ANALYSIS OF MHD FLOW AND HEAT TRANSFER OF LAMINAR FLOW BETWEEN POROUS DISKS

A study is carried out for the two dimensional laminar flow of conducting fluid in presence of magnetic field. The governing non-linear equations of motion are transformed in to dimensionaless form. A solution is obtained by homotopy perturbation method and it is valid for moderately large Reynolds numbers for injection at the wall. Also an efficient algorithm based finite difference scheme is developed to solve the reduced coupled ordinary differential equations with necessary boundary conditions. The effects of Reynolds number, the magnetic parameter and the pradantle number on flow velocity and tempratare distribution is analysed by both the methods and results agree well with previous work for special cases. It is observed that overall effect of magnetic field is same as Hartmann flow. Further the analysis predicts that the heat transfer at the surface of the disks increases with increase in Reynolds number, magnetic parameter and Prandle number. The shear stress at the wall decreases with increase in injection, whearas increase with increase in magnetic parameter. The study of such phenomenon is beneficial in the industry for thermal control in polymeric processing.


INTRODUCTION
The flow of a conducting fluid between two porous disks is of practical importance in lubrication theory, such type of flows have lot of importance and applications in mechanical and manufacturing process, magnetic and storage devices (disk drives), gas engines, crystal growth process and bio-mechanics. MHD effects are used for power generators, light-ion-beam confinements and space crafts. Many authors contributed their efforts to understand these types of problems. The problem was first studied by Batchelor (Batchelor (1967)), who extended the study of Von Karman (Karman (2021)) for a flow over a single disk.Study about flow between rotating disks is done by many authors in 1960s (Lance and Rogers (1962), Root and Lewellen (1966)). The similar type of study with slight modification in the model in which one disk is rotating and one disk is stationary is done by Mellor (Mellor et al. (1968)). In 1979 Wang (Wang and Watson (1979)) and his associates further developed these models with suction or injection by considering the porous disks with rotation. The first MHD analysis about these type of models done by Srivastava and his associates in 1961 (Srivastava and Sharma (1961)). Later many authors contributed the better analysis in MHD in this type models (Stephenson (1969), Chandrasekhara and Rudriah (1971a), Chandrasekhara and Rudriah (1971b)). Further the problem was analysed by Stewartson (Stewartson (1953)) who found perturbation solution.
A recent study of this kind of problems are done by Si Xinhui (Xinhui et al. (2012)) in 2012, authors used homotopy analysis method for laminar flow and heat transfer of viscous fluid between contracting rotating † Corresponding author. Email: nppaimit@yahoo.co.in disks. In 2019 Naresh kumar and his associates (Raju et al. (2019)) studied the heat and mass transfer analysis of MHD couple stress fluid flow through contracting or expanding porous pipes. P Sibanda (Sibanda and Makinde (2012)) and his associates investigate the hydromagnetic steady flow and heat transfer characteristics of an incompressible viscous electrically conducting fluid past a rotating disk in a porous medium with ohmic heating, Hall current and viscous dissipation . Many authors studied this type of models under different conditions and these articles can be found in (Raju et al. (2019), Nazir andMahmood (2011), Osalusi et al. (2007), Gu et al. (2020), Abed et al. (2020), Tufail et al. (2020), Gadamsetty et al. (2020)).
Solving these type of models through existing mathematical tools are also really challenging for the researcher, due to non-linear nature of the equations obtaining analytical solution is not possible in most of the cases, hence the most of the researchers have used numerical methods to solve this type of problems (Bujurke et al. (1995), Sampath and Pai (2019b), Sampath and Pai (2019a)), but numerical methods having their own demerits and most of the time getting solution to this kind of problems is a tedious job to the researchers. However, this class of problems are efficiently solved through semi analytical methods. One such method is homotopy perturbation method (HPM). HPM first proposed by Ji-Huan He in 1998 ( He. (2006)). HPM is the combination of traditional perturbation method and homotopy in topology. Many authors used this method to solve different class of problems (Gupta et al. (2013), Abbas-bandy (2007), Ganji and Sadighi (2006), Rafei and Ganji (2006), Rafei et al. (2007), Siddiqui and Irum (2008) ).
By observing the above literature about this type of study, we got motivated to study the characteristic of MHD flow and heat transfer of laminar flow between porous disks through HPM. The main advantage is that it yields a very rapid convergence of the series solution, only with the few iterations. For simple domains the HPM has advantages over pure numerical results. A single computer program gives the solution for a large range of expansion quantity. The steady state solutions of the Navier-Stokes equations are usually obtained by numerical schemes like finite difference or finite element scheme. The numerical methods are difficult to implement due to non-linearity of the equations, so semi-analytical methods are useful in analysing such problems but in the present study both the methods are implemented efficiently in analysing the problem.

PROBLEM FORMULATION
The problem of the steady laminar flow of an in-compressible viscous fluid between parallel circular non-conducting disks in the presence of a uniform strong magnetic field is investigated for small Reynolds number. The flow under consideration is entirely due to either uniform suction or injection at the disks as shown bellow figure The induced magnetic field is assumed to be small and it can be calculated, assuming the velocity Ur known, from the Maxwell-Ampere's equation: The boundary condition on hr, if the disks are non-conducting, is The boundary conditions on Ur and Uz are the no-slip conditions: and Uz(r, ±h) = ∓U0 = constant (7) The equation for the temperature field, neglecting the viscous dissipation, can be written as The boundary conditions for the temperature field can be written as Using the transformation and Equations (1) and (2), using Eq. (9) and Eq. (10), becomes U0r 2h and Since the right hand side of Eq. (12) is a function of η only, it follows that: Hence, Eq.(11), using Eq. (13), becomes Equation (14) is true for all r, if The boundary conditions on f (η) and θ(η) are

METHOD OF SOLUTION
We adopt two methods to solve the problems considered. Method-I: Homotopy Perturbation Solution: To describe the HPM solution for the system of non-linear differential equations, we consider where D1 and D2 denotes the operator, f (η) and θ(η) are unknown functions, η denote the independent variable and f1, f2 are known functions. D1 and D2 can be written as where L1 and L2 are simple linear part, N 1 and N 2 are remaining part of the Eqs. (22) and (23) respectively. The proper selection of L1, L2, N 1, and N 2 form the governing equations one can get the homotopy equation for Eq. ( 22) and Eq. (23 ) as follows where v0(η) is the initial guess to the Eq. (22) and Eq. (23 ).
We assume the solution of Eq. (24) and Eq. (25) as follows The solution to the considered problems is Eq. (26) and Eq. (27) at q = 1. The slow convergence of the above series Eqs. (26) and (27) at q = 1 requires large number of terms for obtaining an almost exact solution. As we proceed for higher approximations, the computations becomes cumbersome and is difficult to calculate the terms manually. So Mathematica software is used to get higher order terms. By proper selection of linear and non-linear part and applying HPM to solve the consider equations one can get the zeroth and first order solution as follows +5P rR 2 η 9 + 175P r 2 R 2 η 9 Method-II: Finite Difference Solutions: The equations mentioned above Eqs. (18,19 ) subjected Eqs. (20 , 21 ) were solved numerically by FDM to confirm the results obtained by us. Using standard finite difference method, i.e stepping from ηj−1 to ηj, a Crank-Nicolson's scheme was used. These tridiagnal systems are easily solved to update the values on each grid point. Calculations were performed by dividing the interval into 10 4 sub intervals to find the associated parameters. These system of equations were solved using Mathematica.

RESULT AND DISCUSSION
The intention of the authors in this study is to analyse the flow and heat transfer characteristics associated with laminar flow between porous disks in the presence of a uniform magnetic field through semi-numerical technique. In this section we illustrate the numerical and semi-numerical findings in graphical and tabular forms with the interpretations and discussion. To develop better understanding of the conducting fluid between two parallel porous disks and thermal characteristics, we choose to present the shear stress , velocity and temperature across the disks for a range of the Reynolds number R, the magnetic parameter M and the Prandl number P r. The results presented using homotopy perturbation series method by considering 20 terms in the series and also compared the results obtained by HPM with the classical FDM, by dividing the interval in to 10000 sub-intervals. The algebra becomes cumbersome after certain steps we implement these two methods in mathematica software by writing elegant code.  Figures 8 to 14 represents the variation of the temperature for different values of the parameters R, M and P r. Figs. 8 to 10 shows the variation of θ(η) with R with fixed values of M and P r. It is observed from these figures that the temperature is increases in the upper half of the plane and decreases in lower region with increase in R. In Figs. 10 to 12, we fix the values of R , M and plotted θ(η) with the variation in P r. In Figs. 13 to 14, we fix R , P r and plotted temperature profiles with the variation in magnetic parameter for these two cases. Also we observe the similar situation with slight change in position due to increase in the conductivity in the fluid. Table 1 and Table 2 show the heat transfer rate (θ (−1)) and the shear stress(f (−1)) respectively. In this section we compared the results obtained by the HPM with the classical FDM technique and tabulated the values and the results are in good agreement. From Table 1 it is clear that the heat transfer rate decreases with increase in R. As we increase the magnetic parameter, heat transfer rate also increases, however as we increase the Prandtl number heat transfer rate decreases. From the Table 1 it is clear that the shear stress decreases as we increase R and increases with increasing the magnetic parameter.

CONCLUSIONS
The present study has revealed the effects of governing parameters on the flow in presence of magnetic field and thermal effects. The resulting equations with associate boundary conditions are solved by HPM and finite difference method.
The following conclusions are emerging out of the study: 1) The shear stress and heat transfer rate decreases by increasing in injection.
2) The heat transfer rate decreases with increase in P r.
3) The shear stress increases with increase in magnetic parameter. 4)The radial velocity increases near central plane with increase in Reynolds number, where as it decreases with increase in magnetic field and they are parabolic in nature.