complex collaborations like changes in precipitation, scattering, and fluid-solid associations. Partial differential equations are utilized to apply the developing methodology. The arrangement is equivalent to for the recently concentrated on examples of typical differential equations. There are two kinds of partial differential equations: nonlinear and straight. Some certifiable equations, for example, those in electrostatics, heat conduction, transmission lines, quantum mechanics, and wave hypothesis, feature the significance of partial differential equations (PDEs). To make sense of something other than one, two, or three pieces of the partial differential equations, we will check out at the speculative piece of those applications that utilization PDEs in this examination. In all parts of science and development, partial differential equations, or PDEs, are generally utilized. Partial differential equations handle most of genuine frameworks. A condition communicating a connection between a piece of no less than two free factors and the partial helpers of this cutoff concerning these free factors is known as a partial differential condition, or PDE.
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Cao, Ruohan, Jin Su, Jinqian Feng, and Qin Guo. "PhyICNet: Physics-informed interactive learning convolutional recurrent network for spatiotemporal dynamics." Electronic Research Archive 32, no. 12 (2024): 6641–59. https://doi.org/10.3934/era.2024310.
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<p>The numerical solution of spatiotemporal partial differential equations (PDEs) using the deep learning method has attracted considerable attention in quantum mechanics, fluid mechanics, and many other natural sciences. In this paper, we propose an interactive temporal physics-informed neural network architecture based on ConvLSTM for solving spatiotemporal PDEs, in which the information feedback mechanism in learning is introduced between the current input and the previous state of network. Numerical experiments on four kinds of classical spatiotemporal PDEs tasks show that the extended models have superiority in accuracy, long-range learning ability, and robustness. Our key takeaway is that the proposed network architecture is capable of learning information correlation of the PDEs model with spatiotemporal data through the input state interaction process. Furthermore, our method also has a natural advantage in carrying out physical information and boundary conditions, which could improve interpretability and reduce the bias of numerical solutions.</p>
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Boyaval, Sébastien. "A class of symmetric-hyperbolic PDEs modelling fluid and solid continua." ESAIM: Proceedings and Surveys 76 (2024): 2–19. http://dx.doi.org/10.1051/proc/202476002.
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We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.
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Dalir, Nemat. "Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/793685.
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Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.
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Moaddy, K., S. Momani, and I. Hashim. "The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics." Computers & Mathematics with Applications 61, no. 4 (February 2011): 1209–16. http://dx.doi.org/10.1016/j.camwa.2010.12.072.
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Da Prato, Giuseppe, and Vicenţiu D. Rădulescu. "Special issue on stochastic PDEs in fluid dynamics, particle physics and statistical mechanics." Journal of Mathematical Analysis and Applications 384, no. 1 (December 2011): 1. http://dx.doi.org/10.1016/j.jmaa.2011.06.058.
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Sharma, Nishchal. "Deep Learning for Solving Partial Differential Equations: A Review of Literature." International Journal for Research in Applied Science and Engineering Technology 12, no. 10 (October 31, 2024): 588–91. http://dx.doi.org/10.22214/ijraset.2024.64623.
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Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.
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Naowarat, Surapol, Sayed Saifullah, Shabir Ahmad, and Manuel De la Sen. "Periodic, Singular and Dark Solitons of a Generalized Geophysical KdV Equation by Using the Tanh-Coth Method." Symmetry 15, no. 1 (January 3, 2023): 135. http://dx.doi.org/10.3390/sym15010135.
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KdV equations have a lot of applications of in fluid mechanics. The exact solutions of the KdV equations play a vital role in the wave dynamics of fluids. In this paper, some new exact solutions of a generalized geophysical KdV equation are computed with the aid of tanh-coth method. To implement the tanh-coth procedure, we first convert the PDEs to ODEs with the help of wave transformation. Then, using a system of algebraic equations, we obtain several soliton solutions. To verify and clearly illustrate the exact solutions, several graphic presentations are developed by giving the parameter values, which are then thoroughly discussed in the relevant components.
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Zaher, A. Z., Khalid K. Ali, and Kh S. Mekheimer. "Electroosmosis forces EOF driven boundary layer flow for a non-Newtonian fluid with planktonic microorganism: Darcy Forchheimer model." International Journal of Numerical Methods for Heat & Fluid Flow 31, no. 8 (June 30, 2021): 2534–59. http://dx.doi.org/10.1108/hff-10-2020-0666.
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Purpose The study of the electro-osmotic forces (EOF) in the flow of the boundary layer has been a topic of interest in biomedical engineering and other engineering fields. The purpose of this paper is to develop an innovative mathematical model for electro-osmotic boundary layer flow. This type of fluid flow requires sophisticated mathematical models and numerical simulations. Design/methodology/approach The effect of EOF on the boundary layer Williamson fluid model containing a gyrotactic microorganism through a non-Darcian flow (Forchheimer model) is investigated. The problem is formulated mathematically by a system of non-linear partial differential equations (PDEs). By using suitable transformations, the PDEs system is transformed into a system of non-linear ordinary differential equations subjected to the appropriate boundary conditions. Those equations are solved numerically using the finite difference method. Findings The boundary layer velocity is lower in the case of non-Newtonian fluid when it is compared with that for a Newtonian fluid. The electro-osmotic parameter makes an increase in the velocity of the boundary layer. The boundary layer velocity is lower in the case of non-Darcian fluid when it is compared with Darcian fluid and as the Forchheimer parameter increases the behavior of the velocity becomes more closely. Entropy generation decays speedily far away from the wall and an opposite effect occurs on the Bejan number behavior. Originality/value The present outcomes are enriched to give valuable information for the research scientists in the field of biomedical engineering and other engineering fields. Also, the proposed outcomes are hopefully beneficial for the experimental investigation of the electroosmotic forces on flows with non-Newtonian models and containing a gyrotactic microorganism.
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Nazir, Umar, Muhammad Sohail, Muhammad Bilal Hafeez, and Marek Krawczuk. "Significant Production of Thermal Energy in Partially Ionized Hyperbolic Tangent Material Based on Ternary Hybrid Nanomaterials." Energies 14, no. 21 (October 21, 2021): 6911. http://dx.doi.org/10.3390/en14216911.
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Nanoparticles are frequently used to enhance the thermal performance of numerous materials. This study has many practical applications for activities that have to minimize losses of energy due to several impacts. This study investigates the inclusion of ternary hybrid nanoparticles in a partially ionized hyperbolic tangent liquid passed over a stretched melting surface. The fluid motion equation is presented by considering the rotation effect. The thermal energy expression is derived by the contribution of Joule heat and viscous dissipation. Flow equations were modeled by using the concept of boundary layer theory, which occurs in the form of a coupled system of partial differential equations (PDEs). To reduce the complexity, the derived PDEs (partial differential equations) were transformed into a set of ordinary differential equations (ODEs) by engaging in similarity transformations. Afterwards, the converted ODEs were handled via a finite element procedure. The utilization and effectiveness of the methodology are demonstrated by listing the mesh-free survey and comparative analysis. Several important graphs were prepared to show the contribution of emerging parameters on fluid velocity and temperature profile. The findings show that the finite element method is a powerful tool for handling the complex coupled ordinary differential equation system, arising in fluid mechanics and other related dissipation applications in applied science. Furthermore, enhancements in the Forchheimer parameter and the Weissenberg number are necessary to control the fluid velocity.
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Xenos, M. A. "An Euler–Lagrange approach for studying blood flow in an aneurysmal geometry." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2199 (March 2017): 20160774. http://dx.doi.org/10.1098/rspa.2016.0774.
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To numerically study blood flow in an aneurysm, the development of an approach that tracks the moving tissue and accounts for its interaction with the fluid is required. This study presents a mathematical approach that expands fluid mechanics principles, taking into consideration the domain’s motion. The initial fluid equations, derived in Euler form, are expanded to a mixed Euler–Lagrange formulation to study blood flow in the aneurysm during the cardiac cycle. Transport equations are transformed into a moving body-fitted reference frame using generalized curvilinear coordinates. The equations of motion consist of a coupled and nonlinear system of partial differential equations (PDEs). The PDEs are discretized using the finite volume method. Owing to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that velocity is substantially influenced by the pulsating wall. Intensification of polymorphic flow patterns is observed. Increments of Reynolds and Womersley numbers are evident as pulsatility increases. The pressure field reveals areas of a lateral pressure gradient at the aneurysm. As pulsatility increases, the diastolic flow vortex shifts towards the aortic wall, distal to the aneurysmal neck. Wall shear stress is amplified at the shoulders of the moving wall compared with that of the rigid one.
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Ayub, Assad, Tanveer Sajid, Wasim Jamshed, William Rolando Miranda Zamora, Leandro Alonso Vallejos More, Luz Marina Galván Talledo, Nélida Isabel Rodríguez Ortega de Peña, Syed M. Hussain, Muhammad Bilal Hafeez, and Marek Krawczuk. "Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate." Applied Sciences 12, no. 17 (August 31, 2022): 8779. http://dx.doi.org/10.3390/app12178779.
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Background: The infinite shear viscosity model of Carreau fluid characterizes the attitude of fluid flow at a very high/very low shear rate. This model has the capacity for interpretation of fluid at both extreme levels, and an inclined magnetic dipole in fluid mechanics has its valuable applications such as magnetic drug engineering, cold treatments to destroy tumors, drug targeting, bio preservation, cryosurgery, astrophysics, reaction kinetics, geophysics, machinery efficiency, sensors, material selection and cosmology. Novelty: This study investigates and interprets the infinite shear rate of Carreau nanofluid over the geometry of a cylindrical channel. The velocity is assumed to be investigated through imposing an inclined magnetic field onto cylindrical geometry. Activation energy is utilized because it helps with chemical reactions and mass transport. Furthermore, the effects of thermophoresis, the binary chemical process and the Brownian movement of nanoparticles are included in this attempt. Formulation: The mathematics of the assumed Carreau model is derived from Cauchy stress tensor, and partial differential equations (PDEs) are obtained. Similarity transformation variables converted these PDEs into a system of ordinary differential equations (ODEs). Passing this system under the bvp4c scheme, we reached at numerical results of this research attempt. Findings: Graphical debate and statistical analysis are launched on the basis of the obtained computed numerical results. The infinite shear rate aspect of Carreau nanofluid gives a lower velocity. The inclined magnetic dipole effect shows a lower velocity but high energy. A positive variation in activation energy amplifies the concentration field.
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Ullah, Malik Zaka. "Irreversibility Marangoni Tri-Hybrid Nanoflow Analysis for Thermal Enhancement Applications." Nanomaterials 13, no. 3 (January 19, 2023): 423. http://dx.doi.org/10.3390/nano13030423.
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Increasing heat transfer is an important part of industrial, mechanical, electrical, thermal, and biological sciences. The aim of this study is to increase the thermal competency of a conventional fluid by using a ternary hybrid nanofluid. A magnetic field and thermal radiation are used to further improve the thermal conductivity of the base fluid. Irreversibility is analyzed under the influence of the embedded parameters. The basic equations for the ternary hybrid nanofluids are transformed from Partial Differential Equations (PDEs) to Ordinary Differential Equations (ODEs) using the similarity concept. The Marangoni convection idea is used in the mathematical model for the temperature difference between the two media: the surface and fluid. The achieved results are provided and discussed. The results show that ternary hybrid nanofluids are more suitable as heat-transmitted conductors than conventional fluids.
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Reddy, M. G., and S. A. Shehzad. "Molybdenum disulfide and magnesium oxide nanoparticle performance on micropolar Cattaneo-Christov heat flux model." Applied Mathematics and Mechanics 42, no. 4 (March 23, 2021): 541–52. http://dx.doi.org/10.1007/s10483-021-2713-9.
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AbstractThis article intends to illustrate the Darcy flow and melting heat transmission in micropolar liquid. The major advantage of micropolar fluid is the liquid particle rotation through an independent kinematic vector named the microrotation vector. The novel aspects of the Cattaneo-Christov (C-C) heat flux and Joule heating are incorporated in the energy transport expression. Two different nanoparticles, namely, MoS2 and MgO, are suspended into the base-fluid. The governing partial differential equations (PDEs) of the prevailing problem are slackening into ordinary differential expressions (ODEs) via similarity transformations. The resulting mathematical phenomenon is illustrated by the implication of fourth-fifth order Runge-Kutta-Fehlberg (RKF) scheme. The fluid velocity and temperature distributions are deliberated by using graphical phenomena for multiple values of physical constraints. The results are displayed for both molybdenum disulphide and magnesium oxide nanoparticles. A comparative benchmark in the limiting approach is reported for the validation of the present technique. It is revealed that the incrementing material constraint results in a higher fluid velocity for both molybdenum disulphide and magnesium oxide nanoparticle situations.
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Dimitrova, Zlatinka I., and Kaloyan N. Vitanov. "Integrability of Differential Equations with Fluid Mechanics Application: from Painleve Property to the Method of Simplest Equation." Journal of Theoretical and Applied Mechanics 43, no. 2 (June 1, 2013): 31–42. http://dx.doi.org/10.2478/jtam-2013-0012.
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Abstract We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order possesses the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani- Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by ex- act reduction of this PDE possesses the Painleve property. The Painleve property motivated much research on obtaining exact solutions on non- linear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.
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Anantha Kumar, K., B. Ramadevi, V. Sugunamma, and J. V. Ramana Reddy. "Heat transfer characteristics on MHD Powell-Eyring fluid flow across a shrinking wedge with non-uniform heat source/sink." Journal of Mechanical Engineering and Sciences 13, no. 1 (March 29, 2019): 4558–74. http://dx.doi.org/10.15282/jmes.13.1.2019.15.0385.
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This report presents the flow and heat transfer characteristics on magnetohydrodynamic non-Newtonian fluid across a wedge near the stagnation point. The fluid flow is time independent and laminar. The radiation and irregular heat sink/source effects are deemed. The system of nonlinear ODEs is attained from PDEs by choosing the proper similarity transformations. Further, the well-known shooting and Runge-Kutta methods are utilized to acquire the problem’s solution subject to assumed boundary conditions. Figures are outlined to emphasize the impact of several parameters on the fields of velocity and temperature. Further, the rate of heat transfer and friction factor are also anticipated and portrayed with the assistance of table. Results indicate that the curves of velocity diminish with shrinking parameter, magnetic field parameter and material fluid parameter. Also the non-uniform heat source/sink parameters play a crucial role in the heat transfer performance.
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Obeidat, Nazek A., and Mahmoud S. Rawashdeh. "On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics." Advances in Mechanical Engineering 15, no. 1 (January 2023): 168781322211498. http://dx.doi.org/10.1177/16878132221149835.
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In shallow waters, the Wu-Zhang (WZ) system describes the (1+1)-dimensional dispersive long wave in two horizontal directions, which is important for the engineering community. This paper presents proofs for various theorems and shows that the natural decomposition method (NDM) solves systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions, such as the Wu-Zhang system. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. Several examples of linear and nonlinear systems (ODEs and PDEs) are given, including the Wu-Zhang (WZ) system. The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential equations. It also holds great promise for additional real-world applications.
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Zaimuddin, Izzatun Nazurah, and Fazlina Aman. "Nanoparticle Shapes (Sphere, Cylinder and Laminar) Impact with Dusty Carbon Nanotubes-Fluid in Magnetohydrodynamics Radiative Flow." Journal of Nanofluids 11, no. 3 (June 1, 2022): 434–52. http://dx.doi.org/10.1166/jon.2022.1850.
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The idea of dust particles embedded on the MHD radiative flow of single walled carbon nanotubes-fluid (SWCNTs) and multi walled carbon nanotubes-fluid (MWCNTs) with different nanoparticle shapes along water, ethylene glycol and engine oil as based fluids has been investigated. Based on the typical shapes (sphere, cylinder and laminar), the rate of heat transfer is analysed in between fluid phase and dust phase for the velocity and temperature profiles for the first time. The partial differential equations (PDEs) are reformed into ordinary differential equations (ODEs) using similarity transformation and are solved numerically using Runge-Kutta Fehlberg method with shooting technique. With several parameters involved such as volume fraction of dust particle/nanoparticle, magnetic strength, thermal radiation and various nanoparticle shapes, it is found that the involvement of dust particle in carbon nanotubes-fluid (CNTs-fluid) has greater dynamic on heat transfer than a normal fluid. Moreover, the shape differences of nanoparticle resulting in different rate of the heat transfer. The impact of different shapes of nanoparticle are compared using fixed parameter values.
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Layla Esmet Jalil, Nada Hadi Malik, and Mohammed Kayqubad Hussein. "A Comparative Study of Partial Differential Equation Solving Methods and Their Applications." Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa 3, no. 1 (January 2, 2025): 01–15. https://doi.org/10.62383/bilangan.v3i1.375.
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In this review, we undertake an in-depth survey of the traditional as well as modern methods used in finding solutions for partial differential equations (henceforth PDEs). We categorise these equations into three main kinds: elliptic, parabolic, and hyperbolic. We also give illustrative examples of these PDEs and discuss the applications of them in a range of fields. This range extends from fluid dynamics (hydrodynamics), as well as thermal (heat) conduction, to quantum mechanics. Our exploration features a number of analyses used in this regard such as variable splitting or defactorising in addition to the transforms invented by Fourier and Laplace. Not only this but also this survey takes in numerical methods ranging from grid-based (finite difference), mesh-based (finite element) to spectral. Also discussed in this paper is a range of special techniques that ranges from the variational techniques, Green's (fundamental solution) functions to perturbation (also known as Asymptotic expansion in addition to sketching the latest developments with respect to computational methods. This review also sheds light on current challenges that confront addressing complicated PDEs especially those nonlinear and multi-variable. In this regard, the paper calls for more research in order to develop more effective methods. The paper maps out the importance of PDE usages in real-life and its potential for more related discoveries in the future particularly with respect to areas such as machine learning and quantum computing.
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Gladys, T., and G. V. Ramana Reddy. "Soret-Dufour Mechanisms on the Thermal Loading of Catteneo-Christov Theories on Magnetohydrodynamic (MHD) Casson Nanofluid Dynamics Over a Stretching Sheet." Journal of Nanofluids 12, no. 6 (June 1, 2023): 1475–84. http://dx.doi.org/10.1166/jon.2023.1937.
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The dynamics of Casson nanofluid with chemically reactive and thermally conductive medium past an elongated sheet were investigated in this study. The thermal loading of the fluids is considered while experimenting the Cattaneo-Christov theories with MHD boundary layer flow. The Rosseland approximation is used on the radiative heat flux because the fluids are optically thin. Partial differential equations were used in the flow model (PDEs). These PDEs were converted to ordinary differential equations (ODEs). The Runge-kutta method and firing techniques were used to solve the altered equations numerically. Graphs were used to depict the effect of relevant flow parameters, while computations on engineering values of relevance were tabulated. The velocity profile was found to degenerate when the visco-inelastic parameter (Casson) was set to a higher value. The boundary layer distributions degenerate when the unsteadiness parameter (A) is increased. The findings revealed that, the plastic dynamic viscosity of the Casson fluid causes reduction to the velocity profile. This paper is unique because it examined the simultaneous thermal loading of two non-Newtonian fluids (Casson-Williamson) nanofluids with experimentation of Cattaneo-Christov theories. To the very best of our knowledge, no study has explored study of this type in literature.
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Athar, Maria, Yasir Khan, Safia Akram, Khalid Saeed, A. Alameer, and Anwar Hussain. "Consequence of Double-Diffusion Convection and Partial Slip on Magneto-Oldroyd-4 Constants Nanofluids with Peristaltic Propulsion in an Asymmetric Channel." Complexity 2022 (September 30, 2022): 1–20. http://dx.doi.org/10.1155/2022/7634357.
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The double-diffusive convection is a significant physical phenomenon that arises in fluid mechanics. It is primarily associated with a convection process in which two dissimilar density gradients with varying diffusion rates are considered. The primary goal of this study is to investigate the effects of double-diffusivity convection and partial slip with an inclined magnetic field on peristaltic propulsion in an asymmetric channel for Oldroyd-4 constants nanofluids. The flow of an Oldroyd-4 constant nanofluid is mathematically modeled in the presence of double-diffusivity convection and a tilted magnetic field. Lubrication methodology is applied to simplify the highly nonlinear system of partial differential equations (PDEs). The numerical scheme is used to calculate the solution of coupled nonlinear PDEs. Furthermore, the effect of changing the parameters associated with slip, thermophoresis, Brownian motion, Grashof number of nanoparticles, Hartmann number, pumping, and trapping are investigated in this article. It is noticed that the temperature rises as the Brownian motion and thermophoresis constraints increases. This is because the growth in the Brownian motion parameter indicates the increase in the kinetic energy of nanoparticles which results in warming up the nanofluid. Also, concentration falls as the Brownian motion and thermophoresis constraints increases.
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Liu, Yanbing, Liping Chen, Yu Chen, and Jianwan Ding. "An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information." Entropy 26, no. 6 (May 26, 2024): 451. http://dx.doi.org/10.3390/e26060451.
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Physics-informed neural networks (PINNs) have garnered widespread use for solving a variety of complex partial differential equations (PDEs). Nevertheless, when addressing certain specific problem types, traditional sampling algorithms still reveal deficiencies in efficiency and precision. In response, this paper builds upon the progress of adaptive sampling techniques, addressing the inadequacy of existing algorithms to fully leverage the spatial location information of sample points, and introduces an innovative adaptive sampling method. This approach incorporates the Dual Inverse Distance Weighting (DIDW) algorithm, embedding the spatial characteristics of sampling points within the probability sampling process. Furthermore, it introduces reward factors derived from reinforcement learning principles to dynamically refine the probability sampling formula. This strategy more effectively captures the essential characteristics of PDEs with each iteration. We utilize sparsely connected networks and have adjusted the sampling process, which has proven to effectively reduce the training time. In numerical experiments on fluid mechanics problems, such as the two-dimensional Burgers’ equation with sharp solutions, pipe flow, flow around a circular cylinder, lid-driven cavity flow, and Kovasznay flow, our proposed adaptive sampling algorithm markedly enhances accuracy over conventional PINN methods, validating the algorithm’s efficacy.
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Wang, Kang-Jia, Jing-Hua Liu, Jing Si, and Guo-Dong Wang. "Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics." Axioms 12, no. 1 (January 16, 2023): 95. http://dx.doi.org/10.3390/axioms12010095.
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This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.
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Panayotounakos, D. E. "A new technique in constructing closed-form solutions for nonlinear PDEs appearing in fluid mechanics and gas dynamics." Mathematical Problems in Engineering 2, no. 4 (1996): 301–18. http://dx.doi.org/10.1155/s1024123x96000361.
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Shen, Luhang, Daolun Li, Wenshu Zha, Li Zhang, and Jieqing Tan. "Physical Asymptotic-Solution nets: Physics-driven neural networks solve seepage equations as traditional numerical solution behaves." Physics of Fluids 35, no. 2 (February 2023): 023603. http://dx.doi.org/10.1063/5.0135716.
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Deep learning for solving partial differential equations (PDEs) has been a major research hotspot. Various neural network frameworks have been proposed to solve nonlinear PDEs. However, most deep learning-based methods need labeled data, while traditional numerical solutions do not need any labeled data. Aiming at deep learning-based methods behaving as traditional numerical solutions do, this paper proposed an approximation-correction model to solve unsteady compressible seepage equations with sinks without using any labeled data. The model contains two neural networks, one for approximating the asymptotic solution, which is mathematically correct when time tends to 0 and infinity, and the other for correcting the error of the approximation, where the final solution is physically correct by constructing the loss function based on the boundary conditions, PDE, and mass conservation. Numerical experiments show that the proposed method can solve seepage equations with high accuracy without using any labeled data, as conventional numerical solutions do. This is a significant breakthrough for deep learning-based methods to solve PDE.
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Das, NABANITA, Subhrajit Sarma, and Nazibuddin Ahmed. "THERMAL DIFFUSION, AND THERMAL RADIATION EFFECTS ON UNSTEADY MHD CONVECTIVE FLOW PAST AN INCLINED VERTICAL PLATE." Latin American Applied Research - An international journal 53, no. 4 (July 18, 2023): 341–48. http://dx.doi.org/10.52292/j.laar.2023.1066.
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In this investigation, thermal diffusion and thermal radiation effects are studied on an unsteady natural convective flow of an electrically conducting fluid past an inclined vertical plate. A uniform aligned magnetic field is imposed onto the fluid or the plate. The governing PDEs are solved using the Laplace transformation technique. The effect of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number are analyzed graphically. It is observed that velocity field rises for increasing the value of the thermal-diffusion effect. Moreover, fluid concentration gets ososcillate with the effect of Prandtl number and thermal radiation parameter.
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Endalew, Mehari Fentahun, Masitawal Demsie Goshu, and Yimer Chekol Tegegne. "Unsteady MHD Thin Film Flow of a Second-Grade Fluid past a Tilted Plate under the Impact of Thermal Radiation and Chemical Reaction." Journal of Applied Mathematics 2022 (August 25, 2022): 1–12. http://dx.doi.org/10.1155/2022/9491308.
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This paper explores the impact of chemical reaction and thermal radiation on time-dependent hydromagnetic thin-film flow of a second-grade fluid across an inclined flat plate embedded in a porous medium. The thermal radiation based on the Rosseland approximation is incorporated in the energy equation. Uniform applied magnetic field and first-order homogenous chemical reaction are included in the momentum and concentration equations, respectively. The novel mathematical flow model is constructed by using a set of partial differential equations (PDEs). The PDEs are then transformed into an equivalent set of ordinary differential equations (ODEs) and solved by applying the Laplace transform method. However, the time domain solutions are obtained by using the INVLAP subroutine of MATLAB. Physical parameters influencing thin-film velocity, temperature, and concentration are illustrated graphically, while those affecting skin friction, heat, and mass transfer rates are presented in a tabular form. It is found that thin-film velocity and temperature boost with increasing values of thermal radiation, but thin-film velocity decreases with increasing values of chemical reaction and magnetic field. The current investigation is to enhance heat and mass transfer in the design of mechanical systems involving the thin film flow of second-grade fluids over an inclined flat plate after applying thermal radiation and chemical reaction.
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Choudhary, Sushila, and Nihal Chand. "Magnetohydrodynamic Flow and Heat Transfer Analysis on Ethylene Glycol Based Nano Fluid Over a Vertical Permeable Circular Cylinder with Joule Heating and Radiation." Journal of Nanofluids 11, no. 5 (August 1, 2022): 664–74. http://dx.doi.org/10.1166/jon.2022.1873.
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This research’s contribution is towards determining heat transfer characteristics of Ag–C2H6O2 and Cu–C2H6O2 nano fluid over a vertical porous circular cylindrical surface. The mixed convection flow in the presence of electric conductivity, Joule heating and thermal radiation near a stagnation point is considered for investigation. Ethylene glycol is taken as base fluid while copper and silver are nanoparticles. Through similarity transformations, the governing PDEs for momentum, energy, and concentration are turned into ODEs, which are then interpreted using a fourth-order exactness programme (Bvp4c). The parametric impacts on concentration, temperature and velocity are thoroughly discussed graphically while impact on the rate of heat transfer, skin friction and rate of mass transfer is obtained in numeral form. The obtained results are compared to published literature and a comparison between Ag–C2H6O2 and Cu–C2H6O2 nano fluids is demonstrated.
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Hasnain, Jafar, Zaheer Abbas, Mariam Sheikh, and Shaban Aly. "Analysis of dusty Casson fluid flow past a permeable stretching sheet bearing power law temperature and magnetic field." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 6 (August 9, 2019): 3463–80. http://dx.doi.org/10.1108/hff-11-2018-0685.
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Purpose This study aims to present an analysis on heat transfer attributes of fluid-particle interaction over a permeable elastic sheet. The fluid streaming on the sheet is Casson fluid (CF) with uniform distribution of dust particles. Design/methodology/approach The basic steady equations of the CF and dust phases are in the form of partial differential equations (PDEs) which are remodeled into ordinary ones with the aid of similarity transformations. In addition to analytical solution, numerical solution is obtained for the reduced coupled non-linear ordinary differential equations (ODEs) to validate the results. Findings The solution seems to be influenced by significant physical parameters such as CF parameter, magnetic parameter, suction parameter, fluid particle interaction parameter, Prandtl number, Eckert number and number density. The impact of these parameters on flow field and temperature for both fluid and dust phases is presented in the form of graphs and discussed in detail. The effect on skin friction coefficient and heat transfer rate is also presented in tabular form. It has been observed that an increase in the CF parameter curtails the fluid velocity as well as the particle velocity however enhances the heat transfer rate at the wall. Furthermore, comparison of the numerical and analytical solution is also made and found to be in excellent agreement. Originality/value Although the analysis of dusty fluid flow has been widely examined, however, the present study obtained both analytical and numerical results of power law temperature distribution in dusty Casson fluid under the influence of magnetic field which are new and original for such type of flow.
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Rehman, Saif-ur, Nazir Ahmad Mir, Muhammad Farooq, Naila Rafiq, and Shakeel Ahmad. "Analysis of thermally stratified radiative flow of Sutterby fluid with mixed convection." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 236, no. 2 (December 16, 2021): 934–42. http://dx.doi.org/10.1177/09544062211007887.
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In this attempt, we investigate the mixed convection in Sutterby fluid flow based on boundary layer approximation. Heat transport analysis is composed of stratification and thermal radiative phenomena. The system of non-linear PDEs is transformed into coupled ODEs. Convergent series approximations are evaluated via homotopic technique. Influence of various pertinent parameters is sketched and graphically analyzed. It is found that horizontal velocity increments for higher mixed convection parameter. The radiation parameter has a similar relation with temperature whereas the stratification parameter shows opposite behavior for temperature field.
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K, Prasanna Kumar. "An Analytical Approach to Solving Partial Differential Equations in Mathematical Modeling." International Journal for Research in Applied Science and Engineering Technology 12, no. 12 (December 31, 2024): 1651–61. https://doi.org/10.22214/ijraset.2024.66100.
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Partial Differential Equations (PDEs) are quite central in the applied mathematical modeling, and bring out the essence of yielding solution insights of physics, civil and mechanical engineering, biology and even financial related systems. This study focuses on putting into practice first fundamentals of PDEs such as separation of variables, Fourier, transforms, Green’s functions as well as series expansions. The analytical techniques for solving problems of heat conduction, wave propagation and fluid dynamics are derived and applied and various problems are solved. Comparisons to other methods with numbers bring into light primary and secondary advantages and disadvantages of analytical solutions and their computational complexity. The work focuses on the stability/ convergence/ applications/ potential of these techniques and the author suggests mixed analytical-numerical approaches in further research. This work links abstract theories to real-life problems and ensures better prediction as well as innovations for a variety of fields.
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Jawad, Muhammad. "A Computational Study on Magnetohydrodynamics Stagnation Point Flow of Micropolar Fluids with Buoyancy and Thermal Radiation due to a Vertical Stretching Surface." Journal of Nanofluids 12, no. 3 (April 1, 2023): 759–66. http://dx.doi.org/10.1166/jon.2023.1958.
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In current analysis, A numerical approach for magnetohydrodynamics Stagnation point flow of Micropolar fluid due to a vertical stretching Surface is reported. The impact of buoyancy forces is considered. In additions the effects of the thermal radiation and thermal conductivity with non-zero mass flux have been analyzed. we implement the dimensionless variable technique and the systems of coupled non-linear PDEs are transformed into ODEs by using the appropriate similarity technique. Moreover, by using package ND-Solve on Mathematica problem is numerically integrated with the help of shooting technique. Numerical approach for magnetohydrodynamics Stagnation point flow of thermal Radiative Micropolar fluid due to a vertical stretching Surface. The impact of thermophoresis and Brownian motion are considered. We implement the dimensionless variable technique and the systems of coupled non-linear PDEs are transformed into ODEs by using the appropriate similarity technique. To observe the influence of the physical parameters, graphically valuations are performed for numerous emerging parameters like Brownian motion, mixed convection parameters, thermophoresis diffusion, Hartman number, Radiation parameter, Prandtl number, Stretching parameter and other dimension less parameters. These several protuberant parameters of interest are engaged for velocity, temperature and nonlinear micro rotation profile and studied in detail.
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Huang, Adam, and Daniel D. Joseph. "Stability of eccentric core–annular flow." Journal of Fluid Mechanics 282 (January 10, 1995): 233–45. http://dx.doi.org/10.1017/s0022112095000127.
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Perfect core-annular flows are two-phase flows, for example of oil and water, with the oil in a perfectly round core of constant radius and the water outside. Eccentric core flows can be perfect, but the centre of the core is displaced off the centre of the pipe. The flow is driven by a constant pressure gradient, and is unidirectional. This kind of flow configuration is a steady solution of the governing fluid dynamics equations in the cases when gravity is absent or the densities of the two fluids are matched. The position of the core is indeterminate so that there is a family of these eccentric core flow steady solutions. We study the linear stability of this family of flows using the finite element method to solve a group of PDEs. The large asymmetric eigenvalue problem generated by the finite element method is solved by an iterative Arnoldi's method. We find that there is no linear selection mechanism; eccentric flow is stable when concentric flow is stable. The interface shape of the most unstable mode changes from varicose to sinuous as the eccentricity increases from zero.
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Parmar, Amit, Rakesh Choudhary, and Krishna Agarwal. "Magnetohydrodynamics Williamson Fluid Comprising Gyrotactic Microorganisms Flows Through a Permeable Stretching Layer with Variable Fluid Properties." Journal of Nanofluids 9, no. 4 (December 1, 2020): 375–87. http://dx.doi.org/10.1166/jon.2020.1762.
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The present study shows the impacts of Williamson fluid with magnetohydrodynamics flow containing gyrotactic microorganisms under the variable fluid property past permeable stretching sheet. Variable Prandtl number, mass Schmidt number, and gyrotactic microorganisms Schmidt number were all considered. The momentum, energy, mass, and microorganism equations’ governing PDEs are converted into nonlinear coupled ODEs and numerically solved with the bvp4c solver using suitable transformations. The main outcome of this study is that Williamson fluid parameter constantly decreases in velocity profile, however reverse effects can be shown in temperature profile. Also, M parameter and Kp parameter enhance the heat transfer rate, concentration rate and microorganisms boundary layer thickness but declines in momentum boundary layer thickness and velocity profile. The aim of this research is to see how velocity slide, temperature jump, concentration slip, and microorganism slip affect MHD Williamson fluid flow with gyrotactic microorganisms over a leaky surface embedded in spongy medium, with non-linear radiation and non-linear chemical reaction.
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Omole, Ezekiel Olaoluwa, Emmanuel Oluseye Adeyefa, Victoria Iyadunni Ayodele, Ali Shokri, and Yuanheng Wang. "Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies." Axioms 12, no. 9 (September 18, 2023): 891. http://dx.doi.org/10.3390/axioms12090891.
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A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.
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Chepngetich, Winny. "The lie symmetry analysis of third order Korteweg-de Vries equation." Journal of Physical and Applied Sciences (JPAS) 1, no. 1 (November 1, 2022): 38–43. http://dx.doi.org/10.51317/jpas.v1i1.299.
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This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.
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Liu, Zhixiang, Yuanji Chen, Ge Song, Wei Song, and Jingxiang Xu. "Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics." Mathematics 11, no. 19 (October 1, 2023): 4147. http://dx.doi.org/10.3390/math11194147.
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Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.
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Murtaza, Saqib, Poom Kumam, Zubair Ahmad, Muhammad Ramzan, Ibne Ali, and Anwar Saeed. "Computational Simulation of Unsteady Squeezing Hybrid Nanofluid Flow Through a Horizontal Channel Comprised of Metallic Nanoparticles." Journal of Nanofluids 12, no. 5 (June 1, 2023): 1327–34. http://dx.doi.org/10.1166/jon.2023.2020.
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The characteristics of hybrid nanofluid flow contained copper (Cu) and cobalt ferrite (CoFe2O4) nanoparticles (NPs) across a squeezing plate have been computationally evaluated in the present report. In biomedical fields, in very rare cases fluid flow through a static channel. Similarly in industrial sights, we are also often observed that the fluid flows through comprising plates rather than fixed plates (flow in vehicle’s engine between nozzles and piston). CoFe2O4 and Cu nanoparticles are receiving huge attention in medical and technical research due to their broad range of applications. For this purpose, the phenomena have been expressed in the form of the system of PDEs with the additional effect of suction/injection, heat source, chemical reaction, and magnetic field. The system of PDEs is simplified to the dimensionless set of ODEs through similarity replacements. Which further deals with the computational approach parametric continuation method. For the validity and accuracy of the outcomes, the results are confirmed with the existing works. The results are displayed and evaluated through Figures. It is detected that the hybrid nanoliquid has a greater ability for the velocity and energy conveyance rate as related to the nanofluid. Furthermore, the energy profile declines with the consequences of unsteady squeezing term, while enhances with the effects of suction factor, heat absorption and generation, and lower plate stretching sheet.
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Rashidi, M. M., M. Ali, N. Freidoonimehr, B. Rostami, and M. Anwar Hossain. "Mixed Convective Heat Transfer for MHD Viscoelastic Fluid Flow over a Porous Wedge with Thermal Radiation." Advances in Mechanical Engineering 6 (January 1, 2014): 735939. http://dx.doi.org/10.1155/2014/735939.
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The main concern of the present paper is to study the MHD mixed convective heat transfer for an incompressible, laminar, and electrically conducting viscoelastic fluid flow past a permeable wedge with thermal radiation via a semianalytical/numerical method, called Homotopy Analysis Method (HAM). The boundary-layer governing partial differential equations (PDEs) are transformed into highly nonlinear coupled ordinary differential equations (ODEs) consisting of the momentum and energy equations using similarity solution. The current HAM solution demonstrates very good agreement with previously published studies for some special cases. The effects of different physical flow parameters such as wedge angle (β), magnetic field ( M), viscoelastic ( k1), suction/injection ( fw), thermal radiation ( Nr), and Prandtl number (Pr) on the fluid velocity component ( f′( η)) and temperature distribution ( θ( η)) are illustrated graphically and discussed in detail.
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Hou, Enran, Fuzhang Wang, Umar Nazir, Muhammad Sohail, Noman Jabbar, and Phatiphat Thounthong. "Dynamics of Tri-Hybrid Nanoparticles in the Rheology of Pseudo-Plastic Liquid with Dufour and Soret Effects." Micromachines 13, no. 2 (January 27, 2022): 201. http://dx.doi.org/10.3390/mi13020201.
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The rheology of different materials at the micro and macro levels is an area of great interest to many researchers, due to its important physical significance. Past experimental studies have proved the efficiency of the utilization of nanoparticles in different mechanisms for the purpose of boosting the heat transportation rate. The purpose of this study is to investigate heat and mass transport in a pseudo-plastic model past over a stretched porous surface in the presence of the Soret and Dufour effects. The involvement of tri-hybrid nanoparticles was incorporated into the pseudo-plastic model to enhance the heat transfer rate, and the transport problem of thermal energy and solute mechanisms was modelled considering the heat generation/absorption and the chemical reaction. Furthermore, traditional Fourier and Fick’s laws were engaged in the thermal and solute transportation. The physical model was developed upon Cartesian coordinates, and boundary layer theory was utilized in the simplification of the modelled problem, which appears in the form of coupled partial differential equations systems (PDEs). The modelled PDEs were transformed into corresponding ordinary differential equations systems (ODEs) by engaging the appropriate similarity transformation, and the converted ODEs were solved numerically via a Finite Element Procedure (FEP). The obtained solution was plotted against numerous emerging parameters. In addition, a grid independent survey is presented. We recorded that the temperature of the tri-hybrid nanoparticles was significantly higher than the fluid temperature. Augmenting the values of the Dufour number had a similar comportment on the fluid temperature and concentration. The fluid temperature increased against a higher estimation of the heat generation parameter and the Eckert numbers. The impacts of the buoyancy force parameter and the porosity parameter were quite opposite on the fluid velocity.
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Avalos, George, Irena Lasiecka, and Roberto Triggiani. "Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System." gmj 15, no. 3 (September 2008): 403–37. http://dx.doi.org/10.1515/gmj.2008.403.
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Abstract This paper considers an established model of a parabolic-hyperbolic coupled system of two PDEs, which arises when an elastic structure is immersed in a fluid. Coupling occurs at the interface between the two media. Semigroup well-posedness on the space of finite energy for {𝑤, 𝑤𝑡, 𝑢} was established in [Contemp. Math. 440: 15–54, 2007]. Here, [𝑤, 𝑤𝑡] are the displacement and the velocity of the structure, while 𝑢 is the velocity of the fluid. The domain D(A) of the generator A does not carry any smoothing in the 𝑤-variable (its resolvent 𝑅(λ, A) is not compact on this component space). This raises the issue of higher regularity of solutions. This paper then shows that the mechanical displacement, fluid velocity, and pressure terms do enjoy a greater regularity if, in addition to the I.C. {𝑤0, 𝑤1, 𝑢0} ∈ D(A), one also has 𝑤0 in (𝐻2(Ω𝑠))𝑑.
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Panda, R. C., L. Panigrahi, M. K. Nayak, A. J. Chamkha, S. S. Sahoo, and A. K. Barik. "Nanofluid Based Pipe Flow Analysis in Absorber Pipe of Flat Plate Solar Collector: Effects of Inclination and Porosity." Journal of Nanofluids 12, no. 2 (March 1, 2023): 458–64. http://dx.doi.org/10.1166/jon.2023.1979.
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Nanofluid applications in solar collectors are an emerging area for enhanced heat transfer resulting in heat gain for domestic and industrial use. In the present work, the performance of a Flat Plate Solar Collector (FPSC) having water-CuO-based nanofluid has been studied. The effect of the tilting angle of cylindrical pipe and porosity of porous material is investigated for this nanofluid-based FPSC. A numerical approach has been adopted to stimulate the governing equations in the tube. The similarity transformation simplifies the model (PDEs) into ordinary differential equations (ODEs). The governing non-dimensional PDEs along with their appropriate boundary conditions are solved numerically using the 4th order Runge-Kutta method cum shooting technique. The impacts of significant and relevant physical parameters and physical quantities of interest are analyzed. From the present study, it is observed that amplification of tilting angle and curvature parameter ameliorates the heat transfer rate while that of porosity parameter controls it effectively. A similar approach can be employed for other solar collectors to assess the heat transfer augmentation by using nanofluids instead of existing fluids.
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Borisov, V. E., A. V. Ivanov, B. V. Kritsky, and E. B. Savenkov. "Numerical Algorithms for Simulation of a Fluid-Filed Fracture Evolution in a Poroelastic Medium." PNRPU Mechanics Bulletin, no. 2 (December 15, 2021): 24–35. http://dx.doi.org/10.15593/perm.mech/2021.2.03.
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The paper deals with the computational framework for the numerical simulation of the three dimensional fluid-filled fracture evolution in a poroelastic medium. The model consists of several groups of equations including the Biot poroelastic model to describe a bulk medium behavior, Reynold’s lubrication equations to describe a flow inside fracture and corresponding bulk/fracture interface conditions. The geometric model of the fracture assumes that it is described as an arbitrary sufficiently smooth surface with a boundary. Main attention is paid to describing numerical algorithms for particular problems (poroelasticity, fracture fluid flow, fracture evolution) as well as an algorithm for the coupled problem solution. An implicit fracture mid-surface representation approach based on the closest point projection operator is a particular feature of the proposed algorithms. Such a representation is used to describe the fracture mid-surface in the poroelastic solver, Reynold’s lubrication equation solver and for simulation of fracture evolutions. The poroelastic solver is based on a special variant of X-FEM algorithms, which uses the closest point representation of the fracture. To solve Reynold’s lubrication equations, which model the fluid flow in fracture, a finite element version of the closet point projection method for PDEs surface is used. As a result, the algorithm for the coupled problem is purely Eulerian and uses the same finite element mesh to solve equations defined in the bulk and on the fracture mid-surface. Finally, we present results of the numerical simulations which demonstrate possibilities of the proposed numerical techniques, in particular, a problem in a media with a heterogeneous distribution of transport, elastic and toughness properties.
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Abu Arqub, Omar. "Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 11 (July 8, 2019): 4711–33. http://dx.doi.org/10.1108/hff-10-2017-0394.
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Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.
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Lienstromberg, Christina, Stefan Schiffer, and Richard Schubert. "A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case." Archive for Rational Mechanics and Analysis 247, no. 2 (April 2023). http://dx.doi.org/10.1007/s00205-023-01849-w.
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AbstractWe introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $${\mathscr {A}}$$ A -quasiconvexity, we show a $$\Gamma $$ Γ -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.
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Farooq, Umer, Muhammad Irfan, Shaheer Khalid, Ahmed Jan, and Muzamil Hussain. "Computational convection analysis of second grade MHD nanofluid flow through porous medium across a stretching surface." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, February 4, 2024. http://dx.doi.org/10.1002/zamm.202300401.
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AbstractNanofluids significantly influence modern life due to their vast usages in the modern era. Transport analysis of non‐Newtonian fluids with nanoparticles (carbon nanotubes) over a stretching surface is the primary focus of the present investigation. Effects of viscous dissipation, porous medium, magnetic field, and heat source factors are also examined. Boundary layers approximation is used to simulate the underlying collection of partial differential equations (PDEs) that regulate the system. Using non‐similarity transformation, the basic PDEs of the model are converted into dimensionless nonlinear PDEs. Up to the second level of truncation PDEs that can be treated as ordinary differential equations (ODEs) are generated from dimensionless PDEs utilizing local non‐similarity approach. The resultant nonlinear differential equations are computationally solved using bvp4c code of MATLAB technique. In this study, we compile and compare the obtained findings with previous research. Graphs demonstrate how the velocity and temperature profiles depend entirely on a wide range of physical parameters. A rise in the second‐grade fluid parameter upsurges the velocity distribution, while an increment in porosity and magnetic parameter declines the velocity distribution. As magnetic values, Eckert number, porosity parameter, and heat source increase, so does the temperature profile of the nanofluid.
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M. Krishna Reddy and N. Vijayabhaskar Reddy. "AI Based Analysis and Partial Differential Equations." International Journal of Advanced Research in Science, Communication and Technology, July 21, 2024, 86–90. http://dx.doi.org/10.48175/ijarsct-19212.
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The intersection of artificial intelligence (AI) and partial differential equations (PDEs), emphasizing how AI techniques can revolutionize the analysis and solution of PDEs in various scientific and engineering applications. Traditional methods for solving PDEs often face challenges related to computational complexity, high-dimensionality, and nonlinearity. By leveraging advanced AI algorithms, particularly deep learning and neural networks, we propose novel approaches to approximate solutions, reduce computational costs, and handle complex boundary conditions more effectively. The study highlights the advantages of AI-driven methods in terms of accuracy, efficiency, and scalability, presenting case studies from fluid dynamics, quantum mechanics, and financial mathematics. Our findings suggest that AI has the potential to significantly enhance the analytical capabilities and practical applications of PDEs, paving the way for new advancements in both theoretical research and real-world problem solving
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Kyriakoudi, Konstantina C., and Michail A. Xenos. "Magnetohydrodynamic effects on a pathological vessel: An Euler–Lagrange approach." Physics of Fluids 35, no. 12 (December 1, 2023). http://dx.doi.org/10.1063/5.0177036.
Full textAbstract:
For numerically studying blood flow in a pathological vessel under the influence of a magnetic field, it is necessary to develop an approach that tracks the moving tissue and accounts for interactions between the fluid, the arterial wall, and the magnetic field. The current study discusses a mathematical approach of the fluid's motion under the influence of a magnetic field using fluid mechanics principles. A mixed Euler–Lagrange formulation is introduced to mathematically describe the blood flow in the aneurysm during the entire cardiac cycle. Blood is considered a Newtonian, incompressible, and electrically conducting fluid, subjected to a static and uniform magnetic field. Generalized curvilinear coordinates are used to transform the transport equations into body-fitted geometries and provide a manageable form of equations. The system of equations related to motion consists of a coupled and nonlinear system of partial differential equations (PDEs). The discretization of the PDEs is performed using the finite volume method. The addition of the Lorentz force in the momentum PDEs describes the applied uniform magnetic field in the blood flow. Due to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that the magnetic field strongly influences blood flow, reducing the velocity field q¯ and increasing the pressure drop, Δp.