Oscillatory Effect on Magneto - Hydrodynamic Flow and Heat Transfer in a Rotating Horizontal Porous Channel

The rotating flow of electrically conducting viscous incompressible fluid in hydrodynamics has achieved considerable attention owing to its several applications in engineering and physics. In geophysics it is utilized to measure and study the positions and velocities with respect to a fixed co-ordinate system on the surface of earth rotating with regards to an inertial frame in the presence of its magnetic field. The general notion of fluid flow and mass transfer past a porous medium in a rotating environment plays a very significant role in the applications of petrochemical engineering, oceanography, aeronautics, meteorology and geophysics. The motivation for scientific investigation on rotating fluid system is fundamentally originated from geophysical and fluid engineering applications. The rotational fluid theory is applied to determine the viscosity of fluid, in the preparation of turbine and other numerous centrifugal machines. The study of the theory of rotating fluids due to its occurrence in different natural phenomena and its applications in several technological situations is an urgent need which is directly governed by coriolis force. The wide- spread subject matters of atmospheric science; limnology and oceanography contain some features of great importance in rotating fluids. Numerous researchers like Siegman[1], Ganapathy [ 2 ] , Guria [3], Hayat and Hutter[4] and Mazumder[5] have studied on hydrodynamic flow of viscous incompressible electrically conducting fluid in a rotating medium. The investigation of magneto- hydrodynamic fluid flow of a viscous incompressible electrically conducting rotating fluid has been studied by several authors viz. Hayat and Abelman [6], Ghosh and Pop [7] under different conditions and configurations to describe various aspects of the problem concerned. Reddy et.al [8] have analyzed heat transfer in hydro magnetic rotating viscous fluid flow through non- homogeneous porous medium with constant heat source/ sink. Raju et. al. [9] have analyzed an unsteady MHD radiative chemically reactive and rotating fluid past an impulsively started vertical plate with variable temperature and mass diffusion. Philip et.al. [10] Studied MHD rotating heat and mass transfer free convective flow past an exponentially accelerated isothermal plate with fluctuating mass diffusion. Hari krishna et.al. [11] have thrown light upon Hall current effects on an unsteady MHD flow in a rotating parallel plates channel bounded by porous bed on the lower half Darcy Lapwood model. Reddy et. al. [12] studied thermal diffusion and rotating effects on MHD mixed Singh [13] and Seth et.al [14] explored oscillatory MHD couette flow of incompressible electrically conducting viscous fluid in a rotating system under various conditions. Guria et.al [15] considered oscillatory MHD couette flow of electrically conducting fluid between two parallel plates in a rotating system under influence of inclined magnetic field in which the upper plate is kept fixed at rest and the lower plate oscillates non torsionally. Das et.al. [16] have investigated unsteady hydro magnetic couette flow of incompressible viscous and electrically conducting fluid in the rotating system when the flow of fluid within the channel is deduced on account of impulsive movement of one of the plates of the channel. Singh et.al.[17] have studied unsteady hydro magnetic couette flow of viscous incompressible electrically conducting fluid in the rotating system between two parallel plates when one of the plates of the channel is kept uniformly accelerated. Further, Seth et.al. [18] discussed MHD couette flow problem when the lower plate of the channel moves with time dependent velocity and upper plate is held fixed. They investigated two special cases of huge interest. One for impulsive movement of plate and second for uniformly accelerated movement of the plate. In all these investigations, the walls of channels are supposed to be non- porous. The study of fluid flow through porous channels have numerous engineering and geophysical applications in the areas of chemical engineering for the sake of purification and filtration process , in petroleum technology to estimate the movement of natural gas through channel, In agriculture engineering to study the underground water resources, minerals and metallurgical industries, designing of cooling systems together with liquid metals, geothermal reservoirs, underground energy transport , MHD pumps and generators, Accelerators and flow meters etc. Keeping in view of these applications Singh and Sharma [19] studied influence of permeability of porous medium on three dimensional couette flow and heat transfer. Hayat et.al. [20], Singh [21] investigated MHD flow within a parallel channel with porous boundaries under different conditions in non-rotating system. Ahmadi and Manvi [22] studied unsteady MHD flow of conducting fluid through porous medium. Ram and Mishra[23] applied the equations of motion derived by above Ahmadi and Manvi for the purpose of investigation on the MHD flow through porous medium and obtained different results. M.M. Hamza et. al . [24] Studied unsteady heat transfer to MHD oscillatory flow through a porous medium under slip condition. K.D. Singh [25] studied the effect of slip condition on visco-elastic MHD oscillatory forced convection flow in a vertical channel with heat radiation. J.L. Flade et.al. [26] Investigated in influence of suction or injection on MHD oscillatory flow through the porous channel saturated with porous medium. K. M. Joseph [27] studied the chemical reacting fluid on unsteady MHD oscillatory slip flow in planer channel with varying The purpose of the present analysis is to study the effect of oscillation on the flow and heat transfer phenomena of a viscous incompressible electrically conducting fluid in a rotating horizontal porous channel in the presence of uniform transverse magnetic field when the field is fixed relative to the fluid or to the moving plate. The plates of channel are supposed to be porous and the flow within the channel is owing to the oscillatory motion of upper plate. This model has significant utilizations in novel MHD energy system, magneto biofluid and designing MHD devices requiring control of fluid flow.

Here, we consider an oscillatory flow of electrically conducting viscous- incompressible fluid confined between two separated parallel porous plates of infinite length at a distance ‗h‘ apart under the influence of a uniform transverse magnetic field B0 applied in the direction parallel to Zˈ - axis perpendicular the planes of the plates. We consider here two situations of magnetic fields, firstly magnetic field is fixed with respect to fluid and secondly the magnetic field is fixed relative to the moving plate. The heat transfer case of fluid is also studied. A constant injection velocity v0 is applied at the upper oscillating plate as well as lower stationary plate. The upper plate oscillates in its own plane with a velocity about a non-zero uniform mean velocity U0 . The plates of the channel as well as fluid are in the situation of a rotation of a rigid body with constant angular velocity Ω about zˈ- axis. Take the origin at the lower plate lying in xˈ- yˈ plane. The axis of xˈ is chosen in parallel direction of the motion of upper plate. The plates of the channel are infinite in the directions of xˈ and yˈ. The plates are electrically non- conductors therefore all physical quantities, except pressure, will be functions of zˈ and tˈ. For metallic liquids, the magnetic Reynolds number is small enough so the induced magnetic field may be ignored in comparison to applied magnetic field. Initially tˈ< 0, the fluid as well as plates are considered at rest. At the time when tˈ>0, the upper plate begins moving with a velocity proportional to Uˈ (tˈ) in the reference frame of co-ordinates rotating with fluid. The equations of motion, energy and continuity are given in vector form as

With the following results for a moving conductor without Hall current.

The symbols are defined as current density, electric field , the magnetic field and the velocity of fluid respectively. is the unit vector along zˈ direction, µe is the magnetic permeability, σ is the fluid electric conductivity, tˈ represents time. The mathematical model of the problem is depicted in figure 1 , where is the velocity vector is xˈ,yˈ, zˈ directions respectively. such that B0 is a constant. We make here an assumption that no polarization voltage comes into existence i.e. . Equation (6.2.7) is Gaussian law of conservation for electric charge i.e which yields Jyˈ = constant. If we consider the case of vertical component of velocity we ultimately obtain Jzˈ

= 0.

Hence equation (6.2.8) immediately gives and (6.2.2) are rewritten in component form as and The present problem takes the following initial and boundary condition. (The constant plate temperature) (oscillatory plate temperature) Here we notice that equation (6.2.10) holds good only when the magnetic field is fixed with regards to fluid. If the magnetic field is kept fixed relative to the moving plate, equation (6.2.10) is replaced following Raptis and Singh [28] by writing as Now equations (6.2.10) and (6.2.14) can be combined together to get a single equation which as follows: Where λ is defined as λ =0 provided B0 is fixed with respect to fluid. Again λ =1 if B0 is fixed with regards to moving plate. Here the terms used have the following meaning. Kˈ is the permeability of medium, the kinematic viscosity, ρ is the density of fluid, K the thermal conductivity, cp is the specific heat at constant pressure. From equation (6.2.3) on integration, it is clear that wˈzˈ = v0 (constant). approximations on equations (6.2.10) and (6.2.11). Thus, we obtain the relation in the form Substituting wˈzˈ = v0 and values of and from (6.2.16) into (6.2.11) and (6.2.15) we get To non- dimensionalise the equation (6.2.12), (6.2.17) and (6.2.18) we introduce the following non-dimensional quantities (Frequency parameter) M=B0h (Hartmann number) 0 = (Injection/Suction parameter) Ω = (Rotational parameter) Sp = (Permeability parameter) Pr = (Prandtl number parameter) And finally we get And The transformed corresponding boundary conditions are given by For constant plate temperature For oscillating plate temperature T= ϵcost at ζ = 1

By introducing a complex function F = u + iv, equations (6.2.21) and (6.2.20) are combined into a single equation in the form The boundary condition (6.2.22) for F can be given by We consider the solutions following Light hill [29] in the following form And Putting (6.3.3) and (6.3.4) in (6.3.1) and (6.2.19) respectively and comparing harmonic and non- harmonic terms we obtain The relevant corresponding boundary conditions and two different cases of temperature become

T0 = 0 , T1 = 0 for ζ = 1 constant plate temperature T0 = 0 , T1 = 1 for ζ = 1 oscillatory plate temperature for t>0 (6.3.10) The solution of equations (6.3.5) - (6.3.9) are obtained under the boundary conditions (6.3.10) as Where s1 = s2 = s3 = s4 = s5 = s6 =

A1 = 1 A2 = 1 A3 = 1

and Where Now our purpose is to consider the resultant velocities and shear stresses of the steady and unsteady motions of flows of fluid. For the sake of analysis, we write And where (u0,u1) and (v0, v1) are primary and secondary components of velocities of steady and unsteady flow respectively. The expressions for resultant velocities and phase differences for steady and unsteady flows are given by And respectively. Further, the shear stress at the stationary plate for the steady flow is given as The amplitude and phase difference are written as Again for the case of unsteady flow, the shear stress at the stationary plate is governed by

In this section we have discussed the effects of several parameters involved in the present problem graphically under two different cases (a) steady flow case and (b) unsteady flow case.

Figure (2) reveals the influence of rotational parameter Ω on the resultant velocity Ro curve 6 exhibits the effect without rotation (Ω=0) and without porous medium (ks =0). In this situation, the resultant velocity attains the lowest velocity value in both the cases λ =0 and λ =1. Again, we compare curve-6 (ks =100), with curve -1(ks =1), it is clear that the presence of porous matrix without rotation increases Ro. Further, we find that the resultant velocity Ro increases with the increase in the value of rotational parameter Ω. Again observing carefully, it is found that an increase in rotation of frame of reference, the difference between resultant velocities with respect to λ=0 and λ =1 diminishes and ultimately it coincides with the curve-4. Therefore it may be decided that with higher value of Ω the relative status of magnetic field with regards to moving plate and the fluid has no influence. The fig 3 exhibits the influence of permeability parameter (sp) and Hartmann number (M) on the resultant velocity Ro. It is noticed that the resultant velocity Ro remains unaffected by sp but an increase in M increases Ro. Here it is remarkable to notice that the magnetic field fixed with respect to the moving plate (λ =1) gives more to the resultant velocity than the magnetic field fixed relative to fluid (λ =0). The figure 4 depicts the effects of suction/injection parameter (λ 0) onresultant velocity Ro by curve-2. It is observed that a significant increase in R0 is received on account of suctions (λ <5) as shown in curve – 3 and decreases (λ 0 >5) in curve -2.Therefore we infer that the increasing values of λ 0 or decreasing values of λ 0 leads to decrease and increase the resultant value of Ro. It is worth mentioning that resultant velocity Ro for magnetic field fixed relative to the moving plate (λ =1) is always greater than its counter part that is the magnetic field fixed relative to the fluid (λ =0). Again another significant outcome is that in presence or absence of suction/injection, porous media has no valuable effect on the resultant velocity Ro for both λ = 0 and λ =1. Fig 5, exhibits the change of phase angle θ0 for different values of λ 0 and Ω. One most striking result is that the phase angle gets negative value for higher value of rotation parameter (Ω ≥ 25) for the layers (>0.4), a little away from the lower plate. Further, the (permeability parameter) is to decrease the phase angle in both the situation λ =0 and λ = 1. The figure 6 depicts the variation of phase angle θ0 for different values of parameters sp and M. It is noticed that the phase angle steadily decreases for different values permeability parameter λ0. Relative position of magnetic field with respect to fluid (λ =0) has greater phase angle in all cases than its counterpart (λ =0) which has opposite influence on resultant velocity. It is also noted that as the magnetic field enhances, the phase angle increases significantly.

Plots 6(a),6(b),6(c) illustrate the resultant velocity R1 in the case of unsteady fluid motion. On comparing with steady case of motion, it is observed that the influence of parameters sp, λ 0 ,Ω & M involved in the problem remain invariably the same as that of steady flow motion case. In figure 8 the influence of frequency of oscillation  is observed resultant velocity R1. It is seen that frequency parameter  has remarkable effect on magnetic field which is fixed relative to the fluid (λ =0). But in case of its counterpart (λ =1) there is no noticeable effect. Moreover, it is observed that the resultant velocity R1 decreases with respect to increasing value of frequency  for the cases λ =0 and λ =1. Fig. 9 and fig. 10 explain the variation of phase angle θ1 for unsteady flow motion for different values of different parameters like sp,Ω, λ 0 and M involved in the problem. It is seen that the effect of these parameters on the phase angled remains unchanged in case of steady as well as unsteady motions except the magnitude which is approximately twice than the steady case. The figure - 11 depicts the changing of phase angle θ1 for various values of frequency parameter . It is observed that the phase angle θ1 assumes higher values of magnetic field which is fixed relative to the fluid for all values of parameters sp, λ0,  ,Ω and M. It is also significant to note that permeability of medium is responsible for decreasing phase angle whether oscillation is present (≠0) or absent (=0). Fig.12 concerns with temperature distribution for different values of parameters like Prandtl number Pr, frequency of oscillation, and suction parameter λ0 when the plates are kept at constant temperature. It is observed that higher Prandtl number Pr in the presence of injection give contribution to increase the thickness of thermal boundary layer. On the other hand the suction decreases the thermal boundary layer. Further, it is

over thermal boundary layer.

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P.G. Department of Mathematics, Gaya College, Gaya