Numerical study of heat transfer and Hall current effects on the flow of Johnson-Segalman fluid between two eccentric rotating disks

Johnson-Segalman fluid flow is examined in this article between two eccentrically rotating disks. Analysis of velocity and temperature profile is carried out in the presence of Hall current and non-coaxial rotation. The governing non linear momentum and energy equations of Johnson-Segalman fluid constitute a complicated system of equations corresponding to an intricate regime which are solved by using a step wise numerical algorithm (VIM). The graphical interpretations for velocity and temperature profiles have been made for various embedded parameters of interest.


Introduction
The study of non-Newtonian fluids between two parallel disks constitutes an important problem from physical and engineering points of views. Such flows have great applications in industry and modern technology e.g., turbines, turbomachines, spin-stabilized missiles etc. Orthogonal rheometer was introduced by Maxwell and Chartoff [1] to determine the viscosity of the viscoelastic fluids. The orthogonal rheometer had two parallel disks rotating with same angular velocity about two axes normal to the disk [1]. Mohanty [2] looked at flows between velocities −U and U respectively. The schematic diagram showing the 2D view of the geometry with applied magnetic field B 0 is depicted in figure 1.
The boundary conditions at z=−h and h are = -W + - The eccentric rotating disks suggest that the velocity field is the sum of translational and rotational velocities [8][9][10] = -W + The governing equations for the MHD convective flow of an incompressible J-S fluid are , where T is the Cauchy stress tensor for a J-S fluid and is given as with d/dt is the material time derivative, D is the symmetric and W is the skew symmetric part of the velocity gradient, μ is the viscosity, λ s is the relaxation time and a is the slip parameter. The J-S fluid model reduces to the upper convected Maxwell fluid model when μ=0 and a = 1, and reduces to the classical Naiver -Stokes fluid model when λ s =0.
To pursue the forgoing analysis we further require the Maxwell's equations which are given as where J is a current density vector, μ m is the magnetic permeability, E is the total electric field and σ is the electrical conductivity respectively. Furthermore, the generalized Ohm's law in the presence of a Hall current is where e is the electron charge, B 0 is the applied magnetic field, ω e is the cyclotron frequency of electrons, τ e is the electron collision time, n e is the number density of the electron and p e is the electron pressure. In problem under consideration ion-slip and electric field effects are neglected since in the presence of Hall effect magnetic field is strong as compared to the electric field and thus we can neglect the ion slip and electric field effects. Also ω e τ e ≈O(1) and ω i τ i =1 (ω i and τ i are the cyclotron frequency and collision time for ions respectively). Utilizing the equations (6) and (7) into momentum equation (4) and energy equation (5) and after simplification we obtain three equations interms of normal and shear stresses as a a a f g  Integrating equations (10) and (11) we obtain Equations (10) and (11) now take the form with the relevant boundary conditions In the light of above information equations (12), (21) and (22) can be written as a a a f g with the boundary conditions To non-diemensionalize the problem we introduce the following quantities.  l l f f  In proceeding analysis we have suppressed the asteriks for simplicity.

Variational Iteration method
VIM has proven to be a powerful tool for the solution of integro-differential equations. Abbasbandy and Shivanian [46] proposed VIM to solve system of nonlinear integro-differential equations. The results revealed that this method is very effective and promising in comparison with other numerical techniques. In applying VIM [43,47], we construct: (a) A correction functional which is an iterative expresssion that is based on linear and non-linear problems together with Lagrange multiplier which is evaluated according to the differential operator and variational theory, (b) After determining the Lagrange multiplier we then solve the integrodifferential equation(s) iteratively by choosing an initial guess which satisfy the initial or boundary conditions. We will solve the system of equations (30) to ( We will now implement the basic procedure of VIM by introducing the following iterative formulae for equations (34) to (36) known as correction functional as ò l l  It is necessary to determine the Lagrange multipliers V f , V g and V θ through restricted variation by using method of integration by parts. After determining the Lagrange multipliers, an iterative formula, without restricted variation will be used to calculate the next approximations f n , g n and θ n ; n0 for the solutions f (z), g(z) and θ (z). To calculate the Lagrange multipliers, we will take the variation of both sides of equations (37) to (39) which result in The variational condition of f n+1 , g n+1 and θ n+1 requires that δ f n+1 =0, δ g n+1 =0 and δ θ n+1 =0, which implies that the equations (40) to (42) are zero and this will provide us the stationary conditions. We therefore obtain the Lagrange multipliers by solving the corresponding stationary conditions for each of f n+1 , g n+1 and θ n+1 . Equations

Numerical results and discussion
Graphical illustrations corresponding to an approximate solution of the J-S fluid model in the prescribed domain are presented in this section. In non-dimensionalized model, the two disks are at z=−1 and z=1 respectively. Later for the smooth analytic and numerical computation we have calculated the numerical results in the domain   z 1 1 . In figures 2(a)-(b) we are presenting the variation in the velocity components relative to variation in the Hall current. The combination w e τ e is used to characterize an experimental situation that whether it is in the weak- or it is a strong-field limit t  ( ) w 1 .
e e We will confine our attention to the weak-field limit i.e f<1 and will consider the Hall parameter f<1. figure 2(a) shows the axi-symmetric behavior of the horizontal component of velocity and 2(b) depicts that the variation in the vertical component of velocity is negligible for a small change in Hall parameter.
The variation in the horizontal and vertical components of velocity relative to change in rotation parameter are presented in figures 3(a)-(b). The rotation parameter is considered to be  R 1, which implies that the rotation Ω is dominating. We can see the axi-symmetric behavior of the velocity which shows that the magnitude of horizontal velocity decreases by increasing rotation whereas magnitude of vertical velocity increases by increasing rotation.    To note the effect of heat transfer from a moving wall, figures 5(a)-(b) are plotted. Figure 5(a) shows the change in temperature relative to change in the Brinkman number Br which is product of Pr and Ec. The impact of Hall current on the heat transfer is presented in figure 5(b) respectively. Brinkman number causes to increase the thermal permeability so the amount of temperature increases by the increase in Brinkman number. Hall parameter causes to reduce the thermal permeability so by increasing Hall parameter temperature profile decreases i.e to reduce the temperature one can incorporate the effect of Hall current.

Concluding remarks
We made an attempt to explore the rheological features of J-S fluid between two disks rotating about noncoaxial axes. Using a numerical tool VIM, we have presented an approximate solution to the problem under consideration by calculating the velocity and temperature profiles in series of flow parameters, noteworthy of which are the magnetic Reynolds number, the rotation parameter and the Hall effect parameter. The combined effects of magnetic field and Hall current on the flow characteristics are examined in a way that the adequate transition will take place in the rate of rotation, rate of heat transfer and velocity gradient of the main body of the fluid. In conclusion, the results showed that VIM is remarkably efficient and in addition this method has been adopted since it has been proved in literature ( [45,46] and the references therein) that it has more accuracy for the kind of integro-differential problems.