This paper describes the studied effects of thermal radiation and chemical reaction on unsteady MHD non-Newtonian (obeying Walter’s B model) fluid in porous medium. The resulting problems are solved numerically. Graphical results for various interesting parameters are presented. Also the effects of the different parameters on the skin-friction and the heat fluxes are obtained and discussed numerically.
1. Introduction
Thermal convection in porous medium has attracted considerable interest during the last few decades, because it has various applications in geophysics, soil sciences, ground water hydrology, astrophysics, food processing, oceanography, limnology, engineering, and so forth. There are many elasticoviscous fluids that cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s constitutive relations. One such class of fluids is Walter’s B model elasticoviscous fluid having relevance in chemical technology and industry. However, the interest and research activities regarding the boundary layer flow of non-Newtonian fluids have increased considerably in the past few decades and it is one of the thrust areas of current research [1–7]. Of course, this is because of the industrial and engineering applications of non-Newtonian fluids as well as their interesting mathematical challenges in the form of highly nonlinear equations governing the flows.
The effect of magnetic field on thermal instability of Walter’s B model elasticoviscous fluid finds importance in geophysics, particularly, in the study of Earth’s core where the Earth’s mantle, which consists of conducting fluid, behaves like a porous medium which can become convectively unstable as a result of differential diffusion.
Kumar and Srivastava [8] have worked on the effects of chemical reaction on MHD flow of dusty viscoelastic (Walter’s liquid model B) liquid with heat source/sink. Khan et al. [9] have studied the hydromagnetic rotating flows of an Oldroyd-B fluid in a porous medium. Khan et al. [10] have investigated the viscoelastic MHD flow and heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work. The numerical or approximate solutions for both steady and transient flows of Walters’ B fluid have been studied at great length in a diverse range of geometries using a wide spectrum of computational or analytical techniques [11–13]. Effects of thermal diffusion and chemical reaction on MHD flow of dusty viscoelastic (Walter’s liquid model B) fluid have been inspected by Prakash et al. [14]. Abdul Hakeem et al. [15] have found the effect of heat radiation in Walter’s liquid B fluid over a stretching sheet with nonuniform heat source/sink and elastic deformation. Madhurai and Kalpana [16] have discussed thermal effect on unsteady flow of a dusty viscoelastic fluid between two parallel plates under different pressure gradients. Recently, unsteady free convection flow in Walters’ B fluid and heat transfer analysis have been presented by Khan et al. [17].
In the present problem, the effects of design viscoelastic parameter, unsteadiness parameter, magnetic field parameter, porosity parameter, thermal radiation parameter, heating source parameter, and Prandtl number parameter on account of fluid flow, heat transfer, the skin-friction, and the heat fluxes are obtained and discussed numerically.
2. Mathematical Model
Consider unsteady MHD non-Newtonian (obeying Walter’s B model) fluid in porous medium in the presence of thermal radiation and chemical reaction over a vertical rigid plate at y=0. The fluid is assumed to be falling down vertically over a vertical plane. The governing equations are given by the following.
The momentum equations:(1)∂u∂t=μρ∂2u∂y2-k1ρ∂3u∂y2∂t+gβ1T-T∞-σB2ρu-μρκu,the energy equation:(2)∂T∂t=k2ρCp∂2T∂y2-1ρCp∂qr∂y+Q1ρCpT-T∞.The initial conditions are given by(3)uy,t=0,Ty,t=Tw,for t<0.The boundary conditions fort≥0 are(4)u0,t=0,u∞,t=0,uy∞,t=0,T0,t=Tw,T∞,t=T∞.In the above equations uis the velocity fluid phase, t is the time, ρ is the densities of the fluid, g is the acceleration due to gravity, σ is the electrical conductivity, κ=κ01-at4 is the permeability of the porous medium, a magnetic field B=B01-at-1/2 normal to the stretching sheet is applied, T is the fluid temperature, Cp is the specific heat of the fluid at constant pressure, μ=μ01-at3 is the viscosity of the fluid, k1=k01-at4 is Walters’ viscoelasticity parameter, k2=k0∗1-at3 is the thermal conductivity of the fluid, β1=β0b1-at-1 is the expansion coefficient of temperature, Q1=Q0b1-at-1 is the internal heating parameter, ν0=μ0/ρ is the kinematic viscosity of the fluid, Q0 is the volumetric heat generation/absorption rate, Tw=T∞+bT01-at-1 is the wall temperature, a, b, k0, k0∗, and β0 are positive constants, and T∞ is the uniform temperature. Using the Rosseland approximation (Sparrow and Cess [18] and Abdel-Rahman Rashed [19]), the radiative heat flux qr could be expressed by (5)qr=-4σ∗3k∗∂T4∂y,where σ∗ represents the Stefan-Boltzmann constant and k∗ is the Rosseland mean absorption coefficient.
Assume that the temperature difference within the flow is sufficiently small such that T4 could be approached as the linear function of temperature: (6)T4≅4T∞3T-3T∞4.The similarity variables and parameters are as(7)uy,t=b1-at-1f′η,η=b21-at-2y,Ty,t=T∞+bT01-at-1θη.Substituting (5)–(7) in (1)–(4), we obtain (8)f′′′-Γ5f′′′+2ηf1v-Af′+2ηf′′+δθ-M+Sf′=0,1+Rθ′′-Aprθ+2ηθ′+Qθ=0with the boundary conditions (9)f0=0,f′0=0,θ0=1,f′∞=0,f′′∞=0,θ∞=0,where A=a/ν0b4 is the unsteadiness parameter, Γ=k0a/ρν0 is the viscoelastic parameter, δ=gβ0T0/ν0b3 is the temperature buoyancy parameter, M=σB02/μ0b4 is the magnetic field parameter, S=1/κ0b4 is the porosity parameter, pr=μ0Cp/k0∗ is the Prandtl number, R=16σ∗T∞3Tw-T∞3/3k∗k0∗b3T03 is the radiation parameter, and Q=Q0/k0∗b4 is the heating source parameter.
3. Skin-Friction Coefficient and Nusselt Number
A quantity of interest for the present problem is the local skin-friction coefficients Cf and the local Nusselt number Nu, which are defined as (10)Cf=τwy=0μ0b3,Nu=qwy=0k0∗T0b3,where τw and qw are the surface of Walter’s B fluid, which are given by(11)τw=μ∂u∂y-k1∂2u∂y∂ty=0,qw=-k2∂T∂yy=0.
Using (10) and (11), we get(12)Cf=1-3Γf′′0,Nu=-θ′0.
4. Results and Discussion
The system of coupled nonlinear ordinary differential equations (8) subject to boundary conditions (9) is solved numerically by using the shooting technique with forth order of Runge-Kutta algorithm. In order to gain physical insight the velocity and temperature profiles have been discussed.
The velocity f′(η) and the temperature θ(η) profiles increase with the increase of viscoelastic parameter Γ, shown in Figures 1(a) and 1(b).
Effect of Γ on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Figures 2, 3, 4, and 6 show the effect of unsteadiness parameter, magnetic field parameter, porosity parameter, and Prandtl number on the velocity f′(η) and the temperature θ(η) profiles. We found that A, M, S, and pr increase with decrease of the velocity and the temperature profiles.
Effect of A on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Effect of M on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Effect of S on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Figures 5 and 7 confirm that the velocity profile decreases and the temperature profile increases when thermal radiation parameter and heating parameter increase.
Effect of R on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Effect of pr on (a) velocity profile f′(η) and (b) temperature profile θ(η).
Effect of Q on (a) velocity profile f′(η) and (b) temperature profile θ(η).
The numerical values of the local skin-friction and the local Nusselt number are given in Table 1. For an increase in Γ, we observe that the local skin-friction coefficient decreases and the local Nusselt number increases, while, with the increase in each of A, M, S, and pr, we observe that the local skin-friction coefficient increases. But the local Nusselt number decreases.
Numerical of the values skin-friction coefficient (Cf) and Nusselt number (Nu) with Γ, A, M, S, R, pr, and Q.
Γ
A
M
S
R
pr
Q
Cf
Nu
0.1
0.3
4.0
1.0
1.0
0.71
0.0
1.32385
−0.764965
0.2
1.09487
−0.623587
0.3
0.95622
−0.544546
0.2
0.1
4.0
1.0
1.0
0.71
0.0
0.99741
−0.466889
0.3
1.09487
−0.623587
0.5
1.18435
−0.759761
0.2
0.3
2.0
1.0
1.0
0.71
0.0
0.91739
−0.588509
4.0
1.09487
−0.623587
6.0
1.24735
−0.661887
0.2
0.3
4.0
0.1
1.0
0.71
0.0
1.01884
−0.607028
1.0
1.09487
−0.623587
2.0
1.17359
−0.642623
0.2
0.3
4.0
1.0
0.2
0.71
0.0
1.10639
−0.658349
1.0
1.09487
−0.623587
10.0
1.08057
−0.580003
0.2
0.3
4.0
1.0
1.0
0.71
0.0
1.09487
−0.623587
7.0
1.23924
−1.04038
10.0
1.30247
−1.21198
0.2
0.3
4.0
1.0
1.0
0.71
−0.5
1.13492
−0.743195
0.0
1.09487
−0.623587
0.5
1.05331
−0.495614
With an increase in each of R and Q, we observe that the local skin-friction coefficient decreases but the local Nusselt number increases.
5. Conclusions
We computationally investigate the effects of thermal radiation and chemical reaction on unsteady MHD non-Newtonian (obeying Walter’s B model) fluid in porous medium. The following is observed.
The velocity and the temperature profiles decrease with the increase of each of the magnetic field and porosity parameters.
With the increase of each of thermal radiation and heating source parameters, the velocity profile decreases while the temperature profile increases.
For an increase of each of the magnetic field and porosity parameters, the local skin-friction coefficient increases while the local Nusselt number decreases.
With the increase of each of thermal radiation and heating source parameters, the local skin-friction coefficient decreases but the local Nusselt number increases.
Competing Interests
The authors declare that they have no competing interests.
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