Heat Transfer to MHD Oscillatory Viscoelastic Flow in a Channel Filled with Porous Medium

Abstract

The combined effect of a transverse magnetic field and radiative heat transfer on unsteady flow of a conducting optically thin viscoelastic fluid through a channel filled with saturated porous medium and nonuniform walls temperature has been discussed. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. Closed-form analytical solutions are constructed for the problem. The effects of the radiation and the magnetic field parameters on velocity profile and shear stress for different values of the viscoelastic parameter with the combination of the other flow parameters are illustrated graphically, and physical aspects of the problem are discussed.

1. Introduction

The flow of an electrically conducting fluid has important applications in many branches of engineering science such as magnetohydrodynamics (MHD) generators, plasma studies, nuclear reactor, geothermal energy extraction, electromagnetic propulsion, and the boundary layer control in the field of aerodynamics. In the light of these applications, MHD flow in a channel has been studied by many authors; some of them are Nigam and Singh [1], Soundalgekar and Bhat [2], Vajravelu [3], and Attia and Kotb [4]. A survey of MHD studies in the technological fields can be found in Moreau [5]. The flow of fluids through porous media is an important topic because of the recovery of crude oil from the pores of the reservoir rocks; in this case, Darcy’s law represents the gross effect. Raptis et al. [6] have analysed the hydromagnetic free convection flow through a porous medium between two parallel plates. Aldoss et al. [7] have studied mixed convection flow from a vertical plate embedded in a porous medium in the presence of a magnetic field. Makinde and Mhone [8] have considered heat transfer to MHD oscillatory flow in a channel filled with porous medium.

In this study, an attempt has been made to extend the problem studied by Makinde and Mhone [8] to the case of viscoelastic fluid characterised by second-order fluid.

The constitutive equation for the incompressible second-order fluid is of the form 𝜎=−𝑝𝐼+𝜇₁𝐴₁+𝜇₂𝐴₂+𝜇₃𝐴₁², (1) where 𝜎is the stress tensor, 𝑝is the hydrostatic pressure, 𝐼is the unit tensor, 𝐴_𝑛(𝑛=1,2)are the kinematic Rivlin-Ericksen tensors, 𝜇₁,𝜇₂,and𝜇₃are the material coefficients describing viscosity, elasticity, and cross-viscosity, respectively. The material coefficients 𝜇₁,𝜇₂,and𝜇₃have taken constants with 𝜇₁and 𝜇₃as positive and 𝜇₂as negative (Markovitz and Coleman [9]). Equation (1) was derived by Coleman and Noll [10] from that of the simple fluids by assuming that stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past.

2. Mathematical Formulation of the Problem

Consider the flow of a conducting optically thin fluid in a channel filled with saturated porous medium under the influence of an externally applied homogeneous magnetic field and radiative heat transfer as shown in Figure 1. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. The 𝑥-axis is taken along the centre of the channel, and the 𝑦-axis is taken normal to it. Then, assuming a Boussinesq incompressible fluid model, the equations governing the motion are given 𝜕𝑢1𝜕𝑡=−𝜌𝜕𝑃𝜕𝑥+𝜐₁𝜕²𝑢𝜕𝑦²+𝜐₂𝜕³𝑢𝜕𝑦²−𝜐𝜕𝑡¹𝑢𝑘−𝜎_𝑒𝐵₀²𝑢𝜌+𝑔𝛽𝑇−𝑇₀, (2)𝜕𝑇=𝑘𝜕𝑡𝜌𝐶_𝑝𝜕²𝑇𝜕𝑦²−1𝜌𝐶_𝑝𝜕𝑞𝜕𝑦, (3) subject to boundary conditions 𝑢=0,𝑇=𝑇_𝑤on𝑦=1,𝑢=0,𝑇=𝑇₀on𝑦=0, (4) where 𝑢 is the axial velocity, 𝑡 is the time, 𝑇 is the fluid temperature, 𝑃 is the pressure, 𝑔 is the gravitational force, 𝑞 is the radiative heat flux, 𝛽 is the co-efficient of volume expansion due to temperature, 𝐶_𝑝is the specific heat at constant pressure, 𝑘is the thermal conductivity, 𝐾is the porous medium permeability co-efficient, 𝐵₀(=𝜇_𝑒𝐻₀) is the electromagnetic induction, 𝜇_𝑒is the magnetic permeability, 𝐻₀ is the intensity of the magnetic field, 𝜎_𝑒 is the conductivity of the fluid, 𝜌 is the fluid density, and 𝜐_𝑖=𝜇_𝑖/𝜌,(𝑖=1,2). It is assumed that both walls of temperature 𝑇₀,𝑇_𝑤 are high enough to induce radiative heat transfer. Following Cogley et al. [11], it is assumed that the fluid is optically thin with a relatively low density and the radiative heat flux is given by 𝜕𝑞𝜕𝑦=4𝛼²𝑇₀−𝑇, (5) where 𝛼is the mean radiation absorption co-efficient.

Figure 1 

Geometry of the problem.

The following nondimensional quantities are introduced: Re=𝑈𝑎𝜐₁,𝑥𝑥=𝑎,𝑦𝑦=𝑎,𝑢𝑢=𝑈,𝜃=𝑇−𝑇₀𝑇_𝑤−𝑇₀,𝐻²=𝑎²𝜎_𝑒𝐵₀²𝜌𝜐₁,𝑡=𝑡𝑈𝑎,𝑃=𝑎𝑃𝜌𝜐₁𝑈,𝐾Da=𝑎²𝑇,Gr=𝑔𝛽_𝑤−𝑇₀𝑎²𝜐₁𝑈,Pe=𝑈𝑎𝜌𝐶_𝑝𝑘,𝑁²=4𝛼²𝑎²𝑘, (6) where 𝑈 is the flow mean velocity.

The dimensionless governing equations together with the appropriate boundary conditions (neglecting the bars for clarity) can be written as Re𝜕𝑢𝜕𝑡=−𝜕𝑃+𝜕𝜕𝑥²𝑢𝜕𝑦²−𝑠²+𝐻²𝑢𝜕+Gr𝑇+𝛾³𝑢𝜕𝑦²,𝜕𝑡 (7)Pe𝜕𝜃=𝜕𝜕𝑡²𝜃𝜕𝑦²+𝑁²𝜃, (8) with 𝑢=0,𝜃=1on𝑦=1,𝑢=0,𝜃=0on𝑦=0, (9) where Gr,𝐻,𝑁,Pe,Re,Da,𝑆(=1/Da), and𝛾=(𝜐₂Re)/𝑎² are Grashoff number, Hartmann number, Radiation parameter, Péclet number, Reynolds number, Darcy number, porous medium shape factor parameter, and viscoelastic parameter, respectively.

3. Method of Solution

In order to solve (7) and (8) for purely oscillatory flow, let −𝜕𝑃𝜕𝑥=𝜆𝑒^𝑖𝜔𝑡,𝑢𝑦,𝑡=𝑢₀𝑦𝑒^𝑖𝜔𝑡,𝜃𝑦,𝑡=𝜃₀𝑦𝑒^𝑖𝜔𝑡, (10) where 𝜆is a constant and 𝜔 is the frequency of oscillation.

Substituting the above expressions into (7) and (8) and using (9), we get 𝑑1+𝑖𝛾𝜔²𝑢₀𝑑𝑦²−𝑚₂²𝑢₀=−𝜆−Gr𝜃₀,𝑑²𝜃₀𝑑𝑦²+𝑚₁²𝜃₀=0, (11)(12) subject to boundary conditions 𝑢₀=0,𝜃₀𝑢=1on𝑦=1,₀=0,𝜃₀=0on𝑦=0, (13) where 𝑚₁=√𝑁²−𝑖𝜔Pe and 𝑚₂=√𝑆²+𝐻²+𝑖𝜔Re.

Equations (11) and (12) are solved, and the solution, for the fluid velocity and temperature are given as follows: 𝑢𝑦,𝑡=𝑀₁𝑒^{(𝑚2𝑦)/𝐿}+𝑀₁−𝜆𝑚₂²𝑒^{−(𝑚2𝑦)/𝐿}+𝜆𝑚₂²+Grsin𝑚₁𝑦𝑚₁²𝐿+𝑚₂²sin𝑚₁𝑒^𝑖𝜔𝑡, (14)𝜃=𝑚𝑦,𝑡sin₁𝑦sin𝑚₁𝑒^𝑖𝜔𝑡, (15) where 𝑀₁=((𝜆/𝑚₂²)𝑒^{−𝑚2/𝐿}−(𝜆/𝑚₂²)−Gr)/(𝑚₁²𝐿+𝑚₂²))/(𝑒^𝑚2/𝐿−𝑒^{−𝑚2/𝐿}) and 𝐿=1+𝑖𝜔𝛾.

The nondimensional shear stress 𝜎at the wall 𝑦=0 is given by 𝜎=𝜎𝜇₁=𝑈/𝑎𝜕𝑢𝜕𝜕𝑦+𝛾²𝑢𝜕𝑦𝜕𝑡_𝑦=0. (16) The rate of heat transfer across the channel’s wall is given as 𝑁𝑢=−𝜕𝜃𝑚𝜕𝑦=−¹cos𝑚₁sin𝑚₁𝑒^𝑖𝜔𝑡. (17)

4. Discussions and Conclusion

The purpose of this study is to bring out the effects of the viscoelastic parameter 𝛾 on the governing flow with the combination of the other flow parameters. The corresponding results for Newtonian fluid can be deduced from the above results by setting 𝛾=0,and it is worth mentioning here that these results coincide with that of Makinde and Mhone [8]. We have considered the real parts of the results throughout for numerical validation. The velocity profile 𝑢 against 𝑦 is plotted in Figures 2–4 to observe the viscoelastic effects for various sets of values of Hartmann number 𝐻and radiation parameter 𝑁(𝐻=0.5,𝑁=1.5;𝐻=0.5,𝑁=2.5;𝐻=1.5,𝑁=2.5) with fixed values of other flow parameters, namely, Pe=2,Re=2,𝑠=1,𝑡=0,Gr=2,𝜆=1,and 𝜔=1. It is evident from Figures 2–4 that the velocity profile is parabolic in nature, and the values of velocity 𝑢 increase with the increasing values of the viscoelastic parameter |𝛾|(𝛾=0,−0.10,−0.20) in comparison with the Newtonian fluid. It is also noted from the figures that the behaviours of the velocity profiles remain the same with the increasing values of the viscoelastic parameter |𝛾|when (i) the values of the radiation parameter 𝑁increase with the fixed values of the magnetic field parameter 𝐻(Figures 2 and 3), (ii) the values of the magnetic field parameter 𝐻increase with fixed values of the radiation parameter 𝑁(Figures 3 and 4), and (iii) both the values of 𝐻and 𝑁increase (Figures 2 and 4).

Figure 2 

Variation of 𝑢 against 𝑦 when 𝐻 = 0.5, 𝑁 = 1.5.

Figure 3 

Variation of 𝑢 against 𝑦 when 𝐻 = 0.5, 𝑁 = 2.5.

Figure 4 

Variation of 𝑢 against 𝑦 when 𝐻 = 1.5, 𝑁 = 2.5.

Figures 5, 6, 7, and 8 exhibit the effects of the viscoelastic parameter |𝛾|on skin friction 𝜎 against magnetic field parameter 𝐻and radiation parameter 𝑁,respectively with Pe=2,Re=2,𝑠=1,𝑡=0,Gr=2,𝜆=1,and𝜔=1. Figures 5 and 6 show that for radiation parameter 𝑁=1.5and 𝑁=2.5, the shear stress decreases with the increasing values of 𝐻for both Newtonian and non-Newtonian fluids, while shear stress increases for increasing values of |𝛾|(𝛾=0,−0.10,−0.20) in comparison with Newtonian fluid. From Figures 7 and 8, it is evident that the shear stress 𝜎 increases firstly and then decreases with the increasing values of 𝑁for both Newtonian and non-Newtonian cases. Also, Figures 7 and 8 depict that the shear stress increases with the increasing values of the viscoelastic parameter |𝛾|in comparison with Newtonian fluid when the values of the magnetic field parameter 𝐻increase.

Figure 5 

Variation of 𝜎 against 𝐻 when 𝑁 = 1.5.

Figure 6 

Variation of 𝜎 against 𝐻 when 𝑁 = 2.5.

Figure 7 

Variation of 𝑠 against 𝑁 when 𝐻 = 1.5.

Figure 8 

Variation of 𝑠 against 𝑁 when 𝐻 = 2.5.

It has also been observed that the temperature field is not significantly affected by the viscoelastic parameter.