Статті в журналах: "Stationary micropolar fluids equations"

1

Duarte-Leiva, Cristian, Sebastián Lorca, and Exequiel Mallea-Zepeda. "A 3D Non-Stationary Micropolar Fluids Equations with Navier Slip Boundary Conditions." Symmetry 13, no. 8 (July 26, 2021): 1348. http://dx.doi.org/10.3390/sym13081348.

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Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with Navier boundary conditions without friction for the velocity field and homogeneous Dirichlet boundary conditions for the angular velocity. Using the Galerkin method, we prove the existence of weak solutions and establish a Prodi–Serrin regularity type result which allow us to obtain global-in-time strong solutions at finite time.

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2

Kocić, Miloš, Živojin Stamenković, Jelena Petrović, and Jasmina Bogdanović-Jovanović. "MHD micropolar fluid flow in porous media." Advances in Mechanical Engineering 15, no. 6 (June 2023): 168781322311784. http://dx.doi.org/10.1177/16878132231178436.

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The analysis of mass and heat transfer in magnetohydrodynamic (MHD) flows has significant applications in heat exchangers, cooling nuclear reactors, designing energy systems and casting and injection processes of different types of fluids. On the other hand, extraction of crude oil, the flow of human or animal blood, as well as other polymer fluids or liquid crystals are just some examples of micropolar fluid flows. Due to the broad application spectrum of the theory of micropolar fluid flows, and the significance the impact the external magnetic field has on the flow of these fluids, this paper considers the stationary flow of a micropolar fluid between two plates under the influence of an external magnetic field which is perpendicular to the direction of the flow. Stationary plates are maintained at constant and different temperatures, while the whole problem is considered in the non-inductive approximation. The equation system used to define the physical problem under consideration is reduced to the system of differential equations that have been solved analytically and the solutions of which are of general nature. In addition to the solutions for velocity, microrotation and temperature, the paper gives solutions for shear stress at plates, the Nusselt number and flow rate. The provided solutions have been applied in order to reach some general conclusions about the influence of the magnetic field and physical characteristics of a micropolar fluid and the characteristics of porous media on the nature of micropolar fluid flows in porous media by means of chart analysis. General conclusions, obtained in the result analysis in this paper, give us the opportunity to understand the flows of micropolar fluids and highlight their significance.

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3

Eldabe, N. T., and M. Y. Abou-Zeid. "The Wall Properties Effect on Peristaltic Transport of Micropolar Non-Newtonian Fluid with Heat and Mass Transfer." Mathematical Problems in Engineering 2010 (2010): 1–40. http://dx.doi.org/10.1155/2010/898062.

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The problem of the unsteady peristaltic mechanism with heat and mass transfer of an incompressible micropolar non-Newtonian fluid in a two-dimensional channel. The flow includes the viscoelastic wall properties and micropolar fluid parameters using the equations of the fluid as well as of the deformable boundaries. A perturbation solution is obtained, which satisfies the momentum, angular momentum, energy, and concentration equations for case of free pumping (original stationary fluid). Numerical results for the stream function, temperature, and concentration distributions are obtained. Several graphs of physical interest are displayed and discussed.

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4

WENG, HUEI CHU, CHA'O-KUANG CHEN, and MIN-HSING CHANG. "Stability of micropolar fluid flow between concentric rotating cylinders." Journal of Fluid Mechanics 631 (July 17, 2009): 343–62. http://dx.doi.org/10.1017/s0022112009007150.

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In this study, the theory of micropolar fluids is employed to study the stability problem of flow between two concentric rotating cylinders. The field equations subject to no-slip conditions (non-zero velocity and microrotation velocity components) at the wall surfaces are solved. The analytical solutions of the velocity and microrotation velocity fields as well as the shear stress difference, couple stress and strain rate for basic flow are obtained. The equations with respect to non-axisymmetric disturbances are derived and solved by a direct numerical procedure. It is found that non-zero wall-surface microrotation velocity makes the flow faster and more unstable. Moreover, it tends to reduce the limits of critical non-axisymmetric disturbances. The effect on the stability characteristics can be magnified by increasing the microstructure or couple-stress parameter or the microinertia parameter for the cases of corotating cylinders and a stationary outer cylinder or by decreasing the radius ratio or the microinertia parameter for the case of counterrotating cylinders.

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5

Xing, Xin, and Demin Liu. "Numerical Analysis and Comparison of Three Iterative Methods Based on Finite Element for the 2D/3D Stationary Micropolar Fluid Equations." Entropy 24, no. 5 (April 29, 2022): 628. http://dx.doi.org/10.3390/e24050628.

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In this paper, three iterative methods (Stokes, Newton and Oseen iterative methods) based on finite element discretization for the stationary micropolar fluid equations are proposed, analyzed and compared. The stability and error estimation for the Stokes and Newton iterative methods are obtained under the strong uniqueness conditions. In addition, the stability and error estimation for the Oseen iterative method are derived under the uniqueness condition of the weak solution. Finally, numerical examples test the applicability and the effectiveness of the three iterative methods.

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6

Salemovic, Dusko, Aleksandar Dedic, and Bosko Jovanovic. "Micropolar fluid between two coaxial cylinders (numerical approach)." Theoretical and Applied Mechanics 48, no. 2 (2021): 159–69. http://dx.doi.org/10.2298/tam210823012s.

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The paper describes the flow of a suspension which is a mixture of two phases: liquid and solid granules. The continuum model with microstructure is introduced, which involves two independent kinematic quantities: the velocity vector and the micro-rotation vector. The physical analogy is based on the movement of the suspension between two coaxial cylinders. The inner cylinder is stationary and the outer one rotates with constant angular velocity. This physical analogy enabled a mathematical model in a form of two coupled differential equations with variable coefficients. The aim of the paper is to present the numerical aspect of the solution for this complex mathematical model. It is assumed that the solid granules are identically oriented and that under the influence of the fluid they move translationally or rotate around the symmetry axis but the direction of their symmetry axes does not change. The solution was obtained by the ordinary finite difference method, and then the corresponding sets of points (nodes) were routed by interpolation graphics.

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7

Burmasheva, N. V., and E. Yu Prosviryakov. "Exact solutions to the NAVIER–STOKES equations for unidirectional flows of micropolar fluids in a mass force field." Diagnostics, Resource and Mechanics of materials and structures, no. 3 (June 2024): 41–63. http://dx.doi.org/10.17804/2410-9908.2024.3.041-063.

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The paper presents a family of exact solutions to the Navier-Stokes equation system used to describe inhomogeneous unidirectional flows of a viscous fluid taking into account couple stresses. Despite the presence of only one non-zero component of the velocity vector, this component depends on time and two spatial coordinates. In view of the incompressibility equation, which is a special case of the mass conservation law, there is no dependence on the third spatial coordinate. The resulting redefined system of equations is considered in a non-stationary formulation. The construction of a family of exact solutions for the resulting redefined equation system begins with the analysis of the homogeneous Couette-type solution as the simplest in this class. Further, the structure of the solution gradually becomes more complicated, i.e. the profile of the only non-zero component of the velocity vector is represented as a polynomial depending on one variable (horizontal coordinate). The polynomial coefficients functionally depend on the second (vertical) coordinate and time. It is shown that, due to the strong nonlinearity and heterogeneity of the equation under study, the sum of its individual solutions is not a solution. It is also shown that, in the linearly independent basis of the power functions of the horizontal coordinate, which determine the above-mentioned polynomial, the equation in question decomposes into a chain of the simplest homogeneous and inhomogeneous parabolic partial differential equations. These equations are integrated sequentially, the order of integration being described separately. The results reported in this study extend the family of previously presented exact solutions to describing unidirectional unsteady flows.

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8

Arnaud, M. M., G. M. de Araùjo, M. M. Freitas, and E. F. L. Lucena. "ON A SYSTEM OF EQUATIONS OF A NON-NEWTONIAN MICROPOLAR FLUID IN THE STATIONARY FORM." Far East Journal of Applied Mathematics 97, no. 4 (December 2, 2017): 125–42. http://dx.doi.org/10.17654/am097040125.

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9

Chen, James, James D. Lee, and Chunlei Liang. "Constitutive equations of Micropolar electromagnetic fluids." Journal of Non-Newtonian Fluid Mechanics 166, no. 14-15 (August 2011): 867–74. http://dx.doi.org/10.1016/j.jnnfm.2011.05.004.

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10

IDO, Yasushi. "Basic Equations of Micropolar Magnetic Fluids." Transactions of the Japan Society of Mechanical Engineers Series B 70, no. 696 (2004): 2065–70. http://dx.doi.org/10.1299/kikaib.70.2065.

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11

Yang, Hujun, Xiaoling Han, and Caidi Zhao. "Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms." Mathematics 10, no. 14 (July 15, 2022): 2469. http://dx.doi.org/10.3390/math10142469.

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This article studies the 3D incompressible micropolar fluids with rapidly oscillating terms. The authors prove that the trajectory statistical solutions of the oscillating fluids converge to that of the homogenized fluids provided that the oscillating external force and angular momentum possess some weak homogenization. The results obtained indicate that the trajectory statistical information of the 3D incompressible micropolar fluids has a certain homogenization effect with respect to spatial variables. In addition, our approach is also valid for a broad class of evolutionary equations displaying the property of global existence of weak solutions without a known result of global uniqueness, including some model hydrodynamic equations, MHD equations and non-Newtonian fluids equations.

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12

Stamenkovic, Zivojin, Milos Kocic, Jasmina Bogdanovic-Jovanovic, and Jelena Petrovic. "Nano and micropolar MHD fluid flow and heat transfer in inclined channel." Thermal Science, no. 00 (2023): 170. http://dx.doi.org/10.2298/tsci230515170k.

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Magnetohydrodynamic (MHD) fluid flows attract a lot of attention in the extrusion of polymers, in the theory of nanofluids, as well as in the consideration of biological fluids. The considered problem in the paper is the flow and heat transfer of nano and micropolar fluid in inclined channel. Fluid flow is steady, while nano and micropolar fluids are incompressible, immiscible, and electrically conductive. The upper and lower channel plates are electrically insulated and maintained at constant and different temperatures. External applied magnetic field is perpendicular to the fluid flow and considered problem is in induction-less approximation. The equations of the considered problem are reduced to ordinary differential equations, which are analytically solved in closed form. The influence of characteristics parameters of nano and micropolar fluids on velocity, micro-rotation and temperature fields are graphically shown and discussed. The general conclusions given through the analysis of graphs can be used for better understanding of the flow and heat transfer of nano and micropolar fluid, which have a great practical application. Fluids with nanoparticles innovated the modern era, due to their comprehensive applications in nanotechnology and manufacturing processes, while the theory of micropolar fluids explains the flow of biological fluids and various types of liquid metals and crystals.

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13

Rahman, M. M., and T. Sultana. "Radiative Heat Transfer Flow of Micropolar Fluid with Variable Heat Flux in a Porous Medium." Nonlinear Analysis: Modelling and Control 13, no. 1 (January 25, 2008): 71–87. http://dx.doi.org/10.15388/na.2008.13.1.14590.

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A two-dimensional steady convective flow of a micropolar fluid past a vertical porous flat plate in the presence of radiation with variable heat flux has been analyzed numerically. Using Darcy-Forchheimer model the corresponding momentum, microrotation and energy equations have been solved numerically. The local similarity solutions for the flow, microrotation and heat transfer characteristics are illustrated graphically for various material parameters. The effects of the pertinent parameters on the local skin friction coefficient, plate couple stress and the heat transfer are also calculated. It was shown that large Darcy parameter leads to decrease the velocity while it increases the angular velocity as well as temperature of the micropolar fluids. The rate of heat transfer in weakly concentrated micropolar fluids is higher than strongly concentrated micropolar fluids.

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14

Kocić, Miloš, Živojin Stamenković, Jelena Petrović, and Jasmina Bogdanović-Jovanović. "Control of MHD Flow and Heat Transfer of a Micropolar Fluid through Porous Media in a Horizontal Channel." Fluids 8, no. 3 (March 8, 2023): 93. http://dx.doi.org/10.3390/fluids8030093.

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The problem considered in this paper is a steady micropolar fluid flow in porous media between two plates. This model can be used to describe the flow of some types of fluids with microstructures, such as human and animal blood, muddy water, colloidal fluids, lubricants and chemical suspensions. Fluid flow is a consequence of the constant pressure gradient along the flow, while two parallel plates are fixed and have different constant temperatures during the fluid flow. Perpendicular to the flow, an external magnetic field is applied. General equations of the problem are reduced to ordinary differential equations and solved in the closed form. Solutions for velocity, microrotation and temperature are used to explain the influence of the external magnetic field (Hartmann number), the characteristics of the micropolar fluid (coupling and spin gradient viscosity parameter) and the characteristics of the porous medium (porous parameter) using graphs. The results obtained in the paper show that the increase in the additional viscosity of micropolar fluids emphasizes the microrotation vector. Moreover, the analysis of the effect of the porosity parameter shows how the permeability of a porous medium can influence the fluid flow and heat transfer of a micropolar fluid. Finally, it is shown that the influence of the external magnetic field reduces the characteristics of micropolar fluids and tends to reduce the velocity field and make it uniform along the cross-section of the channel.

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15

Cruz, Felipe W. "Global strong solutions for the incompressible micropolar fluids equations." Archiv der Mathematik 113, no. 2 (April 6, 2019): 201–12. http://dx.doi.org/10.1007/s00013-019-01319-4.

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16

Kim, Jae-Myoung, and Seungchan Ko. "Some Liouville-type theorems for the stationary 3D magneto-micropolar fluids." Acta Mathematica Scientia 44, no. 6 (October 1, 2024): 2296–306. http://dx.doi.org/10.1007/s10473-024-0614-0.

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17

IDO, Yasushi, and Takahiko TANAHASHI. "Fundamental equations for magnetic fluids by micropolar theory. 2nd report: Constitutive equations." Transactions of the Japan Society of Mechanical Engineers Series B 56, no. 525 (1990): 1392–99. http://dx.doi.org/10.1299/kikaib.56.1392.

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18

Hassanien, I. A. "Mixed Convection in Micropolar Boundary-Layer Flow Over a Horizontal Semi-Infinite Plate." Journal of Fluids Engineering 118, no. 4 (December 1, 1996): 833–38. http://dx.doi.org/10.1115/1.2835517.

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A boundary layer analysis is presented to study the effects of buoyancy-induced streamwise pressure gradients on laminar forced convection heat transfer to micropolar fluids from a horizontal semi-infinite flat plate. The transformed boundary-layer equations have been solved numerically. The effects of the buoyancy force, material parameters, and viscous dissipative heat on the friction factor, total heat transfer, displacement thickness, and wall couple stress, as well as the details of the velocity, microrotation, and temperature fields are discussed. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced heat transfer rate as compared with Newtonian fluids. Also the micropolar properties of the fluid are found to play an important role in controlling flow separation. Furthermore, it is observed that, for high values of the buoyancy and material parameters, the flow and thermal fields are significantly affected by the presence of viscous dissipation heat.

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19

Khalid, Asma, Ilyas Khan, and Sharidan Shafie. "Free convection flow of micropolar fluids over an oscillating vertical plate." Malaysian Journal of Fundamental and Applied Sciences 13, no. 4 (December 26, 2017): 654–58. http://dx.doi.org/10.11113/mjfas.v13n4.738.

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An analytical investigation is carried out to study the unsteady free convection flow of micropolar fluids over an oscillating vertical plate. Wall couple stress is engaged at the bounding plate with isothermal temperature. Problem is modelled in terms of coupled partial differential equations together with some physical conditions and then written in non-dimensional form. Exact solutions are obtained using the Laplace transform technique. Analytical results of velocity, microrotation and temperature are plotted in graphs and discussed for various embedded parameters. Excellent validation of present results is achieved with existing results in literature. It is observed that, the velocity is smaller for micropolar fluids than for Newtonian fluids.

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20

BRESCH, DIDIER, and JÉRÔME LEMOINE. "STATIONARY SOLUTIONS FOR SECOND GRADE FLUIDS EQUATIONS." Mathematical Models and Methods in Applied Sciences 08, no. 05 (August 1998): 737–48. http://dx.doi.org/10.1142/s0218202598000330.

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We discuss the existence and uniqueness for stationary second grade fluids. We give results in Lr(Ω)3 where r > 3, for an open set Ω which is only of class [Formula: see text]. This relies on a fixed point method.

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21

K.C., Durga Jang, and Dipendra Regmi. "Global regularity criteria for the 2D Magneto-micropolar Equations with Partial Dissipation." Nepali Mathematical Sciences Report 40, no. 1-2 (December 31, 2023): 55–70. http://dx.doi.org/10.3126/nmsr.v40i1-2.61498.

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The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field.These equations have been the focus of numerous analytical, experimental, and numerical investigations.One fundamental problem concerning these equations is whether their classical solutions are globally regular for all time or if they develop finite time singularities.The global regularity problem can be particularly challenging when there is only partial dissipation. In this paper, we study the 2D incompressible magneto-micropolar equations with partial dissipation prove two new regularity results. The first result addresses a weak solution, and the second result establishes global regularity criteria. As a consequence, we can single out one special partial dissipation case and establish the global regularity if (∂y u1, ∂y u2) ∈ L∞ [0, T ], R2. The proofs of our main results rely on anisotropic Sobolev-type inequalities and the appropriate combination and cancellation of terms.

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22

VIAGGIU, STEFANO. "GENERATING ANISOTROPIC FLUIDS FROM VACUUM ERNST EQUATIONS." International Journal of Modern Physics D 19, no. 11 (September 2010): 1783–95. http://dx.doi.org/10.1142/s0218271810018025.

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Starting with any stationary axisymmetric vacuum metric, we build anisotropic fluids. With the help of the Ernst method, the basic equations are derived together with the expression for the energy–momentum tensor and with the equation of state compatible with the field equations. The method is presented by using different coordinate systems: the cylindrical coordinates ρ, z and the oblate spheroidal ones. A class of interior solutions matching with stationary axisymmetric asymptotically flat vacuum solutions is found in oblate spheroidal coordinates. The solutions presented satisfy the three energy conditions.

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23

Eringen, A. C. "A mixture theory for geophysical fluids." Nonlinear Processes in Geophysics 11, no. 1 (February 25, 2004): 75–82. http://dx.doi.org/10.5194/npg-11-75-2004.

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Abstract. A continuum theory is developed for a geophysical fluid consisting of two species. Balance laws are given for the individual components of the mixture, modeled as micropolar viscous fluids. The continua allow independent rotational degrees of freedom, so that the fluids can exhibit couple stresses and a non-symmetric stress tensor. The second law of thermodynamics is used to develop constitutive equations. Linear constitutive equations are constituted for a heat conducting mixture, each species possessing separate viscosities. Field equations are obtained and boundary and initial conditions are stated. This theory is relevant to an atmospheric mixture consisting of any two species from rain, snow and/or sand. Also, this is a continuum theory for oceanic mixtures, such as water and silt, or water and oil spills, etc.

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24

Srinivas, J., J. V. Ramana Murthy, and Ali J. Chamkha. "Analysis of entropy generation in an inclined channel flow containing two immiscible micropolar fluids using HAM." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 3/4 (May 3, 2016): 1027–49. http://dx.doi.org/10.1108/hff-09-2015-0354.

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Purpose – The purpose of this paper is to examine the flow, heat transfer and entropy generation characteristics for an inclined channel of two immiscible micropolar fluids. Design/methodology/approach – The flow region consists of two zones, the flow of the heavier fluid taking place in the lower zone. The flow is assumed to be governed by Eringen’s micropolar fluid flow equation. The resulting governing equations are then solved using the homotopy analysis method. Findings – The following findings are concluded: first, the entropy generation rate is more near the plates in both the zones as compared to that of the interface. This indicates that the friction due to surface on the fluids increases entropy generation rate. Second, the entropy generation rate is more near the plate in Zone I than that of Zone II. This may be due to the fact that the fluid in Zone I is more viscous. This indicates the more the viscosity of the fluid is, the more the entropy generation. Third, Bejan number is the maximum at the interface of the fluids. This indicates that the amount of exergy (available energy) is maximum and irreversibility is minimized at the interface between the fluids. Fourth, as micropolarity increases, entropy generation rate near the plates decreases and irreversibility decreases. This indicates an important industrial application for micropolar fluids to use them as a good lubricant. Originality/value – The problem is original as no work has been reported on entropy generation in an inclined channel with two immiscible micropolar fluids.

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25

Liang, Zhilei, and Dehua Wang. "Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows." Mathematical Models and Methods in Applied Sciences 30, no. 12 (October 23, 2020): 2445–86. http://dx.doi.org/10.1142/s0218202520500475.

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A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

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26

Sava, Valeriu Al. "A spatial decay estimate of the flow equations of micropolar fluids." International Journal of Engineering Science 24, no. 3 (January 1986): 449–52. http://dx.doi.org/10.1016/0020-7225(86)90099-6.

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27

Chandrawat, Rajesh Kumar, Varun Joshi, and O. Anwar Bég. "Ion Slip and Hall Effects on Generalized Time-Dependent Hydromagnetic Couette Flow of Immiscible Micropolar and Dusty Micropolar Fluids with Heat Transfer and Dissipation: A Numerical Study." Journal of Nanofluids 10, no. 3 (September 1, 2021): 431–46. http://dx.doi.org/10.1166/jon.2021.1792.

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The hydrodynamics of immiscible micropolar fluids are important in a variety of engineering problems, including biofluid dynamics of arterial blood flows, pharmacodynamics, Principle of Boundary layers, lubrication technology, short waves for heat-conducting fluids, sediment transportation, magnetohydrodynamics, multicomponent hydrodynamics, and electrohydrodynamic. Motivated by the development of biological fluid modeling and medical diagnosis instrumentation, this article examines the collective impacts of ion slip, viscous dissipation, Joule heating, and Hall current on unsteady generalized magnetohydrodynamic (MHD) Couette flow of two immiscible fluids. Two non-Newtonian incompressible magnetohydrodynamic micropolar and micropolar dusty (fluid-particle suspension) fluids are considered in a horizontal duct with heat transfer. No-slip boundary conditions are assumed at the channel walls and constant pressure gradient. Continuous shear stress and fluid velocity are considered across the interface between the two immiscible fluids. The coupled partial differential equations are formulated for fluids and particle phases and the velocities, temperatures, and microrotation profiles are obtained. Under the physically realistic boundary and interfacial conditions, the Modified cubic-Bspline differential quadrature approach (MCB-DQM) is deployed to obtain numerical results. The influence of the magnetic, thermal, and other pertinent parameters, i.e. Hartmann magnetic number, Eckert (dissipation) number, Reynolds number, Prandtl number, micropolar material parameters, Hall and ion-slip parameters, particle concentration parameter, viscosity ratio, density ratio, and time on velocity, microrotation, and temperature characteristics are illustrated through graphs. The MCB-DQM is found to be in good agreement with accuracy and the skin friction coefficient and Nusselt number are also explored. It is found that fluids and particle velocities are reduced with increasing Hartmann numbers whereas they are elevated with increment in ion-slip and Hall parameters. Temperatures are generally enhanced with increasing Eckert number and viscosity ratio. The simulations are relevant to nuclear heat transfer control, MHD energy generators, and electromagnetic multiphase systems in chemical engineering.

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28

Benariba, Aboubakeur, Ahmed Bouzidane, and Marc Thomas. "Analytical analysis of a rigid rotor mounted on three hydrostatic pads lubricated with micropolar fluids." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 233, no. 6 (October 23, 2018): 859–69. http://dx.doi.org/10.1177/1350650118806374.

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Analytical analysis of rigid rotor set vertically, supported by a new hydrostatic squeeze film damper consisting in three hydrostatic pads fed through three capillary restrictors operating with micropolar lubricant is presented. The modified Reynolds equation is obtained using the micropolar lubrication theory and solving it analytically. The calculation of the flow rate, dimensionless vibratory amplitude and amplitude of transmitted forces are determined by resolving the equations of rotor motion using nonlinear methods. It has been observed that a rigid rotor operating with micropolar lubricant shows an increase in the value of transmitted forces at high speeds and a reduction in the value of vibratory response at critical speed when compared to a corresponding similar rigid rotor operating with Newtonian lubricant.

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29

Rafique, Anwar, Misiran, Khan, Baleanu, Nisar, Sherif, and Seikh. "Hydromagnetic Flow of Micropolar Nanofluid." Symmetry 12, no. 2 (February 6, 2020): 251. http://dx.doi.org/10.3390/sym12020251.

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Similar to other fluids (Newtonian and non-Newtonian), micropolar fluid also exhibits symmetric flow and exact symmetric solution similar to the Navier–Stokes equation; however, it is not always realizable. In this article, the Buongiorno mathematical model of hydromagnetic micropolar nanofluid is considered. A joint phenomenon of heat and mass transfer is studied in this work. This model indeed incorporates two important effects, namely, the Brownian motion and the thermophoretic. In addition, the effects of magnetohydrodynamic (MHD) and chemical reaction are considered. The fluid is taken over a slanted, stretching surface making an inclination with the vertical one. Suitable similarity transformations are applied to develop a nonlinear transformed model in terms of ODEs (ordinary differential equations). For the numerical simulations, an efficient, stable, and reliable scheme of Keller-box is applied to the transformed model. More exactly, the governing system of equations is written in the first order system and then arranged in the forms of a matrix system using the block-tridiagonal factorization. These numerical simulations are then arranged in graphs for various parameters of interest. The physical quantities including skin friction, Nusselt number, and Sherwood number along with different effects involved in the governing equations are also justified through graphs. The consequences reveal that concentration profile increases by increasing chemical reaction parameters. In addition, the Nusselt number and Sherwood number decreases by decreasing the inclination.

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30

Chu, Li Ming, Jaw-Ren Lin, Yuh-Ping Chang, and Chung-Chun Wu. "Elastohydrodynamic lubrication of circular contacts at pure squeeze motion with micropolar lubricants." Industrial Lubrication and Tribology 68, no. 6 (September 12, 2016): 640–46. http://dx.doi.org/10.1108/ilt-10-2015-0139.

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Purpose This paper aims to explore pure squeeze elastohydrodynamic lubrication (EHL) motion of circular contacts with micropolar lubricants under constant load. The proposed model can reasonably calculate the pressure distributions, film thicknesses and normal squeeze velocities during the pure squeeze process. Design/methodology/approach The transient modified Reynolds equation is derived in polar coordinates using micropolar fluids theory. The finite difference method and the Gauss–Seidel iteration method are used to solve the transient modified Reynolds equation, the elasticity deformation equation, load balance equation and lubricant rheology equations simultaneously. Findings The simulation results reveal that the effect of the micropolar lubricant is equivalent to enhancing the lubricant viscosity. As the film thickness is enlarged, the central pressure and film thickness for micropolar lubricants are larger than those of Newtonian fluids under the same load in the elastic deformation stage. The greater the coupling parameter (N), the greater the maximum central pressure. However, the smaller the characteristic length (L), the greater the maximum central pressure. The time needed to achieve maximum central pressure increases with increasing N and L. Originality/value A numerical method for general applications was developed to investigate the effects of the micropolar lubricants at pure squeeze EHL motion of circular contacts under constant load.

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31

Ahmad, Farooq, A. Othman Almatroud, Sajjad Hussain, Shan E. Farooq, and Roman Ullah. "Numerical Solution of Nonlinear Diff. Equations for Heat Transfer in Micropolar Fluids over a Stretching Domain." Mathematics 8, no. 5 (May 25, 2020): 854. http://dx.doi.org/10.3390/math8050854.

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A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat transfer is associated with two different sets of boundary conditions, the isothermal wall and isoflux boundary. On the basis of porosity of medium, similarity functions are utilized to avail a set of ordinary differential equations. The non-linear coupled ODE’s have been solved with a very stable and reliable numerical scheme that involves Simpson’s Rule and Successive over Relaxation method. The accuracy of the results is improved by making iterations on three different grid sizes and higher order accuracy in the results is achieved by Richardson extrapolation. This study provides realistic and differentiated results with due considerations of micropolar fluid theory. The micropolar material parameters demonstrated reduction in the bulk fluid speed, thermal distribution and skin friction coefficient but increase in local heat transfer rate and couple stress. The spin behavior of microstructures is also exhibited through microrotation vector N ( η ) .

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32

Uddin, Ziya, Manoj Kumar, and Souad Harmand. "Influence of thermal radiation and heat generation/absorption on MHD heat transfer flow of a micropolar fluid past a wedge considering hall and ion slip currents." Thermal Science 18, suppl.2 (2014): 489–502. http://dx.doi.org/10.2298/tsci110712085u.

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In this paper a numerical model is developed to examine the effect of thermal radiation on magnetohydrodynamic heat transfer flow of a micropolar fluid past a non-conducting wedge in presence of heat source/sink. In the model it is assumed that the fluid is viscous, incompressible and electrically conducting. The Hall and ion slip effects have also been taken into consideration. The model contains highly non-linear coupled partial differential equations which have been converted into ordinary differential equation by using the similarity transformations. These equations are then solved numerically by Shooting technique along with the Runge-Kutta-Fehlberg integration scheme for entire range of parameters with appropriate boundary conditions. The effects of various parameters involved in the problem have been studied with the help of graphs. Numerical values of skin friction coefficients and Nusselt number are presented in tabular form. The results showed that the micropolar fluids are better to reduce local skin drag as compared to Newtonian fluids and the presence of heat sink increases the heat transfer rate.

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33

IDO, Yasushi, and Takahiko TANAHASHI. "Fundamental equations for magnetic fluids by micropolar theory. 1st report: Strain tensors and balance equations." Transactions of the Japan Society of Mechanical Engineers Series B 56, no. 525 (1990): 1385–91. http://dx.doi.org/10.1299/kikaib.56.1385.

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34

Tangsali, Param R., Nagaraj N. Katagi, Ashwini Bhat, and Manjunath Shettar. "Analysis of Magnetohydrodynamic Free Convection in Micropolar Fluids over a Permeable Shrinking Sheet with Slip Boundary Conditions." Symmetry 16, no. 4 (March 29, 2024): 400. http://dx.doi.org/10.3390/sym16040400.

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The convective micropolar fluid flow over a permeable shrinking sheet in the presence of a heat source and thermal radiation with the magnetic field directed towards the sheet has been studied in this paper. The mathematical formulation considers the partial slip condition at the sheet, allowing a realistic representation of the fluid flow near the boundary. The governing equations for the flow, heat, and mass transfer are formulated using the conservation laws of mass, momentum, angular momentum, energy, and concentration. The resulting nonlinear partial differential equations are transformed into a system of ordinary differential equations using suitable similarity transformations. The numerical solutions are obtained using robust computational techniques to examine the influence of various parameters on the velocity, temperature, and concentration profiles. The impact of slip effects, micropolar fluid characteristics, and permeability parameters on the flow features and heat transfer rates are thoroughly analyzed. The findings of this investigation offer valuable insights into the behavior of micropolar fluids in free convection flows over permeable shrinking sheets with slip, providing a foundation for potential applications in various industrial and engineering processes. Key findings include the observation that the velocity profile overshoots for assisting flow with decreasing viscous force and rising magnetic effects as opposed to opposing flow. The thermal boundary layer thickness decreases due to buoyant force but shows increasing behavior with heat source parameters. The present result agrees with the earlier findings for specific parameter values in particular cases.

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35

Chandrawat, Rajesh Kumar, Varun Joshi, and O. Anwar Bég. "Numerical Study of Interface Tracking for the Unsteady Flow of Two Immiscible Micropolar and Newtonian Fluids Through a Horizontal Channel with an Unstable Interface." Journal of Nanofluids 10, no. 4 (December 1, 2021): 552–63. http://dx.doi.org/10.1166/jon.2021.1805.

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The dynamics of the interaction between immiscible fluids is relevant to numerous complex flows in nature and industry, including lubrication and coating processes, oil extraction, physicochemical separation techniques, etc. One of the most essential components of immiscible flow is the fluid interface, which must be consistently monitored. In this article, the unsteady flow of two immiscible fluids i.e., an Eringen micropolar and Newtonian liquid is considered in a horizontal channel. Despite the no-slip and hyper-stick shear stress condition at the channel edge, it is accepted that the liquid interface is dynamic, migrating from one position to the next and possibly get absolute change; as a result, The CS (continuum surface) model is integrated with the single moment equation based on the VOF (volume of fluid) approach to trace the interface. The immiscible fluids are considered to flow under three applied pressure gradients (constant, decaying, and periodic) and flow is analyzed under seamless shear stress over the entire interface. The modified cubic b-spline differential quadrature method (MCB-DQM) is used to solve the modeled coupled partial differential equations for the fluid interface evolution. The advection and tracking of the interface with time, wave number, and amplitude are illustrated through graphs. It is observed that the presence of micropolar parameters affects the interface with time. The novelty of the current study is that previous studies (which considered the smooth and unstable movement of the micropolar fluid, the steady stream of two immiscible fluids, and interface monitoring through different modes) are extended and generalized to consider the time-dependent flow of two immiscible fluids namely Eringen micropolar and Newtonian with a moving interface in a horizontal channel. For the decaying pressure gradient case, which requires more time to achieve the steady-state, the peak of the waves resembles those for the constant pressure gradient case. The interface becomes steady for a more extensive time when a constant pressure gradient is applied. The interface becomes stable quickly with time as the micropolar parameter is decreased for the constant pressure gradient case i.e., weaker micropolar fluids encourage faster stabilization of the interface. With periodic pressure gradient, the interface takes more time to stabilize, and the crest of the waves is significantly higher in amplitude compared to the constant and decaying pressure cases. The simulations demonstrate the excellent ability of MCB-DQM to analyze complex interfacial immiscible flows.

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36

Cheruku, Vasavi, and B. Ravindra Reddy. "Numerical Study in Effect of Thermal Slip on Two Fluid Flow in a Vertical Channel." Transactions on Energy Systems and Engineering Applications 4, no. 2 (July 17, 2023): 1–18. http://dx.doi.org/10.32397/tesea.vol4.n2.517.

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The present study investigates the effect of thermal slip on an immiscible flow of micropolar and viscous fluids in a vertical channel. The left boundary is subjected to thermal slip with appropriate boundary and interface conditions, resulting in a linked system of nonlinear partial differential equations. The ND Solve technique in Mathematica software is used to implement the Runge-Kutta method of the sixth order. The velocity, temperature, and concentration equations are then calculated. The mass, heat, and velocity transmission rates at the boundaries were recorded for all the variations in the governing parameters. In addition, the impact of relevant parameters on various physical properties of micropolar and viscous fluids is analyzed through graphical means. The results are then discussed in detail. Thermal slip, Grashof number, molecular number, magnetic parameter, and Reynolds number are crucial factors that significantly affect heat and mass transfer in fluid flow. The effect of the increased thermal slip is noted to result in a decrease in both the velocity profile and temperature. It was also observed that higher values of Grashof and molecular Grashof numbers led to increased velocity and angular velocity.

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37

Nabwey, Hossam A., Ahmed M. Rashad, and Waqar A. Khan. "Slip Microrotation Flow of Silver-Sodium Alginate Nanofluid via Mixed Convection in a Porous Medium." Mathematics 9, no. 24 (December 14, 2021): 3232. http://dx.doi.org/10.3390/math9243232.

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In the previous decennium, considerable applications ofnanoparticles have been developed in the area of science. Nanoparticles with micropolar fluid suspended in conventional fluids can increase the heat transfer. Micropolar fluids have attracted much research attention because of their use in industrial processes. Exotic lubricants, liquid crystal solidification, cooling of a metallic plate in a bath, extrusion of metals and polymers, drawing of plastic films, manufacturing of glass and paper sheets, and colloidal suspension solutions are just a few examples. The primary goal of this studywas to see how radiation and velocity slip affect the mixed convection of sodium alginate nanofluid flow over a non-isothermal wedge in a saturated porous media.In this communication, theTiwari and Das model was employed to investigate the micropolarnanofluid flow via mixed convection over aradiated wedge in a saturated porous medium with the velocity slip condition. Nanoparticles of silver (Ag) wreused in asodium alginate base fluid. The intended system of governing equations is converted to a set of ordinary differential equations and then solved applying the finite difference method. Variousfluid flows, temperatures, and physical quantities of interest were examined. The effects of radiation on the skin friction are negligible in the case of forced and mixed convection, whereas radiation increases the skin friction in free convection. It is demonstrated that the pressure gradient, solid volume fraction, radiation, and slip parameters enhance the Nusselt number, whereas the micropolar parameter reduces the Nusselt number.

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38

Erofeev, V. I., A. V. Shekoyan, and M. V. Belubekyan. "SPATIALLY-LOCALIZED NONLINEAR MAGNETOELASTIC WAVES IN A MICROPOLAR ELECTRICAL CONDUCTING MEDIUM." Problems of strenght and plasticity 81, no. 4 (2019): 402–15. http://dx.doi.org/10.32326/1814-9146-2019-81-4-402-415.

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A nonlinear model of an electrically conducting micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending-torsion tensor. In both tensors, linear and nonlinear terms are taken into account in rotation gradients and displacement gradients (geometric nonlinearity). The components of the bending-torsion tensor, which have identical indices, describe torsional deformations, and the rest - bending deformations. The stress state of the medium is described by two asymmetric tensors: stress tensor and moment stress tensor. It is assumed, as it is usual in magnetoelasticity, that the action of the electromagnetic field on the deformation field occurs through the Lorentz forces. From the system of Maxwell equations follow the equations for electrical and magnetic inductions, which, together with the electromagnetic equations of state, must be added to the equations of the dynamics of a micropolar medium. Within the framework of the proposed model, a one-dimensional nonlinear shear-rotation magnetoelastic wave is considered. The nonlinear term is selected and taken into account in the equations of dynamics, making the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of a sharp drop in the moving profile. The dispersion, on the contrary, smoothes the differences due to the difference in the phase velocities of the harmonic components of the waves. The combined effect of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing the shape. Only those cases are physically feasible when there is no constant component in the deformation wave. Stationary waves can be both periodic and aperiodic. The latter are spatially localized waves - solitons. It is shown that the behavior of "subsonic" and “supersonic” solitons will be qualitatively different.

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39

Naduvinamani, N. B., and G. B. Marali. "Dynamic Reynolds equation for micropolar fluids and the analysis of plane inclined slider bearings with squeezing effect." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 221, no. 7 (July 1, 2007): 823–29. http://dx.doi.org/10.1243/13506501jet286.

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The general dynamic Reynolds equation of sliding-squeezing surfaces with micro-polar fluids is derived for the assessment of dynamic characteristics of bearings with general film thickness. The detailed analysis is presented for the plane inclined slider bearings by using perturbation method. Two Reynolds-type equations corresponding to steady performance and perturbed characteristics are obtained. The closed form solution of these equations is obtained. The numerical computations of the results show that, the micropolar fluids provide an improved characteristics for both steady-state and the dynamic stiffness and damping characteristics. It is found that the maximum steady-load-carrying capacity is function of coupling parameter and is achieved at smaller values of profile parameter for larger values of the coupling parameter.

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40

Hasnain, Jafar, and Zaheer Abbas. "Entropy generation analysis on two-phase micropolar nanofluids flow in an inclined channel with convective heat transfer." Thermal Science 23, no. 3 Part B (2019): 1765–77. http://dx.doi.org/10.2298/tsci170715221h.

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This article deals the entropy generation due to mixed convective flow of two nonmiscible and electrically conducting fluids streaming through an inclined channel by considering convective boundary conditions at the walls of channel. Micropolar fluid is flowing adjacent to the upper wall of the channel and fluid flowing between the non- Newtonain fluid layer and lower plate of channel is water based nanofluid. The transformed dimensionless coupled equations are solved numerically via shooting technique. The numerical results are plotted to analyze the effects of various emerging parameters. This study shows that an increase in magnetic parameter and Brinkman number causes an increase in entropy generation whereas entropy generation reduces with increase in micropolar parameter and nanoparticle volume fraction.

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41

Nadeem, S., M. Y. Malik, and Nadeem Abbas. "Heat transfer of three-dimensional micropolar fluid on a Riga plate." Canadian Journal of Physics 98, no. 1 (January 2020): 32–38. http://dx.doi.org/10.1139/cjp-2018-0973.

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In this article, we deal with prescribed exponential surface temperature and prescribed exponential heat flux due to micropolar fluids flow on a Riga plate. The flow is induced through an exponentially stretching surface within the time-dependent thermal conductivity. Analysis is performed inside the heat transfer. In our study, two cases are discussed here, namely prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). The governing systems of the nonlinear partial differential equations are converted into nonlinear ordinary differential equations using appropriate similarity transformations and boundary layer approach. The reduced systems of nonlinear ordinary differential equations are solved numerically with the help of bvp4c. The significant results are shown in tables and graphs. The variation due to modified Hartman number M is observed in θ (PEST) and [Formula: see text] (PEHF). θ and [Formula: see text] are also reduced for higher values of the radiation parameter Tr. Obtained results are compared with results from the literature.

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42

VADASZ, PETER. "Coriolis effect on gravity-driven convection in a rotating porous layer heated from below." Journal of Fluid Mechanics 376 (December 10, 1998): 351–75. http://dx.doi.org/10.1017/s0022112098002961.

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Linear stability and weak nonlinear theories are used to investigate analytically the Coriolis effect on three-dimensional gravity-driven convection in a rotating porous layer heated from below. Major differences as well as similarities with the corresponding problem in pure fluids (non-porous domains) are particularly highlighted. As such, it is found that, in contrast to the problem in pure fluids, overstable convection in porous media is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr<1). Moreover, it is also established that in the porous-media problem the critical wavenumber in the plane containing the streamlines for stationary convection is not identical to the critical wavenumber associated with convection without rotation, and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless it is evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e. the higher the viscosity the less stable the fluid. Finite-amplitude results obtained by using a weak nonlinear analysis provide differential equations for the amplitude, corresponding to both stationary and overstable convection. These amplitude equations permit one to identify from the post-transient conditions that the fluid is subject to a pitchfork bifurcation in the stationary convection case and to a Hopf bifurcation associated with the overstable convection. Heat transfer results were evaluated from the amplitude solution and are presented in terms of Nusselt number for both stationary and overstable convection. They show that rotation has in general a retarding effect on convective heat transfer, except for a narrow region of small values of the parameter containing the Prandtl number where rotation enhances the heat transfer associated with overstable convection.

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43

Eltayeb, I. A. "Convective instabilities of Maxwell–Cattaneo fluids." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20160712. http://dx.doi.org/10.1098/rspa.2016.0712.

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Motivated by the need to understand better the dynamics of non-Fourier fluids, we examine the linear and weakly nonlinear stabilities of a horizontal layer of fluid obeying the Maxwell–Cattaneo relationship of heat flux and temperature using three different forms of the time derivative of the heat flux. Linear stability mode regime diagrams in the parameter plane have been established and used to summarize the linear instabilities. The energy balance of the system is used to identify the mechanism by which the Maxwell–Cattaneo effect (i) introduces overstability, (ii) leads to preferred stationary modes with the critical Rayleigh and wavelengths either both increasing or both decreasing, (iii) gives rise to instabilities in a layer heated from above, and (iv) enhances heat transfer. A formal weakly nonlinear analysis leads to evolution equations for the amplitudes of linear instability modes. It is shown that the amplitude of the stationary mode obeys an equation of the Landau–Stuart type. The two equally excitable overstable modes obey two equations of the same type coupled by an interaction term. The evolution of the different amplitudes leads to supercritical stability, supercritical instability or subcritical instability depending on the model and parameter values. The results are presented in regime diagrams.

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44

Chen, Mingtao, Bin Huang, and Jianwen Zhang. "Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum." Nonlinear Analysis: Theory, Methods & Applications 79 (March 2013): 1–11. http://dx.doi.org/10.1016/j.na.2012.10.013.

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45

Adeniyan, Adetunji, Gbeminiyi M. Sobamowo, and Samsondeen O. Kehinde. "Impacts of Slips on Peristaltic flow and Heat transfer of micropolar fluids in an asymmetric channel." International Journal of Mathematical Analysis and Optimization: Theory and Applications 7, no. 2 (March 2022): 107–29. http://dx.doi.org/10.52968/28308561.

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The study of peristaltic motion is an area of increasing research interest in industrial, biological and engineering interest. In this study, effects of slips on the peristaltic flow and heat transfer of micropolar fluids in an asymmetric channel are investigated analytically. The developed non-linear coupled partial differential equations are converted into non-linear coupled ordinary differential equations using similarity transformation. The ordinary differential equations are solved for the cases when the thermal viscosity parameter is zero and non-zero. Exact solutions are gotten for the cases of linear and non-linear when the thermal viscosity parameter is zero and non-zero, respectively. The obtain results depict that viscous and thermal slips enhances the flow of the bolus as it is being transported through the digestive system. Also, the effect of microrotation helps in reducing the pressure gradient for the flow.

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46

DUAN, RENJUN, SEIJI UKAI, TONG YANG, and HUIJIANG ZHAO. "OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH POTENTIAL FORCES." Mathematical Models and Methods in Applied Sciences 17, no. 05 (May 2007): 737–58. http://dx.doi.org/10.1142/s021820250700208x.

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For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

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47

Ishigaki, Yusuke, and Yoshihiro Ueda. "Stability of stationary solutions to outflow problem for compressible viscoelastic system in one dimensional half space." AIMS Mathematics 9, no. 11 (2024): 33215–53. http://dx.doi.org/10.3934/math.20241585.

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<p>The system of equations describing motion of compressible viscoelastic fluids is considered in a one dimensional half space under the outflow boundary condition. We investigate the existence and stability of stationary solutions. It is shown that the stationary solution exists for large Mach number and small number of propagation speed of elastic wave. We next show that the stationary solution is asymptotically stable, provided that the initial perturbation is sufficiently small.</p>

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48

Chandrawat, Rajesh Kumar, and Varun Joshi. "Numerical Solution of the Time-Depending Flow of Immiscible Fluids with Fuzzy Boundary Conditions." International Journal of Mathematical, Engineering and Management Sciences 6, no. 5 (October 1, 2021): 1315–30. http://dx.doi.org/10.33889/ijmems.2021.6.5.079.

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Fluid flow modeling using fuzzy boundary conditions is one of the viable areas in biofluid mechanics, drug suspension in pharmacology, as well as in the cytology and electrohydrodynamic analysis of cerebrospinal fluid data. In this article, a fuzzy solution for the two immiscible fluid flow problems is developed, which is motivated by biomechanical flow engineering. Two immiscible fluids, namely micropolar and Newtonian fluid, are considered with fuzzy boundary conditions in the horizontal channel. The flow is considered unsteady and carried out by applying a constant pressure gradient in the X-direction of the channel. The coupled partial differential equations are modeled for fuzzy profiles of velocity and micro-rotation vectors then the numerical results are obtained by the modified cubic B - spline differential quadrature method. The evolution of membership grades for velocity and microrotation profiles has been depicted with the fuzzy boundaries at the channel wall. It is observed that Micropolar fluid has a higher velocity change than Newtonian fluid, and both profiles indicate a declining nature toward the interface.

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49

Lin, Hongxia, Sen Liu, Heng Zhang, and Qing Sun. "Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation." Nonlinearity 37, no. 5 (March 18, 2024): 055001. http://dx.doi.org/10.1088/1361-6544/ad3098.

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Abstract This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.

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50

Célérier, M. N. "Fully integrated interior solutions of GR for stationary rigidly rotating cylindrical perfect fluids." Journal of Mathematical Physics 64, no. 2 (February 1, 2023): 022501. http://dx.doi.org/10.1063/5.0131945.

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In an important series of articles published during the 1970s, Krasiński [Acta Phys. Pol. B 5, 411 (1974); 6, 223 (1975); J. Math. Phys. 16, 125 (1975); Rep. Math. Phys. 14, 225 (1978)] displayed a class of interior solutions of the Einstein field equations sourced by a stationary isentropic rotating cylinder of a perfect fluid. However, these solutions depend on an unspecified arbitrary function, which leads the author to claim that the equation of state of the fluid could not be obtained directly from the field equations but had to be added by hand. In the present article, we use a double ansatz, which we have developed in 2021 and implemented at length into a series of recent papers displaying exact interior solutions for a stationary rotating cylindrically symmetric fluid with anisotropic pressure. This ansatz allows us to obtain here a fully integrated class of solutions to the Einstein equations, written with the use of very simple analytical functions, and to show that the equation of state of the fluid follows naturally from these field equations.

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