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The intent of the current work is to study the time-dependent MHD flow of two immiscible Newtonian and micropolar fluids in a porous pipe in presence of heat transfer. The numerical solutions for the flow under consideration are obtained using Crank–Nicolson approach. The classical no-slip condition at the boundary of the pipe is used along with the appropriate interface conditions at the interface, to solve the initial boundary value problem governing the flow. The results for fluid velocities, microrotation and temperatures with varying physical parameters are displayed through graphs and are discussed as well. The Nusselt number is also studied for the considered flow, and the results are tabulated.

In this investigation, we have studied the problem of peristaltic flow with heat transfer through the gap between coaxial inclined tubes where the inner tube is rigid and the outer tube has sinusoidal wave travelling down its wall. The problem has been formulated in cylindrical coordinate system. The equations governing the flow have been simplified under the long wavelength and low Reynolds number assumptions. The exact solution is obtained for the temperature profile. The perturbation solutions for the velocity and pressure gradient are obtained for small couple stress parameter. Pressure difference per wavelength and frictional forces on the tube walls have been computed numerically. Results are demonstrated for various flow parameters. The better pumping results occur in vertical tube, while less pumping is seen in horizontal tube. The size of trapped bolus is small in triangular wave as compared to other waves. The present study has a wide range of applications in bio-medical engineering like the transport phenomenon in peristaltic micro pumps.

This article is intended to study the peristaltic motion of a Prandtl nanoliquid through an inclined tapered asymmetric channel. The simultaneous effects such as magnetic field, thermal radiation and chemical reactions have been considered. The geometrical model is considered as tapered asymmetric channel because this situation is observed in the flow of uterine fluid in the uterus. The equations governing the flow are simplified under the assumptions of long wavelength and low Reynolds number. The simplified equations are complex in nature, so that the numerical solutions are presented for the simplified nonlinear partial differential equations considering slip and convective boundary conditions using computational software Mathematica via shooting method. The sundry parameters on the flow quantities have been discussed in detail through graphical and tabular forms. The observed results show that rise in the magnetic effects leads to a reduction in velocity. The radiation parameter decreases the temperature and there is an increment in the pressure gradient with an increase in energy Grashof number. This study is encouraged by exploring the nanofluid dynamics in peristaltic transport as symbolized by heat transport in biological flows, novel pharmacodynamic pumps and gastrointestinal motility enhancement.

This paper deals with an unsteady flow of a micropolar fluid sandwiched between Newtonian fluids through a horizontal channel. The governing time-dependent partial differential equations are solved numerically by using the Crank-Nicolson finite difference approach. The continuity of velocity and shear stress is considered at the fluid–fluid interfaces. It is observed that the fluid velocities increase with time; eventually, a steady state is reached at a certain time instant. The velocity decreases with increasing micropolarity parameter in the micropolar fluid region and remains almost unchanged in both Newtonian fluid regions.

The aim of this article is to study the unsteady flow of immiscible couple stress fluid sandwiched between Newtonian fluids through a horizontal channel. The fluids and plates are initially at rest. At an instant of time, a constant pressure gradient is applied along the horizontal direction to generate the flow. The time-dependent partial differential equations are solved numerically using the finite difference method. The continuity of velocities and shear stresses at the fluid-fluid interfaces has been considered. The obtained results are displayed through graphs and are discussed for various fluid parameters pertaining the flow. The volume flow rate is also obtained numerically for diverse fluid parameters and is presented through a table. It is noticed that fluid velocities increased with time and reached a steady state after a certain time level. Also, the presence of couple stresses reduced the fluid velocities. Volume flow rate increased with Reynolds number and is reduced by increase of ratio of viscosities.

This paper is devoted to the study of the peristaltic flow of a magnetohydrodynamic Walters B fluid through porous medium in a vertical asymmetric channel with heat transfer. The flow is investigated in the wave frame of reference moving with constant velocity with the wave. The non-linear governing partial differential equations are solved using perturbation technique by taking wave number as a small parameter. The expressions for pressure difference and frictional forces are calculated using numerical integration. The graphs are plotted to discuss the variation of velocity, pressure gradient, temperature, heat transfer coefficient, pressure difference, frictional force and trapping for the different flow parameters. It is observed from our analysis that the trapping bolus and pumping rate increase with increase of heat generation parameter. In all situations, the behavior of Hartmann number and Darcy number is always opposite.

This paper investigates the influence of heat transfer on the peristaltic transport of an incompressible magnetohydrodynamic second grade fluid through a porous medium in an inclined asymmetric channel. The flow is investigated in the wave frame of reference moving with the velocity of the wave. The solutions are obtained for the stream function, temperature and pressure gradient using the regular perturbation method by taking the wave number as the perturbation parameter. The pressure difference and frictional force are calculated through numerical integration. The influence of the inclination angle on the flow is discussed, and also the graphical results are presented for different wave forms. The present investigation results agree well with the existing literature. It is found that the velocity, temperature and trapping bolus increase with an increase of the inclination angle. The velocity increases with an increase of the heat generation parameter and Darcy number near the centre of the channel. An increasing Grashof number increases the pressure gradient. There is an increasing effect on the pumping with an increase of the inclination angle and Grashof number. The temperature decreases with an increase of the Hartmann number.

This investigation deals with the influence of heat and mass transfer on magnetohydrodynamic peristaltic flow of Pseudoplastic fluid in a vertical asymmetric channel through porous medium. The flow is examined in the wave frame of reference moving with the velocity of the wave. For formulation of the problem, long wavelength and low Reynolds number assumptions are taken into account. The perturbation solution has been derived for the stream function and pressure gradient. The pressure difference is calculated numerically. The influence of various parameters of interest on velocity, pressure gradient, temperature, concentration, pressure difference, heat transfer coefficient and trapping has been investigated graphically. The graphical results are also discussed for four different wave shapes. It is found that, in all the cases the magnetic parameter and Darcy number have opposite effect on the flow variables. The size of the trapped bolus increases from porous medium to non-porous medium and it decreases with increase of magnetic effects.

This study deals with the MHD steady flow and heat transfer of micropolar and Newtonian fluids, flowing immiscibly through a circular pipe. The pipe is assumed to be filled with uniform porous media. The micropolar and Newtonian fluids occupy core and peripheral regions, respectively. The equations governing the flow are coupled and non-linear. The solutions for velocity, microrotation and temperature are acquired numerically employing finite difference method. At fluid–fluid interface, continuity of velocities, shear stresses, temperatures and heat fluxes are considered. The results for velocity, microrotation and temperature are displayed graphically.

This study deals with the unsteady flow and heat transfer of micropolar and Newtonian fluids, flowing immiscibly through a circular pipe. The micropolar and Newtonian fluids occupy core and peripheral regions, respectively. Initially, the pipe and fluids in both regions are at rest; after an instant of time, a constant pressure gradient is applied to generate the flow. The equations governing the flow are time dependent, coupled and nonlinear. The solutions for velocity, microrotation and temperature are acquired numerically employing Crank–Nicolson finite difference approach. Volume flow rate is also obtained numerically and presented in tabular form. At fluid–fluid interface, continuity of velocities, shear stresses, temperatures and heat fluxes are considered. The results for velocity, microrotation and temperature are displayed graphically. It is seen that the fluid velocities, microrotation and temperatures are increasing with time and attain a steady state after a particular time level. The micropolarity parameter has a decreasing effect on the fluid velocities and temperatures.

The unsteady flow of two immiscible micropolar and Newtonian fluids through a horizontal channel is considered. In addition to the classical no-slip and hyper-stick conditions at the boundary, it is assumed that the fluid velocities and shear stresses are continuous across the fluid-fluid interface. Three cases for the applied pressure gradient are considered to study the problem: one with constant pressure gradient and the other two cases with time-dependent pressure gradients, viz. periodic and decaying pressure gradient. The Crank-Nicolson approach has been used to obtain numerical solutions for fluid velocity and microrotation for diverse sets of fluid parameters. The nature of fluid velocities and microrotation with various values of pressure gradient, Reynolds number, ratio of viscosities, micropolarity parameter and time is illustrated through graphs. It has been observed that micropolarity parameter and ratio of viscosities reduce the fluid velocities.

In the present paper, the effects of magnetic field and heat transfer on the peristaltic flow of an incompressible couple stress fluid through porous medium in an inclined asymmetric channel have been studied under the long wavelength approximation. The exact solutions of the resultant governing equations have been obtained for the stream function, pressure gradient, temperature and heat transfer coefficients. The pressure difference and frictional forces have been computed numerically. The effects of Hartmann number, Darcy number, Grashof number, couple stress parameter, heat generation parameter and inclination angle on the heat characteristics, velocity characteristics, pumping characteristics and trapping phenomena are discussed in detail. It is found that the pressure gradient increases from horizontal channel to vertical channel. The best pumping can be seen at higher Hartmann number. The size of trapped bolus decreases with the increase of couple stress parameter and the strength of the magnetic flied. Increase of heat generation parameter increases the pressure gradient, temperature and the size of the bolus.

The intention of this investigation is to study the effects of heat transfer and inclined magnetic field on the peristaltic flow of Williamson fluid in an asymmetric channel through porous medium. The governing two-dimensional equations are simplified under the assumption of long wavelength approximation. The simplified equations are solved for the stream function, temperature, and axial pressure gradient by using a regular perturbation method. The expression for pressure rise is computed numerically. The profiles of velocity, pressure gradient, temperature, heat transfer coefficient and stream function are sketched and interpreted for various embedded parameters and also the behavior of stream function for various wave forms is discussed through graphs. It is observed that the peristaltic velocity increases from porous medium to non-porous medium, the magnetic effects have increasing effect on the temperature, and the size of the trapped bolus decreases with the increasing of magnetic effects while the trend is reversed with the increasing of Darcy number. Moreover, limiting solutions of our problem are in close agreement with the corresponding results of the Newtonian fluid model.

In the present article, the exact solutions for fundamental flows namely Couette, Poiseuille and generalized Couette flows of an incompressible couple stress fluid between parallel plates are obtained using slip boundary conditions. The effect of various parameters on velocity for each problem is discussed. It is found that, for each of the problems, the solution in the limiting case as couple stresses approaches to zero is similar to that of classical viscous Newtonian fluid. The results indicate that, the presence of couple stresses decreases the velocity of the fluid.

We establish, in this paper, the closed form analytical solutions of steady fully developed flows of couple stress fluid between two concentric cylinders, generated due to the constant pressure gradient or the translatory motion of the outer cylinder or both, using the slip boundary conditions. The classical solutions for Newtonian fluid in the hydrodynamic case appear as a limiting case of our solutions. The velocity profiles of the flows are presented and the effect of various parameters on velocity is discussed. The results indicate that the presence of couple stresses decreases the velocity of the fluid.

This article describes the effect of heat transfer on the magnetohydrodynamic peristaltic flow of Walters B fluid in a vertical symmetric channel. The flow is investigated in a wave frame of reference moving with the velocity of the wave. A perturbation method is used to obtain the approximate analytical solution for the stream function, temperature, and pressure gradient. The pressure difference and frictional force are obtained numerically. The variation of velocity, pressure gradient, temperature, stream function, pressure difference, and frictional force are sketched for various embedded parameters. The behavior of the Nusselt number and skin friction is presented in tables for the different parameters involved. Graphical results are also presented for four different wave shapes.

In this paper, we study the run up flow of an incompressible micropolar fluid between two horizontal infinitely long parallel plates. Initially a flow of the fluid is induced by a constant pressure gradient until steady state is reached. After the steady state is reached, the pressure gradient is suddenly withdrawn while the two plates are impulsively started with different velocities in their own plane. Using the Laplace transform technique and adopting the state space approach, we obtain the velocity and microrotation components in Laplace transform domain. A standard numerical inversion procedure is used to find the velocity and microrotation in space-time domain for various values of time, distance, material parameters and pressure gradient. The variation of velocity and microrotation components is studied and the results are illustrated through graphs. It is observed that the micropolarity parameter has a decreasing effect on velocity component. It is also found that as the gyration parameter increases there is a decrease in microrotation component and an increase in velocity component.

The flow of an incompressible micropolar fluid over a suddenly moved plate is considered under isothermal conditions. State-space technique is used to find the solution of the problem. Inversion of Laplace transform is carried out using a numerical approach. The variation of velocity and microrotation fields is studied with respect to various flow parameters and the results are presented through graphs.

Stokes’ first and second problems for an incompressible couple stress fluid are considered under isothermal conditions. The problems are solved through the use of Laplace transform technique. Inversion of the Laplace transform of the velocity component in each case is carried out using a standard numerical approach. Velocity profiles are plotted and studied for different times and different values of couple stress Reynolds number. The results are presented through graphs in each case.

The intrauterine fluid flow due to myometrial contractions is peristaltic type motion and the myometrial contractions may occur in both symmetric and asymmetric directions. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitude, and phase due to the variation of channel width, wave amplitudes and phase differences. In this paper, we study the effects of heat and mass transfer on the peristaltic transport of magnetohydrodynamic couple stress fluid through homogeneous porous medium in a vertical asymmetric channel. The flow is investigated in the wave frame of reference moving with constant velocity with the wave. The governing equations of couple stress fluid have been simplified under the long wave length approximation. The exact solutions of the resultant governing equations have been derived for the stream function, temperature, concentration, pressure gradient, and heat transfer coefficients. The pressure difference and frictional forces at both the walls are calculated using numerical integration. The influence of diverse flow parameters on the fluid velocity, pressure gradient, temperature, concentration, pressure difference, frictional forces, heat transfer coefficients, and trapping has been discussed. The graphical results are also discussed for four different wave shapes. It is noticed that increasing of couple stresses and heat generation parameter increases the size of the trapped bolus. The heat generation parameter increases the peristaltic pumping and temperature.

In this paper, we investigate the unsteady flow of an incompressible micropolar fluid between infinite parallel plates using state space approach when one of the plates is set to move suddenly while the other is at rest. Analytical expressions of the fluid velocity and microrotation are obtained in the Laplace transform domain. A standard numerical inversion technique is used to invert the Laplace transform of the velocity and microrotation. The effect of various material parameters on flow variables is discussed and the results are presented through graphs.

In the present paper, we have studied the effects of endoscope and heat transfer on the peristaltic flow of second grade fluid through an inclined tube. The endoscope is a solid circular cylinder which is inserted in a peristaltic tube, and the flow takes place through the gap between endoscope and the peristaltic tube. The endoscope is maintained at a temperature [Formula: see text], while the outer tube has a sinusoidal wave traveling down its wall and is exposed to temperature [Formula: see text]. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The equations governing the flow of second grade fluid are modeled in cylindrical coordinates. Using perturbation method, the solutions are obtained for the stream function, pressure gradient and temperature fields. The pressure difference and frictional force at both the walls are calculated using numerical integration. The graphical results are presented to interpret the effect of various physical parameters of interest. It is found that, velocity increases with an increase in inclination angle and the best pumping rate appear in the vertical tube as compared to the horizontal tube. It is also found that, the heat generation parameter has an increasing effect on the velocity of the fluid.

In this paper, we have studied the flow of incompressible fluids in a straight square duct through the porous medium. The couple stress fluid model and Jeffrey fluid model are considered separately to study the flow properties. The governing partial differential equations have been solved numerically using finite difference method in each case. In both the cases, the variation of different flow parameters on the fluid velocity is illustrated graphically and the numerical results for the volume flow rate have been presented through tables. It is observed that, the velocity and volume flow rate decrease with an increase in couple stress parameter and porosity parameter, while the velocity and volume flow rate increase with an increase in Jeffrey parameter and pressure gradient.

In this paper, we study the influence of heat transfer on the peristaltic transport of an incompressible magnetohydrodynamic second grade fluid in vertical symmetric and asymmetric channels. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in the wave frame of reference moving with velocity of the wave. Perturbation solutions are obtained for the stream function, temperature and pressure gradient under long wave length assumption. Pressure difference and frictional force are discussed through numerical integration. The influence of various parameters of interest on the flow are discussed and also the graphical results are obtained for different wave forms.