Journal articles: 'PDEs with coupled fluxes'

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Assefa, Migbar, Xin Lai, Lisheng Liu, and Yang Liao. "Peridynamic Formulation for Coupled Thermoelectric Phenomena." Advances in Materials Science and Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/9836741.

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Modeling of heat and electrical current flow simultaneously in thermoelectric convertor using classical theories do not consider the influence of defects in the material. This is because traditional methods are developed based on partial differential equations (PDEs) and lead to infinite fluxes at the discontinuities. The usual way of solving such PDEs is by using numerical technique, like Finite Element Method (FEM). Although FEM is robust and versatile, it is not suitable to model evolving discontinuities. To avoid such shortcomings, we propose the concept of peridynamic theory to derive the balance of energy and charge equations in the coupled thermoelectric phenomena. Therefore, this paper presents the transport of heat and charge in thermoelectric material in the framework of peridynamic (PD) theory. To illustrate the reliability of the PD formulation, numerical examples are presented and results are compared with those from literature, analytical solutions, or finite element solutions.

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Ullah, Asad, Nahid Fatima, Khalid Abdulkhaliq M. Alharbi, Samia Elattar, Ikramullah Ikramullah, and Waris Khan. "A Numerical Analysis of the Hybrid Nanofluid (AgTiO2Water) Flow in the Presence of Heat and Radiation Fluxes++." Energies 16, no. 3 (January 22, 2023): 1220. http://dx.doi.org/10.3390/en16031220.

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The hydrothermal characteristics of (Ag+TiO2+H2O) hybrid nanofluid three dimensional flow between two vertical plates, in which the right permeable plate stretches as well as rotates, are investigated by employing varying magnetic, heat and radiation fluxes. The motion is governed by coupled PDEs (nonlinear) obeying suitable boundary conditions. The PDEs coupled system is transformed to a coupled set of nonlinear ODEs employing appropriate similarity transformation relations. The resultant equations are numerically solved through the bv4c solver. The impact of the changing strength of associated parameters on the flow is investigated graphically and through tables. It has been found that the velocity gradient and velocity initially increase and then decrease with increasing Grashof number values in both the suction and injection cases. The enhancing magnetic field first augments and then lowers the velocity gradient in the presence of radiation source of maximum strength. The increasing strength of injection parameter drops the velocity. The temperature distribution in the fluid increases with the increasing Eckert number, radiation flux and heat strength and nanomaterial concentration, and depreciates with the enhancing injection parameter values and Prandtl number. The Cfx increases with a higher magnetic field magnitude and nanomaterial concentration, and declines with an increasing Grashof number. The results obtained are compared with the available literature in the form of tables.

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Zeleke, Migbar Assefa, Lai Xin, and Li Sheng Liu. "Bond Based Peridynamic Formulation for Thermoelectric Materials." Materials Science Forum 883 (January 2017): 51–59. http://dx.doi.org/10.4028/www.scientific.net/msf.883.51.

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Modeling of heat and electrical current flow simultaneously in thermoelectric convertor using classical theories do not consider the influence of defects in the material. This is because traditional methods are developed based on partial differential equations (PDEs) and lead to infinite fluxes and stress fields at the crack tips. The usual way of solving such PDEs is by using numerical technique, like Finite Element Method (FEM). Although FEM is robust and versatile, it is not suitable to model evolving discontinuities since discontinuous fields are mathematically singular at the crack tip and required an external criterion for the prediction of crack growth. In this paper, we follow the concept of peridynamic (PD) theory to overcome the shortcomings above. Therefore, the main aim of this paper is to develop the peridynamic equations for the generalized Fourier’s and Ohm’s laws. Furthermore, we derived the peridynamic equations for the conservation of energy and charge for the coupled thermoelectric phenomena.

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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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Yates, Christian A., and Mark B. Flegg. "The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion." Journal of The Royal Society Interface 12, no. 106 (May 2015): 20150141. http://dx.doi.org/10.1098/rsif.2015.0141.

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Spatial reaction–diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction–diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as ‘compartment-based’ or ‘on-lattice’. These models are characterized by a discretization of the computational domain into a grid/lattice of ‘compartments’. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: ‘how are individual particles transported between the vastly different model descriptions?’ First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations.

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Mishra, S. R., S. Baag, and S. K. Parida. "Entropy Generation Analysis on Magnetohydrodynamic Eyring-Powell Nanofluid Over a Stretching Sheet by Heat Source/Sink." Journal of Nanofluids 11, no. 4 (August 1, 2022): 537–44. http://dx.doi.org/10.1166/jon.2022.1861.

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In this communication, the analysis of the entropy generation on magnetohydrodynamic (MHD) Eyring-Powell nanofluid over a stretching sheet with the effects of heat source/sink is reported. The presence of thermophoresis and Brownian motion are responsible for the enhancement in the properties of heat transfer. With the help of suitable similarity transformation entity, the involved governing partially differential equations (PDEs) are converted into nonlinear coupled ordinary differential equations (ODEs). Further, converted differential equations are solved by numerical methods such as Runge-Kutta fourth order correlated with shooting technique. Influence of various pertinent physical parameters is discussed via velocity, temperature, concentration and entropy profiles. The effect of these variables on the quantities of engineering advance such as Nusselt and Sherwood number are furnished in illustrative form and discussed. Further, the major findings of the outcomes are laid down as follows; the Brownian motion of the particles enhances the fluid temperature whereas thermophoresis retards significantly. The entropy generation overshoots due to the increase in the Reynolds number. Nanofluids with high critical heat fluxes and high-power density have the potential to provide the required cooling effect in military ships, submarines, wave energy converters and high-power laser diodes.

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Mahdy, A., and Ali J. Chamkha. "Unsteady MHD boundary layer flow of tangent hyperbolic two-phase nanofluid of moving stretched porous wedge." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 11 (November 5, 2018): 2567–80. http://dx.doi.org/10.1108/hff-12-2017-0499.

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Purpose The purpose of this paper is to address the thermo-physical impacts of unsteady magneto-hydrodynamic (MHD) boundary layer flow of non-Newtonian tangent hyperbolic nanofluid past a moving stretching wedge. To delineate the nanofluid, the boundary conditions for normal fluxes of the nanoparticle volume fraction are chosen to be vanish. Design/methodology/approach The local similarity transformation is implemented to reformulate the governing PDEs into coupled non-linear ODEs of higher order. Then, numerical solution is obtained for the simplified governing equations with the aid of finite difference technique. Findings Numerical calculations point out that pressure gradient parameter leads to improve all skin friction coefficient, rate of heat transfer and absolute value of rate of nanoparticle concentration. As well as, lager values of Weissenberg number tend to upgrade the skin friction coefficient, while power law index and velocity ratio parameter reduce the skin friction coefficient. Again, the horizontal velocity component enhances with upgrading power law index, unsteadiness parameter, velocity ratio parameter and Darcy number and it reduces with rising values of Weissenberg number. Originality/value A numerical treatment of unsteady MHD boundary layer flow of tangent hyperbolic nanofluid past a moving stretched wedge is obtained. The problem is original.

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Veera Reddy, K., G. Venkata Ramana Reddy, and Ali J. Chamkha. "Effects of Viscous Dissipation and Thermal Radiation on an Electrically Conducting Casson-Carreau Nanofluids Flow with Cattaneo-Christov Heat Flux Model." Journal of Nanofluids 11, no. 2 (April 1, 2022): 214–26. http://dx.doi.org/10.1166/jon.2022.1836.

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The primary goal of this research is to study the Cattaneo-Christov heat flux model on the impacts of mass and energy transit of MHD Casson-Carreu nanofluid through a permeable vertical accelerating plate with Soret and Dufour mechanism. The non-Newtonian fluids flowed over the porous vertical plate to reach the boundary layer in this investigation. In order to understand the physical model, partial differential equations (PDEs) are used. To get a linked nonlinear set of ordinary differential equations (ODEs), we reduced this set of PDEs by using similarity variables. SHAM, a spectrum basis technique, was utilized to solve these modified equations to understand the physical significance. A good method is to utilize SHAM to decouple the coupled nonlinear ODE systems and divide them into linear and nonlinear equation sets since this helps to separate the systems. As a result, the two non-Newtonian fluids (Carreu and Cassin) flow together through the vertical wall and into the boundary layer, where different parameters’ impacts are scrutinized. The current results showed that an upturn in the Casson parameter (β) degenerates the boundary layer velocity and the total thickness. Upturn in the Weissenberg number (We) on the other hand, raises the velocities and temperatures in both directions. Additionally, increasing the Soret and Dufour parameters sped up the velocity graph.

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Harrison, Jonathan U., and Christian A. Yates. "A hybrid algorithm for coupling partial differential equation and compartment-based dynamics." Journal of The Royal Society Interface 13, no. 122 (September 2016): 20160335. http://dx.doi.org/10.1098/rsif.2016.0335.

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Stochastic simulation methods can be applied successfully to model exact spatio-temporally resolved reaction–diffusion systems. However, in many cases, these methods can quickly become extremely computationally intensive with increasing particle numbers. An alternative description of many of these systems can be derived in the diffusive limit as a deterministic, continuum system of partial differential equations (PDEs). Although the numerical solution of such PDEs is, in general, much more efficient than the full stochastic simulation, the deterministic continuum description is generally not valid when copy numbers are low and stochastic effects dominate. Therefore, to take advantage of the benefits of both of these types of models, each of which may be appropriate in different parts of a spatial domain, we have developed an algorithm that can be used to couple these two types of model together. This hybrid coupling algorithm uses an overlap region between the two modelling regimes. By coupling fluxes at one end of the interface and using a concentration-matching condition at the other end, we ensure that mass is appropriately transferred between PDE- and compartment-based regimes. Our methodology gives notable reductions in simulation time in comparison with using a fully stochastic model, while maintaining the important stochastic features of the system and providing detail in appropriate areas of the domain. We test our hybrid methodology robustly by applying it to several biologically motivated problems including diffusion and morphogen gradient formation. Our analysis shows that the resulting error is small, unbiased and does not grow over time.

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Jabeen, K., M. Mushtaq, and R. M. Akram Muntazir. "Analysis of MHD Fluids around a Linearly Stretching Sheet in Porous Media with Thermophoresis, Radiation, and Chemical Reaction." Mathematical Problems in Engineering 2020 (May 7, 2020): 1–14. http://dx.doi.org/10.1155/2020/9685482.

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This paper presents the comparative analysis of MHD boundary layer fluid flow around a linearly stretching surface in the presence of radiative heat flux, heat generation/absorption, thermophoresis velocity, and chemical reaction effects in a permeable surface. The governing equations are highly nonlinear PDEs which are converted into coupled ODEs with the help of dimensionless variables and solved by using semianalytical techniques. The numerical and graphical outcomes are observed and presented via tables and graphs. Also, the Nusselt and Sherwood numbers and skin friction coefficient are illustrated by tables. On observation of heat and mass transfer, it was noticed that Maxwell fluid dominates the other fluids such as Newtonian, Williamson, and Casson fluid due to high rate of thermal conductivity, and hence, Maxwell fluid has better tendency for heat and mass transfer than other Newtonian and non-Newtonian fluids.

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Luongo, Vincenzo, Maria Rosaria Mattei, Luigi Frunzo, Berardino D'Acunto, Kunal Gupta, Shankararaman Chellam, and Nick Cogan. "A transient biological fouling model for constant flux microfiltration." Mathematical Biosciences and Engineering 20, no. 1 (2022): 1274–96. http://dx.doi.org/10.3934/mbe.2023058.

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<abstract><p>Microfiltration is a widely used engineering technology for fresh water production and water treatment. The major concern in many applications is the formation of a biological fouling layer leading to increased hydraulic resistance and flux decline during membrane operations. The growth of bacteria constituting such a biological layer implicates the formation of a multispecies biofilm and the consequent increase of operational costs for reactor management and cleaning procedures. To predict the biofouling evolution, a mono-dimensional continuous free boundary model describing biofilm dynamics and EPS production in different operational phases of microfiltration systems has been well studied. The biofouling growth is governed by a system of hyperbolic PDEs. Substrate dynamics are modeled through parabolic equations accounting for diffusive and advective fluxes generated during the filtration process. The free boundary evolution depends on both microbial growth and detachment processes. What is not addressed is the interplay between biofilm dynamics, filtration, and water recovery. In this study, filtration and biofilm growth modeling principles have been coupled for the definition of an original mathematical model able to reproduce biofouling evolution in membrane systems. The model has been solved numerically to simulate biologically relevant conditions, and to investigate the hydraulic behavior of the membrane. It has been calibrated and validated using lab-scale data. Numerical results accurately predicted the pressure drop occurring in the microfiltration system. A calibrated model can give information for optimization protocols as well as fouling prevention strategies.</p></abstract>

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Batool, Shamaila, A. M. Alotaibi, Waris Khan, Ahmed Hussein Msmali, Ikramullah, and Wali Khan Mashwani. "Homotopic Solution for 3D Darcy–Forchheimer Flow of Prandtl Fluid through Bidirectional Extending Surface with Cattaneo–Christov Heat and Mass Flux Model." Complexity 2021 (December 7, 2021): 1–15. http://dx.doi.org/10.1155/2021/8204928.

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The 3D Prandtl fluid flow through a bidirectional extending surface is analytically investigated. Cattaneo–Christov fluid model is employed to govern the heat and mass flux during fluid motion. The Prandtl fluid motion is mathematically modeled using the law of conservations of mass, momentum, and energy. The set of coupled nonlinear PDEs is converted to ODEs by employing appropriate similarity relations. The system of coupled ODEs is analytically solved using the well-established mathematical technique of HAM. The impacts of various physical parameters over the fluid state variables are investigated by displaying their corresponding plots. The augmenting Prandtl parameter enhances the fluid velocity and reduces the temperature and concentration of the fluid. The momentum boundary layer boosts while the thermal boundary layer mitigates with the rising elastic parameter ( α 2 ) strength. Furthermore, the enhancing thermal relaxation parameter ( γ e )) reduces the temperature distribution, whereas the augmenting concentration parameter ( γ c ) drops the strength of the concentration profile. The increasing Prandtl parameter declines the fluid temperature while the augmenting Schmidt number drops the fluid concentration. The comparison of the HAM technique with the numerical solution shows an excellent agreement and hence ascertains the accuracy of the applied analytical technique. This work finds applications in numerous fields involving the flow of non-Newtonian fluids.

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Ali, Kashif, Wasim Jamshed, Sohail Ahmad, Hina Bashir, Shahzad Ahmad, and El Sayed M. Tag El Din. "A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet." Symmetry 14, no. 10 (September 23, 2022): 1991. http://dx.doi.org/10.3390/sym14101991.

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Nano-fluids have considerable importance in the field of thermal development that relates to several industrial systems. There are some key applications in recent construction systems flow, as well as microscale cooling gadgets and microstructure electric gadgets for thermal migration. The current investigation concludes the study of electrically conducting nano-fluid flow and heat transfer analysis in two-dimensional boundary layer flow over a curved extending surface in the coexisting of magnetic field, heat generation and thermal radiation. The small sized particles of copper (Cu) are taken as nanoparticles and water is assumed to be the base fluid. We used quasi-linearization and central difference approximation to numerically solve the system of coupled equations obtained from the partial differential equations (PDEs) by incorporating the concept of similarity. The impacts of non-dimensional parameters on velocity, concentration and thermal profiles have been discussed with the help of suitable graphs and tables. It has been noticed that the velocity decelerated with the effect of the magnetic field interaction parameter. Thermal radiation caused an increase in temperature.

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Khan, Umair, A. Zaib, A. Ishak, S. Abu Bakar, El-Sayed M. Sherif, and Noor Muhammad. "Radiation effect on three-dimensional stagnation point flow involving copper-aqueous titania hybrid nanofluid induced by a non-Fourier heat flux over a horizontal plane surface." Physica Scripta 97, no. 1 (January 1, 2022): 015002. http://dx.doi.org/10.1088/1402-4896/ac45ab.

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Abstract This research numerically investigates 3D stagnation-point flow (SPF) past a horizontal plane surface conveying copper-aqueous titania hybrid nanofluid induced by non-Fourier heat flux (NFHF) that utilized in heat transfer processes. A Tiwari-Das model is engaged to examine the fluid flow dynamics and the heat transfer features of the hybrid nanofluid with thermal radiation effect. With aid of similarity variables, the leading nonlinear system involving partial differential equations (PDEs) is reduced to a system of ordinary differential equations (ODEs). This set of dimensionless coupled ODEs is then tackled through the bvp4c solver in MATLAB. For hybrid nanofluid, the graphical findings of the pertaining parameters as well as the saddle/nodal indicative parameter are disclosed and explained with the assist of figures and tables. The results illustrate that the rise of hybrid nanoparticles declines the motion of the fluids in both axes of coordinates ( x − and y − directions), while the temperature enhances. In addition, the temperature distribution declines due to relaxation parameter but uplifts due to radiation. Also, the thermal relaxation parameter reduces the temperature. Moreover, the present solution displays an excellent agreement with earlier published works in the limited cases of normal fluid and nanofluid.

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Bu, Wankui, Hui Xu, Ilyas Khan, Sheikh Irfan Ullah Khan, and Anwar Zeb. "Mixed Convection Squeezing Flow of Nanofluids in a Rotating Channel with Thermal Radiation." Journal of Mathematics 2022 (February 18, 2022): 1–15. http://dx.doi.org/10.1155/2022/3885463.

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In the present study, 3-dimensional squeezing movement in a circling conduit under the stimulus effective Prandtl number with the aid of thermal radiation is taken into account. Water and ethylene glycol are the base fluids along with gamma-alumina nanoparticles. The coupled nonlinear system of PDEs is transformed into a system of ODEs with the support of some appropriate resemblance alterations. Then, the explanation was obtained numerically by the Runge–Kutta–Fehlberg (RKF) method. The emerging parameters such as quotient of the electric magnetic field to viscous forces (M), Prandtl number (Pr), and Reynolds number (Re), along with physical parameters such as the Nusselt number and skin friction coefficient, will be integrated graphically. The Prandtl number is important for regulating the momentum and thermal boundary layers. As a result, the effect of the effective Prandtl number on the nanoboundary layer and laminar incompressible flow of γ Al 2 O 3 − H 2 O and γ Al 2 O 3 − C 2 H 6 O 2 nanoparticles is considered. The impact of the radiation parameter (Rd) favors the temperature distribution. Furthermore, the thermal conductance enriches with the enhancement of solid volume fraction.

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Pravin Kashyap, K., Odelu Ojjela, and Samir Kumar Das. "MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink." Nonlinear Engineering 8, no. 1 (January 28, 2019): 523–33. http://dx.doi.org/10.1515/nleng-2018-0036.

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Abstract The present article explores the effects of uniform heat source and first order destructive chemical reaction on an upper convected Maxwell fluid through an expanding or contracting channel with the porous slip condition at the upper plate. It is assumed that the fluid is sucked or injected through the upper plate. The temperature and concentration at the plates is maintained constant. Using suitable similarity transformations, nonlinear coupled ODEs are developed from the governing PDEs. The subsequent ODEs are converted into a first order system and integrated via shooting method. The effect of various prominent parameters on heat, flow and mass transfer characteristics are studied in detail through graphs and tables. The present results suggest that the presence of chemical reaction and heat source yields in the reduction of concentration and of the enhancement of temperature the fluid. It is also observed that the wall expansion shows an increasing effect on the radial velocity component, but the slip parameter exhibits an opposing effect. The viscous case has been studied as a special case where the present results are found to be close to the earlier ones. The flow of such nonlinear viscoelastic fluids has important applications in separation processes like petroleum and medical industries.

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Naqvi, Syed Muhammad Raza Shah, Umar Farooq, Hassan Waqas, Taseer Muhammad, and Muhammad Abid. "Solar radiation effects on unsteady bio convectional flow of viscoelastic nanofluid confined by a wedge." Advances in Mechanical Engineering 14, no. 11 (November 2022): 168781322211250. http://dx.doi.org/10.1177/16878132221125060.

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Nanofluid suspension comprises a chip-sized rigid particle trapped in the convection fluid. These forms of fluids are known as nanofluids. Nanofluid is more user-friendly in engineering including fuel cells, cooling systems, and a wide range of applications of technologies to improve. The idea of this work is to analyze the Bio convectional stream characteristics and Solutal boundary characteristics of nanofluid flow via a wedge. Furthermore, the consequences of motile bacteria and heat conductivity are considered. When the boundary layer is estimated, the governing equations will be seen. Coupled PDEs are reduced into nonlinear ODEs using the required similarity vector, and the resultant structures are shown using the MATLAB computational tool bvp4c firing (Labotto IIIA formula). The implications of fluid velocity, temperature area, nanoparticles concentration, and microorganism concentration on the induced parameters are shown in graphical and numerical values. The velocity field is booming up for higher fluid parameter values and depressed for larger buoyancy ratio parameter estimations. For increasing levels of both the thermal conductivity parameter and the thermal Biot number, the temperature field rises. The microorganism’s field has dropped for Peclet number values and increased for Microorganisms Biot number values. The concentration field is lowered for the Lewis number while it is increased for the concentration Biot number. The current implications are novel and original for the investigation of flow and heat transfer over a wedge in a viscoelastic nanofluid with thermal conductivity, motile microorganisms, and bioconvection.

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Asifa, Talha Anwar, Poom Kumam, Zahir Shah, and Kanokwan Sitthithakerngkiet. "Significance of Shape Factor in Heat Transfer Performance of Molybdenum-Disulfide Nanofluid in Multiple Flow Situations; A Comparative Fractional Study." Molecules 26, no. 12 (June 18, 2021): 3711. http://dx.doi.org/10.3390/molecules26123711.

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In this modern era, nanofluids are considered one of the advanced kinds of heat transferring fluids due to their enhanced thermal features. The present study is conducted to investigate that how the suspension of molybdenum-disulfide (MoS2) nanoparticles boosts the thermal performance of a Casson-type fluid. Sodium alginate (NaAlg) based nanofluid is contained inside a vertical channel of width d and it exhibits a flow due to the movement of the left wall. The walls are nested in a permeable medium, and a uniform magnetic field and radiation flux are also involved in determining flow patterns and thermal behavior of the nanofluid. Depending on velocity boundary conditions, the flow phenomenon is examined for three different situations. To evaluate the influence of shape factor, MoS2 nanoparticles of blade, cylinder, platelet, and brick shapes are considered. The mathematical modeling is performed in the form of non-integer order operators, and a double fractional analysis is carried out by separately solving Caputo-Fabrizio and Atangana-Baleanu operators based fractional models. The system of coupled PDEs is converted to ODEs by operating the Laplace transformation, and Zakian’s algorithm is applied to approximate the Laplace inversion numerically. The solutions of flow and energy equations are presented in terms of graphical illustrations and tables to discuss important physical aspects of the observed problem. Moreover, a detailed inspection on shear stress and Nusselt number is carried out to get a deep insight into skin friction and heat transfer mechanisms. It is analyzed that the suspension of MoS2 nanoparticles leads to ameliorating the heat transfer rate up to 9.5%. To serve the purpose of achieving maximum heat transfer rate and reduced skin friction, the Atangana-Baleanu operator based fractional model is more effective. Furthermore, it is perceived that velocity and energy functions of the nanofluid exhibit significant variations because of the different shapes of nanoparticles.

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Klein, Richard I., and Jonathan Arons. "Radiation Gas Dynamics of Polar Cap Accretion onto Magnetized Neutron Stars." Symposium - International Astronomical Union 125 (1987): 246. http://dx.doi.org/10.1017/s0074180900160826.

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We present results of the first self-consistent, time-dependent, 2-D calculations of the accretion of plasma onto polar caps of high luminosity (L*>1036erg-s−1) magnetized neutron stars. We follow the temporal and spatial evolution of three fluids, electrons, ions and photons in a superstrong (B=3×1012 Gauss) dipole magnetic field where radiation pressure dominates plasma pressure by solving coupled 2-D equations of radiation hydrodynamics. We have included several physical processes in the radiation-plasma coupling in superstrong magnetic fields (Klein, et al., 1984, Santa Cruz Workshop on High Energy Transients, and Arons, this conference). We solve the resulting system of coupled 2-D PDEs on a Cray XMP-48 by applying implicit finite-difference techniques with iterative operator splitting methods. We present results for two models of 5×1037 erg-s and 1.5×1038 erg-s−1 super-Eddington luminosity on one polar cap, each having initial mass flux independent of co-latitude of a field lines footprint. We find (a) Radiation develops a broad transverse fan beam that emerges from an annulus 0.2–0.5km above the polar cap. (b) The beam profile is determined by advective trapping of radiation in optically thick (τ11,τ⊥ ≈103) flow. Here the time for diffusion of radiation up through the accretion column is ≫ the time for downward advection. (c) There is a three fluid nonequilibrium with Ti≫Tγ≥Te. (d) Maximum photon temperature of ≈ 10–20 keV in the fan beam is in the observed range. (e) Cyclotron emission ≫ bremsstrahlung as a source of photons. (f) At early times (≪lms) radiation pressure strongly decelerates flow to 10−3 of freefall in central regions of accretion column resulting in a density mound, but plasma freefalls down the sides of the column. (g) Analytical models have reasonable agreement with numerical calculations; velocity and energy density roughly Gaussian transversally and exponential vertically, until the onset of “photon bubbles” after several dynamical times (∼lms). (h) Multiple “photon bubbles” rising subsonically in the accretion column form in the high luminosity model. We believe the photon bubbles to be a possible consequence of overstable convection in super-Eddington flows. These photon bubbles could be observable as 10–100μs fluctuations in the emergent flux and, thus, be an important diagnostic for inhomogeneous structure of the column.

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Lasiecka,, I., and GC Gaunaurd,. "Mathematical Control Theory of Coupled PDEs." Applied Mechanics Reviews 56, no. 1 (January 1, 2003): B3. http://dx.doi.org/10.1115/1.1523354.

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Bhrawy, A. H., M. A. Alghamdi, and Eman S. Alaidarous. "An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions." Abstract and Applied Analysis 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/295936.

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One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

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Gropp, William D. "Solving PDEs on loosely-coupled parallel processors." Parallel Computing 5, no. 1-2 (July 1987): 165–73. http://dx.doi.org/10.1016/0167-8191(87)90015-9.

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Angiuli, Luciana, and Luca Lorenzi. "On coupled systems of PDEs with unbounded coefficients." Dynamics of Partial Differential Equations 17, no. 2 (2020): 129–63. http://dx.doi.org/10.4310/dpde.2020.v17.n2.a3.

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Forest, M. G., D. W. McLaughlin, D. J. Muraki, and O. C. Wright. "Nonfocusing Instabilities in Coupled, Integrable Nonlinear Schrödinger pdes." Journal of Nonlinear Science 10, no. 3 (June 2000): 291–331. http://dx.doi.org/10.1007/s003329910012.

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Engwer, Christian, and Sebastian Westerheide. "An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries." Computational Methods in Applied Mathematics 21, no. 3 (June 1, 2021): 569–91. http://dx.doi.org/10.1515/cmam-2020-0056.

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Abstract The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.

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Asmolov, Evgeny S., Tatiana V. Nizkaya, and Olga I. Vinogradova. "Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers." Mathematics 10, no. 9 (May 1, 2022): 1503. http://dx.doi.org/10.3390/math10091503.

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Catalytic swimmers self-propel in electrolyte solutions thanks to an inhomogeneous ion release from their surface. Here, we consider the experimentally relevant limit of thin electrostatic diffuse layers, where the method of matched asymptotic expansions can be employed. While the analytical solution for ion concentration and electric potential in the inner region is known, the electrostatic problem in the outer region was previously solved but only for a linear case. Additionally, only main geometries such as a sphere or cylinder have been favoured. Here, we derive a non-linear outer solution for the electric field and concentrations for swimmers of any shape with given ion surface fluxes that then allow us to find the velocity of particle self-propulsion. The power of our formalism is to include the complicated effects of the anisotropy and inhomogeneity of surface ion fluxes under relevant boundary conditions. This is demonstrated by exact solutions for electric potential profiles in some particular cases with the consequent calculations of self-propulsion velocities.

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Xiao, Lishun, Shengjun Fan, and Dejian Tian. "A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems." ESAIM: Probability and Statistics 24 (2020): 207–26. http://dx.doi.org/10.1051/ps/2019023.

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In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.

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Aghili, Arman. "Non-homogenous KdV and coupled sub-ballistic fractional PDEs." New Trends in Mathematical Science 3, no. 5 (August 25, 2017): 107–17. http://dx.doi.org/10.20852/ntmsci.2017.189.

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29

Save, Yogesh Dilip, H. Narayanan, and Sachin B. Patkar. "Solution of PDEs-electrically coupled systems with electrical analogy." Integration 46, no. 4 (September 2013): 427–40. http://dx.doi.org/10.1016/j.vlsi.2012.10.002.

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30

Yuan, Gonglin, and Xiangrong Li. "A Numerical Algorithm for the Coupled PDEs Control Problem." Computational Economics 53, no. 2 (October 10, 2017): 697–707. http://dx.doi.org/10.1007/s10614-017-9757-6.

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31

Auriol, Jean, and Florent Di Meglio. "Minimum time control of heterodirectional linear coupled hyperbolic PDEs." Automatica 71 (September 2016): 300–307. http://dx.doi.org/10.1016/j.automatica.2016.05.030.

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Li, Jian, and Yungang Liu. "Adaptive stabilization for ODE systems coupled with parabolic PDES." Journal of Systems Science and Complexity 29, no. 4 (May 27, 2016): 959–77. http://dx.doi.org/10.1007/s11424-016-5094-4.

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33

Teboul, S., L. Blanc-Feraud, G. Aubert, and M. Barlaud. "Variational approach for edge-preserving regularization using coupled PDEs." IEEE Transactions on Image Processing 7, no. 3 (March 1998): 387–97. http://dx.doi.org/10.1109/83.661189.

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34

Aksikas, I., A. Alizadeh Moghadam, and J. F. Forbes. "Optimal linear–quadratic control of coupled parabolic–hyperbolic PDEs." International Journal of Control 90, no. 10 (September 29, 2016): 2152–64. http://dx.doi.org/10.1080/00207179.2016.1237046.

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35

Oh, Sahuck. "An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials." Mathematics 7, no. 1 (January 16, 2019): 90. http://dx.doi.org/10.3390/math7010090.

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We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved.

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Li, Miao, and Tower Wang. "M2-branes coupled to antisymmetric fluxes." Journal of High Energy Physics 2008, no. 07 (July 22, 2008): 093. http://dx.doi.org/10.1088/1126-6708/2008/07/093.

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37

Mamandi, Ahmad, and Mohammad H. Kargarnovin. "Nonlinear Dynamic Analysis of a Timoshenko Beam Resting on a Viscoelastic Foundation and Traveled by a Moving Mass." Shock and Vibration 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/242090.

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The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. Primarily, the beam’s nonlinear governing coupled PDEs of motion for the lateral and longitudinal displacements as well as the beam’s cross-sectional rotation are derived using Hamilton’s principle. On deriving these nonlinear coupled PDEs the stretching effect of the beam’s neutral axis due to the beam’s fixed end conditions in conjunction with the von-Karman strain-displacement relations is considered. To obtain the dynamic responses of the beam under the act of a moving mass, derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic responses are obtained using mode summation technique. Furthermore, after verification of our results with other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocity of the moving mass, damping coefficient, and stiffnesses of the foundation including linear and cubic nonlinear parts, respectively. It is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundation damping will produce significant effect in the beam’s dynamic response.

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Ghousein, Mohammad, and Emmanuel Witrant. "Adaptive Boundary Observer Design for coupled ODEs-Hyperbolic PDEs systems." IFAC-PapersOnLine 53, no. 2 (2020): 7605–10. http://dx.doi.org/10.1016/j.ifacol.2020.12.1359.

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39

Aslimani, Abderrahim, Imad El Ghazi, Mohamed El Kadiri, and Sabah Haddad. "On the potential theory of some systems of coupled PDEs." Commentationes Mathematicae Universitatis Carolinae 57, no. 2 (July 5, 2016): 135–54. http://dx.doi.org/10.14712/1213-7243.2015.165.

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40

Hu, Long, Florent Di Meglio, Rafael Vazquez, and Miroslav Krstic. "Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs." IEEE Transactions on Automatic Control 61, no. 11 (November 2016): 3301–14. http://dx.doi.org/10.1109/tac.2015.2512847.

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41

Auriol, Jean, and Florent Di Meglio. "Two-Sided Boundary Stabilization of Heterodirectional Linear Coupled Hyperbolic PDEs." IEEE Transactions on Automatic Control 63, no. 8 (August 2018): 2421–36. http://dx.doi.org/10.1109/tac.2017.2763320.

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42

Prasath, V. B. Surya, and D. Vorotnikov. "On a System of Adaptive Coupled PDEs for Image Restoration." Journal of Mathematical Imaging and Vision 48, no. 1 (October 3, 2012): 35–52. http://dx.doi.org/10.1007/s10851-012-0386-3.

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43

Alizadeh Moghadam, Amir, Ilyasse Aksikas, Stevan Dubljevic, and J. Fraser Forbes. "Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs." Automatica 49, no. 2 (February 2013): 526–33. http://dx.doi.org/10.1016/j.automatica.2012.11.016.

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44

Nawaz, Rashid, Samreen Farid, and Samia Bushnaq. "Applications of New Iterative Method to fractional non linear coupled ITO system." Boletim da Sociedade Paranaense de Matemática 40 (January 31, 2022): 1–10. http://dx.doi.org/10.5269/bspm.47787.

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In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The results obtained by the proposed method are compared with that of Homotopy Perturbation Method (HPM). It is shown that the proposed method is accurate for strongly nonlinear fractional coupled system of PDEs.

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45

Tian, Lei, Yongxue Zhang, Jianyong Yin, Liang Lv, and Jianjun Zhu. "A simplified model for the gas-vapor bubble dynamics." Journal of the Acoustical Society of America 152, no. 4 (October 2022): 2117–27. http://dx.doi.org/10.1121/10.0014695.

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This paper presents a full numerical model accounting for the heat transfer and phase-change by combining the modified Keller–Miksis equation with the second order term of compressibility of liquid, partial differential equations (PDEs), and Hertz–Knudsen–Langmuir equation. Then, a simplified model for studying the dynamics of the cavitation bubble or bubble excited by the acoustic waves is proposed. The major contribution is to simplify the full model with PDEs to a set of coupled ordinary differential equations (ODEs). Specifically, two energy PDEs are converted to three ODEs by coupling the boundary conditions. The comparison among the full model and other simplified models is used to validate the accuracy and superiority of the simplified model, from which the application range of the proposed simplified model can be determined.

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Thomas, Sunil G., Hector M. Klie, Adolfo A. Rodriguez, and Mary F. Wheeler. "A Parallel Stochastic Framework for Reservoir Characterization and History Matching." Journal of Applied Mathematics 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/535484.

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The spatial distribution of parameters that characterize the subsurface is never known to any reasonable level of accuracy required to solve the governing PDEs of multiphase flow or species transport through porous media. This paper presents a numerically cheap, yet efficient, accurate and parallel framework to estimate reservoir parameters, for example, medium permeability, using sensor information from measurements of the solution variables such as phase pressures, phase concentrations, fluxes, and seismic and well log data. Numerical results are presented to demonstrate the method.

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Javeed, Shumaila, Sidra Riaz, Khurram Saleem Alimgeer, M. Atif, Atif Hanif, and Dumitru Baleanu. "First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models." Symmetry 11, no. 6 (June 12, 2019): 783. http://dx.doi.org/10.3390/sym11060783.

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In this work, we establish the exact solutions of some mathematical physics models. The first integral method (FIM) is extended to find the explicit exact solutions of high-dimensional nonlinear partial differential equations (PDEs). The considered models are: the space-time modified regularized long wave (mRLW) equation, the (1+2) dimensional space-time potential Kadomtsev Petviashvili (pKP) equation and the (1+2) dimensional space-time coupled dispersive long wave (DLW) system. FIM is a powerful mathematical tool that can be used to obtain the exact solutions of many non-linear PDEs.

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Hamzah, Amir Syafiq Syamin Syah, and Ali Hassan Mohamed Murid. "Nonlinear Partial Dierential Equations Model Related to Oxidation Pond Treatment System: A Case Study of mPHO at Taman Timor Oxidation Pond, Johor Bahru." MATEMATIKA 34, no. 2 (December 2, 2018): 293–311. http://dx.doi.org/10.11113/matematika.v34.n2.1038.

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This study presents a mathematical model examining wastewater pollutant removalthrough an oxidation pond treatment system. This model was developed to describethe reaction between microbe-based product mPHO (comprising Phototrophic bac-teria (PSB)), dissolved oxygen (DO) and pollutant namely chemical oxygen demand(COD). It consists of coupled advection-diusion-reaction equations for the microor-ganism (PSB), DO and pollutant (COD) concentrations, respectively. The couplingof these equations occurred due to the reactions between PSB, DO and COD to pro-duce harmless compounds. Since the model is nonlinear partial dierential equations(PDEs), coupled, and dynamic, computational algorithm with a specic numericalmethod, which is implicit Crank-Nicolson method, was employed to simulate the dy-namical behaviour of the system. Furthermore, numerical results revealed that theproposed model demonstrated high accuracy when compared to the experimental data.Keywords Oxidation pond; nonlinear PDEs; PSB; implicit Crank-Nicolson.

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Vaddireddy, Harsha, and Omer San. "Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach." Fluids 4, no. 2 (June 15, 2019): 111. http://dx.doi.org/10.3390/fluids4020111.

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Advances in machine learning (ML) coupled with increased computational power have enabled identification of patterns in data extracted from complex systems. ML algorithms are actively being sought in recovering physical models or mathematical equations from data. This is a highly valuable technique where models cannot be built using physical reasoning alone. In this paper, we investigate the application of fast function extraction (FFX), a fast, scalable, deterministic symbolic regression algorithm to recover partial differential equations (PDEs). FFX identifies active bases among a huge set of candidate basis functions and their corresponding coefficients from recorded snapshot data. This approach uses a sparsity-promoting technique from compressive sensing and sparse optimization called pathwise regularized learning to perform feature selection and parameter estimation. Furthermore, it recovers several models of varying complexity (number of basis terms). FFX finally filters out many identified models using non-dominated sorting and forms a Pareto front consisting of optimal models with respect to minimizing complexity and test accuracy. Numerical experiments are carried out to recover several ubiquitous PDEs such as wave and heat equations among linear PDEs and Burgers, Korteweg–de Vries (KdV), and Kawahara equations among higher-order nonlinear PDEs. Additional simulations are conducted on the same PDEs under noisy conditions to test the robustness of the proposed approach.

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Wang, Ji, and Miroslav Krstic. "Observer of Coupled Hyperbolic PDEs for Deep-sea Construction Vessel Vibrations." IFAC-PapersOnLine 53, no. 2 (2020): 7641–46. http://dx.doi.org/10.1016/j.ifacol.2020.12.1365.

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Journal articles on the topic 'PDEs with coupled fluxes'

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