Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

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Abstract

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy–Brinkman–Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.

Introduction

The theory of microfluids as developed by Eringen [1], which includes the effect of local rotary inertia, the couple stresses and inertial spin. The theory can be used to explain the flow of colloidal fluids, liquid crystal, etc. Eringen [2] also developed the theory of micropolar fluids for the case where only microrotational effects and microrotational inertia exist. He [3] extended the theory of thermomicropolar fluids and derived the constitutive laws for fluids with microstructure. Simple problem on the flow of such fluids were studied by a number of researchers and a review of this work was given by Ariman et al. [4]. The flow and heat transfer for an electrically conducting polar fluid past a porous plate through porous medium in the presence of magnetic field with viscous heating have been an active area of research in view of its applications in many engineering problems such as oil exploration, geothermal energy extractions and the boundary layer control in aerodynamics where neutral fluid seeding in the form of rigid microinclusions.

A great number of porous MHD studies have been carried out examining the effect of magnetic field on hydrodynamics flow and heat transfer in various configurations e.g. in channels, past plate etc. Raptis and Kafousias [5] studied the effect of magnetic field on heat transfer in flow through porous medium bounded by an infinite vertical plate. Kim [6] has examined the influence of magnetic field on unsteady convection flow of polar fluids past a vertical moving porous plate in a porous medium Also very recently Beg et al. [7] used the network thermodynamic simulation approach to study the hydromagnetic convection flow from an isothermal sphere to a non-Darcian porous medium with heat generation or absorption effects. These studies shows that effect of magnetic field can be used to control heating process of electrically conducting fluid flow.

Modeling of viscous dissipation effects on a flow of a fluid saturated porous medium was considered by several author and they have modeled this effect in different ways. The Darcy’s law was agreed by most of the authors. Murthy and Singh [8] have modeled the viscous dissipation effect on the flow of an incompressible fluid in a saturated porous medium. They applied the Forchheimer–Darcy model for the momentum equation by including two terms in the energy equation to represent the viscous dissipation effect. These terms correspond to the first and second order velocity terms in the momentum equation (Forchheimer–Darcy model). A contradiction occurring in the non-Darcy term in the approach of Murthy and Singh [8] was resolved by Nield [9]. But Al-Hadhrami et al [10] point out that the Nield model has also some drawback. It does not hold as the permeability tends to infinity. Later on, a new model for viscous dissipation in a porous medium was proposed by Al-Hadhrami et al. [10] which is probably adequate for most practical purposes. But till now various authors [11], [12] are taking only one term with velocity derivative for viscous dissipation effect in Darcy medium which is wrong. They have ignored the recent developments in the field of porous media, so results presented by them are doubtful.

In the present work we consider the case of free convection flow of a micropolar fluid past a semi-infinite, steadily moving porous plate through porous media with varying suction velocity normal to the plate in the presence of viscous dissipation effect. The Darcy–Brinkman–Forchheimer model which includes the effects of boundary and inertia forces is employed. The viscous dissipation effect is modeled in according to Al-Hadhrami et al. [10]. The mathematical model of problem is highly non linear whose analytical solution is very hard to find out, so the only choice left is approximate numerical solution.

A variety of numerical techniques [13], [14], [15] are used by researcher to solve these types of problems. Out of all the numerical methods developed so far, Finite Element Method [16] has been found the most general method not only to solve the problems of nonlinear heat transfer but also to solve the various problems in different areas of engineering and sciences. Although FEM is most general numerical method but the discretization, meshing and re-meshing of complex geometries of the problems are very difficult and expensive. To overcome these problems, a number of meshless methods have been developed in last two decades. Among all the meshless methods, the element free Galerkin method has been successfully used to solve various types of problems in different areas such as fracture mechanics [17], static and dynamics fracture [18], heat transfer [19], electromagnetic field [20], wave propagation [21]. Recently Sharma et al. [22] has applied element free Galerkin method to study boundary layer flow and heat transfer past a semi-infinite vertical porous moving plate.

Therefore, in this study, a meshless element free Galerkin method (EFGM) has been used as a tool for the numerical simulation. Penalty method has been applied to enforce the essential boundary conditions. The EFGM results have been obtained using cubicspline weight function. Numerical results of the local skin friction coefficient and the local Nusselt number as well as the velocity, microrotation and temperature profiles are presented for different physical parameters.

Section snippets

Mathematical model

An unsteady MHD convection of a micropolar fluid through a porous medium past an infinite vertical porous plate in the presence of viscous dissipation, with periodic temperature when variable suction velocity distribution fluctuating with time is applied. The x-axis is directed along the infinite plate and y-axis is transverse to this. A magnetic field B0 of uniform strength is applied transversely to the direction of the flow and magnetic Reynolds number is assumed to be small so that the

Transformation of model

To obtain the non-dimensional form of the governing equations, we now introduce the following dimensionless variables:U=uV0,V=vV0,Y=V0yν,t=V02tν,Up=upV0,n=nνV02,ω=νV02N,j=V02jν2,θ=T-TTw-T.Substituting relation (8) into the Eqs. (2), (3), (4), following dimensionless partial differential equations are obtained:Ut-(1+δAent)UY=(1+K)2UY2+KωY-M+1+KDa+FrDaUU+Grθ,ωt-(1+δAent)ωY=1+K22ωY2-Kj2ω+UY,θt-(1+δAent)θY=1Pr2θY2+Ec(1+K)1DaU2+FrDaU3+(1+K)UY2-Qθ,where K = k/ρν, Ec=V02c

Review of element free galerkin method

The element free Galerkin method (EFGM) requires moving least square (MLS) interpolation functions to approximate an unknown function, which is made up of three components: a weight function associated with each node, a basis function and a set of coefficients that depends on position. The weight function is non-zero over a small neighborhood at a particular node, called support of the node. Using MLS approximation, the unknown velocity component U is approximated over the domain [0, ∞] asU(Y)Uh

Weight function description

The weight function is non-zero over a small neighborhood of YI, called the domain of influence of node I. The choice of weight function w(Y  YI) affects the resulting approximation in EFG and other meshless methods. Singh et al. [23] has studied on these weight functions and find that cubicspline weight function gives more accurate results as compared to others. Therefore, in the present work, a cubicspline weight function [23] has been used.

Cubicspline weight function:w(r-rI)=w(r)=23-4r2+4r3r1

Enforcement of essential boundary conditions

Due to lack of Kronecker delta property in EFG, shape function ΦI poses some difficulty in the imposition of essential boundary condition. To remove this problem, different numerical techniques have been proposed to enforce the essential boundary condition in EFG method such as Lagrange multiplier technique, modified variational principle approach, penalty approach, etc. In this work, penalty method [25] has been used for the enforcement of essential boundary condition.

Result and discussion

In order to obtain some physical insight into the problem, numerical calculations is carried out for various values of the parameters that describe the flow characteristics using element free Galerkin method and the results are reported in terms of graphs. A selected set of results has been obtained covering the ranges 0.01  Da  5.0, 0.0  Fr  10, 0.0  ϕ  1.0, 0.5  Up  5.0, 0.0  K  5.0, 0.0  M  2.0, 0.0  s  1.0, Ec = 0.01, Gr = 2.0, n = 0.1 and Pr = 0.71. For computational purposes, the spatial domain under

Conclusions

This study presents a numerical solution of unsteady magneto-hydrodynamics boundary layer flow and heat transfer of an incompressible, electrically conducting micropolar fluid over an infinite vertical permeable moving plate in the presence of a transverse magnetic field, heat absorption and viscous dissipation. The main finding can be summarized as:

  • (i)

    The velocity increases with the increase of plate velocity or Darcy number but it decreases as each of magnetic parameter, Forchhimer number and

Acknowledgment

One of the authors (Rajesh Sharma) would like to thank Ministry of Human Resource Development (MHRD), Government of India, for its financial support through the award of a research grant.

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