Microrotation Profiles Research Articles

Numerical investigations of asymmetric flow of a micropolar fluid between two porous disks

Numerical solution is presented for the two-dimensional flow of a micropolar fluid between two porous coaxial disks of different permeability for a range of Reynolds number Re (−300 ≤ Re < 0) and permeability parameter A (1.0 ≤ A ≤ 2.0). The main flow is superimposed by the injection at the surfaces of the two disks. Von Karman’s similarity transformations are used to reduce the governing equations of motion to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on the finite difference method is employed to solve these ODEs and Richardson’s extrapolation is used to obtain higher order accuracy. The results indicate that the parameters Re and A have a strong influence on the velocity and microrotation profiles, shear stresses at the disks and the position of the viscous/shear layer. The micropolar material constants c1, c2, c3 have profound effect on microrotation as compared to their effect on streamwise and axial velocity profiles. The results of micropolar fluids are compared with the results for Newtonian fluids.

Studying effect of MHD on thin films of a micropolar fluid

This paper deals with the study of the effect of MHD on thin films of a micropolar fluid. These thin films are considered for three different geometries, namely: (i) flow down an inclined plane, (ii) flow on a moving belt and (iii) flow down a vertical cylinder. The transformed boundary layer governing equations of a micropolar fluid and the resulting system of coupled non-linear ordinary differential equations are solved numerically by using shooting method. Numerical results were presented for velocity and micro-rotation profiles within the boundary layer for different parameters of the problem including micropolar fluid parameters, magnetic field parameter, etc., which are also discussed numerically and illustrated graphically.

Numerical study of asymmetric laminar flow of micropolar fluids in a porous channel

A numerical study is presented for the two-dimensional flow of a micropolar fluid in a porous channel. The channel walls are of different permeability. The fluid motion is superimposed by the large injection at the two walls and is assumed to be steady, laminar and incompressible. The micropolar model due to Eringen is used to describe the working fluid. The governing equations of motion are reduced to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form by using an extension of Berman’s similarity transformations. A numerical algorithm based on finite difference discretization is employed to solve these ODEs. The results obtained are further improved by Richardson’s extrapolation for higher order accuracy. Comparisons with the previously published work are performed and are found to be in a good agreement. It has been observed that the velocity and microrotation profiles change from the most asymmetric shape to the symmetric shape across the channel as the parameter R or the permeability parameter A are varied between their extreme values. The results indicate that larger the injection velocity at a wall relative to the other is, smaller will be the shear stress at it than that at the other. The position of viscous layer has been found to be more sensitive to the permeability parameter A than to the parameter R. The micropolar fluids reduce shear stress and increase couple stress at the walls as compared to the Newtonian fluids.

Unsteady Boundary Layer Flow Induced by Accelerating Motion Near the Rear Stagnation Point in a Micropolar Fluid

The unsteady boundary layer flow of a micropolar fluid induced by a two-dimensional body, which is started impulsively from rest, is studied in this paper. The variation with time t of the external stream V(t) is assumed to be of the form V(t) 1 - exp(-atm), where a = 0 means a coefficient of acceleration and m is an arbitrary integral value. The problem is formulated for the flow at the rear stagnation point on an infinite plane wall. Numerical solutions of the unsteady boundary layer equations are obtained using an implicit finite-difference scheme known as the Keller's box method. Results are given for the velocity and microrotation profiles, as well as for the dimensionless time elapsed before the boundary layer begins to separate from the wall. It is found that the dimensionless time elapsed before separation takes place is lower for a micropolar fluid (K 0) than for a Newtonian fluid (K 0), where K denotes the micropolar or material parameter.

The unsteady boundary layer flow past a circular cylinder in micropolar fluids

PurposeThe purpose of this paper is to study the unsteady boundary layer flow of a micropolar fluid past a circular cylinder which is started impulsively from rest.Design/methodology/approachThe nonlinear partial differential equations consisting of three independent variables are solved numerically using the 3D Keller‐box method.FindingsNumerical solutions for the velocity profiles, wall skin friction and microrotation profiles are obtained and presented for various values of time t and material parameter K with the boundary condition for microrotation n=0 (strong concentration of microelements) and n=1/2 (weak concentration of microelements). The results are presented along the points on the cylinder surface, starting from the forward to the rear stagnation point, for small time up to the time when the boundary layer flow separates from the cylinder.Originality/valueIt is believed that this is the first paper that uses the 3D Keller‐box method to study the unsteady boundary layer flow of micropolar fluids. In the last four decades, there has been overhelming interest shown by researchers in micropolar fluids and still many problems are unsolved. The paper shows not only the fundamental importance of this problem, but also the implications for situations of practical interest.

Moving wedge and flat plate in a micropolar fluid

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and boundary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solve numerically using a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocity and microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law parameter ( m), the ratio of the boundary velocity to the free stream velocity parameter ( λ) and the material parameter ( K). Important features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when the boundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction compared to Newtonian fluids.

Series solutions of unsteady boundary layer flow of a micropolar fluid near the forward stagnation point of a plane surface

In this paper, the unsteady boundary-layer flow of a micropolar fluid started impulsively from rest near the forward stagnation point of a two-dimensional plane surface is studied by means of an analytic approach, namely homotopy analysis method. This approach gives accurate approximations uniformly valid for all dimensionless time. Besides, analytic results are given for the reduced velocity and microrotation profiles, as well as for the skin friction coefficient when the material parameter K takes the value K=0 (Newtonian fluid), 1, 3, 5 and 10. To the best of our knowledge, such a kind of series solutions has been never reported.

Flow of a micropolar fluid on a continuous moving surface

The present paper deals with the analysis of steady boundary layer flow and heat transfer of a micropolar fluid on an isothermal continuously moving plane surface. It is assumed that the microinertia density is variable and not constant, as in many other published papers. Also, the viscous dissipation effect is taken into account. The basic partial differential equations are reduced to a system of nonlinear ordinary differential equations, which is solved numerically using the Keller-box method. Numerical results are obtained for the skin friction coefficient, local Nusselt number, as well as velocity, temperature and microrotation profiles. Results are shown in graphical form and the numerical values for the skin friction coefficient and local Nusselt number are given in the form of tables. The effects of material parameter K , Prandtl number Pr and Eckert number E c on the flow and heat transfer characteristics are discussed.

Steady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface

PurposeTo study the steady mixed convection boundary‐layer flow of a micropolar fluid near the region of the stagnation point on a double‐infinite vertical flat plate is studied. The results of this paper are important for the researchers in the area of micropolar fluids.Design/methodology/approachFor the case considered the problem reduces to a system of ordinary differential equations, which is solved numerically using the Keller‐box method. This method is very efficient for solving boundary‐layer problems and it can easily be applied to other general situations than that presented in this paper. Any PhD student can learn and apply it very easily.FindingsRepresentative results for the velocity, microrotation and temperature profiles, as well as for the reduced skin friction coefficient and the local Nusselt number have been obtained for the case of strong concentration, Prandtl number of 0.7, some values of the material parameter K and the mixed convection parameter λ(≥0). Both assisting and opposing flow cases are considered. Results for the reduced skin friction coefficient and reduced local Nusselt number as well as for the reduced velocity, temperature and microrotation profiles are given in tables and figures. The obtained results are compared with ones from the open literature and it is found that they are in excellent agreement. The important conclusion is, we have been able to show that for opposing flow solutions are possible are possible only for a limited range of values of the mixed convection parameter λ.Research limitations/implicationsThe results of this paper are valid only in the small region around the stagnation point on a vertical surface and they are not applicable outside this region.Practical implicationsThe theory of micropolar fluids and also the results of the present paper can be used to explain the characteristics in certain fluids such as exotic lubricants, colloidal suspensions or polymeric fluids, liquid crystals, and animal blood.Originality/valueThe paper is very well prepared, presented and readable. We believe that the results are original and important from both theoretical and application point of views.

Steady two‐dimensional asymmetric stagnation point flow of a micropolar fluid

AbstractNumerical solutions are presented for the steady two‐dimensional boundary‐layer flow of a micropolar fluid near an asymmetric plane stagnation point. The transformed non‐similar boundary‐layer equations are solved numerically using an implicit finite‐difference method known as Keller‐box method. Numerical results are given for the reduced velocity and microrotation profiles, as well as for the reduced skin friction coefficient when the material parameter K takes the value K = 0 (Newtonian fluid), 0.5, 1, 1.5, 2, 2.5, and 3. Important features of these flow characteristics are shown on graphs and in tables. Comparisons with existing solutions for both symmetric and asymmetric stagnation point flow cases are given and it is shown that these results are in very good agreement. Discussion of the flow characteristics is also made.

Unsteady boundary layer flow of a micropolar fluid near the rear stagnation point of a plane surface

The growth of the boundary layer flow of a viscous and incompressible micropolar fluid started impulsively from rest near the rear stagnation point of a two-dimensional plane surface is studied theoretically. The transformed non-similar boundary-layer equations are solved numerically using a very efficient finite-difference method known as Keller-box method. This method may present well-behaved solutions for the transient (small time) solution up to the separation boundary layer flow. Numerical results are given for the reduced velocity and microrotation profiles, as well as for the skin friction coefficient when the material parameter K takes the values K=0 (Newtonian fluid), 0.5, 1, 1.1, 1.5, 2, 2.5 and 3 with the boundary condition for microrotation n=0 (strong concentration of microelements) and n=1/2 (weak concentration of microelements), respectively. Important features of these flow characteristics are shown on graphs and in tables.

Unsteady boundary layer flow of a micropolar fluid near the forward stagnation point of a plane surface

The growth of the boundary-layer flow of a micropolar fluid started impulsively from rest near the forward stagnation point of a two-dimensional plane surface is studied theoretically. The transformed non-similar boundary-layer equations are solved numerically using a very efficient finite-difference method known as Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. Numerical results are given for the reduced velocity and microrotation profiles, as well as for the skin friction coefficient when the material parameter K takes the value K=0 (Newtonian fluid), 0.5, 1, 1.5, 2, 2.5 and 3. Important features of these flow characteristics are shown on graphs and in tables.

MIXED CONVECTION OF MICROPOLAR FLUIDS ALONG A VERTICAL WAVY SURFACE WITH A DISCONTINUOUS TEMPERATURE PROFILE

This article presents numerical solutions for solving the problem of a mixed convective micropolar fluid flow and heat transfer along a vertical wavy surface with a discontinuous temperature profile. The overall surface is equally divided into a heated section succeeded by an unheated section alternately. The problems in the present study have been formulated by using a simple transposition theorem and the cubic spline collocation method. Eringen has applied the spline alternating direction implicit (SADI) procedure to solve the governing momentum, angular momentum, and energy equations those formulated. Along the wavy surface, the velocity, temperature, and microrotation profiles are presented. The influences of micropolar parameters R, u , geometry, and Gr/Re 2 number on the skin friction coefficient and Nusselt number have been studied in this work. The results demonstrate that the skin friction coefficient consists of a mixture of two harmonics in micropolar fluids and in Newtonian fluids. As the vortex viscosity parameter (R) increases, the heat transfer rate decreases, but the skin friction increases. In addition, when the spin gradient viscosity parameter ( u ) increases, the skin friction decreases. Comparisons between a Newtonian fluid and a micropolar fluid are also discussed.

Nonsimilar incompressible laminar boundary-layer flows in micropolar fluids

The solution of the steady laminar incompressible nonsimilar boundary-layer problem for micropolar fluids over two-dimensional and axisymmetric bodies has been presented. The partial differential equations governing the flow have been transformed into new co-ordinates having finite range. The resulting equations have been solved numerically using implicit finite-difference scheme. The computations have been carried out for a cylinder and a sphere. The results indicate that the separation in micropolar fluids occurs at earlier streamwise locations as compared to Newtonian fluids. The skin friction and velocity profiles depend on the shape of the body and are almost insensitive to microrotation or coupling parameter, provided the coupling parameter is small. On the other hand, the microrotation profiles and microrotation gradient depend on the microrotation parameter and they are insensitive to the coupling parameter.

Similar solutions for the incompressible laminar boundary layer with pressure gradient in micropolar fluids

This paper presents the similarity solution for the steady incompressible laminar boundary layer flow of a micropolar fluid past an infinite wedge. The governing equations have been solved numerically using fourth orderRunge-Kutta-Gill method. The results indicate the extent to which the velocity and microrotation profiles, and the surface shear stress are influenced by coupling, microrotation, and pressure gradient parameters. The important role played by the standard length of the micropolar fluid in determining the structure of the boundary layer has also been discussed.