Exact solutions of unsteady MHD free convection in a heat absorbing fluid flow past a flat plate with ramped wall temperature
Abstract
Unsteady MHD free convection and mass transfer from a viscous, incompressible, electrically conducting and heat absorbing fluid flow past a vertical infinite flat plate is investigated. The flow is induced by a general timedependent movement of the vertical plate, and the cases of ramped temperature and isothermal plates are studied. Exact solutions of the governing equations are obtained. The Sherwood number, Nusselt number and skin friction coefficients are obtained for both ramped temperature and isothermal plates. Some applications of practical interest are discussed for different types of plate motions. The numerical values of species concentration, fluid temperature and fluid velocity are displayed graphically whereas the numerical values of Sherwood number, the Nusselt number and skin friction are presented in tabular form, for different parameter values for both ramped and isothermal plates.
Keywords
Nusselt Number Skin Friction Sherwood Number Vertical Plate Heat Absorption1 Introduction
The investigation of the effects of a magnetic field on the flow of a viscous, incompressible and electrically conducting fluid is important in many practical applications, such as in MHD power generators and boundary layer flow control. Due to this fact, a large number of researchers have contributed to the literature on the flow of fluids in the presence of a magnetic field. Hayat et al. [1] investigated the flow of a thirdgrade fluid on an oscillating porous plate in the presence of a transverse magnetic field. They obtained an analytic solution of the governing nonlinear boundary layer equations. Hayat et al. [2] also obtained the exact solution of an oscillatory boundary layer flow bounded by two horizontal flat plates, one of which was oscillating in its own plate and the other was at rest. Seth et al. [3] obtained the exact solution for the effects of Hall current on the rotating Hartmann flow in the presence of an inclined magnetic field.
Magnetohydrodynamic free convection finds applications in fluid engineering problems such as MHD pumps, accelerators and flow meters, plasma studies, nuclear reactors, geothermal energy extraction, etc. Free convective flow past a vertical plate in the presence of a transverse magnetic field has been studied by several researchers. Kim [4] studied the magnetohydrodynamic convective heat transfer past a semiinfinite vertical porous moving plate with variable suction. The combined effects of thermal and mass diffusion on the unsteady free convection flow of a viscous incompressible fluid over an infinite vertical porous plate was investigated by Takhar et al. [5]. Ahmed et al. [6] considered the effects of thermal diffusion on a threedimensional MHD free convection flow of a viscous incompressible fluid over a vertical plate embedded in a porous medium.
The combined effects of convective heat and mass transfer on the flow of a viscous, incompressible and electrically conducting fluid has many engineering and geophysical applications such as in geothermal reservoirs, drying of porous solids, thermal insulation, enhanced oil recovery, cooling of nuclear reactor and underground energy transports. The hydromagnetic free convection flow with mass transfer effect has been studied extensively by many researchers. Hossain and Mandal [7] discussed the mass transfer effects on the unsteady hydromagnetic free convection flow past an accelerated vertical porous plate. Jha [8] investigated the hydromagnetic free convection flow through a porous medium with mass transfer. Elbashbeshy [9] studied the heat and mass transfer along a vertical plate with variable surface tension and concentration in the presence of a magnetic field. Chamkha and Khaled [10] investigated the hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid saturated porous medium. Chen [11] studied the combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous dissipation. Afify [12] discussed the MHD free convective heat and mass transfer flow over a stretching sheet in the presence of suction/injection with thermal diffusion and diffusion thermo effects. Eldabe et al. [13] studied the unsteady motion of an MHD viscous incompressible fluid with heat and mass transfer through porous medium near a moving vertical plate.
Heat absorption/generation effects have significant impact on the heat and mass transfer flow of a viscous, incompressible and electrically conducting fluid. Chamkha and Khaled [14] investigated heat generation/absorption effects on hydromagnetic combined heat and mass transfer flow from an inclined plate. The effects of a heat source/sink on unsteady MHD convection through porous medium with combined heat and mass transfer was studied by Kamel [15]. Chamkha [16] solved the problem of unsteady MHD convective heat and mass transfer past a semiinfinite vertical permeable moving plate with heat absorption. Makinde [17] discussed the hydromagnetic boundary layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux.
In most of the above investigations, the solutions were obtained by assuming the velocity and temperature at the interface to be continuous and well defined. There are, however, several problems of physical interest that may require nonuniform or arbitrary wall conditions. Several researchers (e.g., [18, 19, 20, 21]) have investigated problems of free convection from a vertical plate with stepdiscontinuities in the surface temperature. Chandran et al. [22] studied the unsteady natural convection flow in a viscous incompressible fluid near a vertical plate with ramped wall temperature. The MHD natural convection flow past an impulsively moving vertical plate with ramped wall temperature in the presence of thermal diffusion with heat absorption was studied by Seth and Ansari [23]. Recently, Seth et al. [24] studied the unsteady natural convection flow of a viscous incompressible electrically conducting fluid past an impulsively moving vertical plate in a porous medium with ramped wall temperature taking into account the effects of thermal radiation. They compared the results of natural convection near a ramped temperature plate with those of natural convection near an isothermal plate.
The aim of the present paper is to study the hydromagnetic free convective heat and mass transfer flow of a viscous, incompressible, electrically conducting and heat absorbing fluid past a vertical infinite flat plate. The fluid flow is induced by a general timedependent movement of the infinite plate. The governing equations are solved analytically, and a general solution valid for any timedependent movement of the plate is obtained. Some particular cases that highlight the applications of the general solution are discussed.
2 Formulation of the problem
The system of differential equations (5)(7) together with the initial and boundary conditions (9a)(9e) describes our model for the MHD free convective heat and mass transfer flow of a viscous, incompressible, electrically conducting and heat absorbing fluid past a vertical flat plate with ramped wall temperature.
3 Solution of the problem
Here, $erfc(x)$, ${\mathcal{L}}^{1}$ and $H(t1)$ are respectively the complimentary error function, the inverse Laplace transform operator and the Heaviside unit step function.
Equations (10)(14) represent the analytical solutions for the flow induced by a general timedependent movement of the vertical flat plate. To gain some practical understanding of the flow dynamics, some particular cases of timedependent movements of the plate are discussed below.
3.1 Plate movement with uniform velocity
The solutions (11), (13), (17), (19), (20), (21), (23) and (24), in the absence of heat absorption ($\varphi =0$) and mass transfer ($Gm=0$), are in agreement with the results obtained by Seth et al. [24] when ${K}_{1}=0$, $N=0$. The results are also consistent with those of Seth and Ansari [23] for a nonporous medium ${K}_{1}=0$ in the absence of mass transfer and with the results reported by Chandran et al. [22] in the absence of a magnetic field $M=0$, mass transfer and heat absorption.
3.2 Plate movement with single acceleration
3.3 Plate movement with periodic acceleration
4 Results and discussion
Skin friction when $\mathit{f}\mathbf{(}\mathit{t}\mathbf{)}\mathbf{=}\mathit{H}\mathbf{(}\mathit{t}\mathbf{)}$ for $\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.6}$
t  ϕ  Pr  M  Gr  Gm  ${\mathit{\tau}}_{\mathbf{1}}$  ${\mathit{\tau}}_{\mathbf{1}\mathit{i}}$ 

0.3  1  0.71  3  2  2  −0.80740428  −0.06468570 
0.5  1  0.71  3  2  2  −0.49752200  0.17704678 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  3  0.71  3  2  2  −0.29496426  −0.03525641 
0.7  5  0.71  3  2  2  −0.31937448  −0.09571177 
0.7  1  0.50  3  2  2  −0.26830104  0.26126499 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  1  7.00  3  2  2  −0.43648929  −0.25489651 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  1  0.71  5  2  2  −1.01876598  −0.61694534 
0.7  1  0.71  7  2  2  −1.58008231  −1.25625623 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  1  0.71  3  4  2  0.06194951  1.14312581 
0.7  1  0.71  3  6  2  0.41121836  2.03298282 
0.7  1  0.71  3  2  2  −0.28731934  0.25326881 
0.7  1  0.71  3  2  4  0.82070361  1.36129176 
0.7  1  0.71  3  2  6  1.92872655  2.46931471 
Skin friction when $\mathit{f}\mathbf{(}\mathit{t}\mathbf{)}\mathbf{=}\mathit{t}\mathit{H}\mathbf{(}\mathit{t}\mathbf{)}$ for $\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.6}$
t  ϕ  Pr  M  Gr  Gm  ${\mathit{\tau}}_{\mathbf{2}}$  ${\mathit{\tau}}_{\mathbf{2}\mathit{i}}$ 

0.3  1  0.71  3  2  2  0.24349996  0.98621853 
0.5  1  0.71  3  2  2  0.12077153  0.79534031 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  3  0.71  3  2  2  −0.04858666  0.21112119 
0.7  5  0.71  3  2  2  −0.07299688  0.15066583 
0.7  1  0.50  3  2  2  −0.02192344  0.50764260 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  1  7.00  3  2  2  −0.19011168  −0.00851890 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  1  0.71  5  2  2  −0.56908866  −0.16726802 
0.7  1  0.71  7  2  2  −0.97488806  −0.65106197 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  1  0.71  3  4  2  0.30832712  1.38950342 
0.7  1  0.71  3  6  2  0.65759597  2.27936043 
0.7  1  0.71  3  2  2  −0.04094174  0.49964642 
0.7  1  0.71  3  2  4  1.06708121  1.60766936 
0.7  1  0.71  3  2  6  2.17510416  2.71569231 
Figure 2 shows that species concentration $C(\eta ,t)$ increases with t and decreases with an increase in Sc. Since the Schmidt number Sc is the ratio of viscosity to mass diffusivity, an increase in Sc implies a decrease in the mass diffusion rate. Thus it follows that species concentration increases with an increase in time or mass diffusion rate.
Skin friction when $\mathit{f}\mathbf{(}\mathit{t}\mathbf{)}\mathbf{=}\mathbf{cos}\mathit{\omega}\mathit{t}\mathit{H}\mathbf{(}\mathit{t}\mathbf{)}$ for $\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.6}$
t  ϕ  Pr  M  Gr  Gm  ${\mathit{\tau}}_{\mathbf{3}}$  ${\mathit{\tau}}_{\mathbf{3}\mathit{i}}$ 

0.3  1  0.71  3  2  2  −0.51381907  0.22889951 
0.5  1  0.71  3  2  2  0.23463588  0.90920466 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  3  0.71  3  2  2  0.97245968  1.23216753 
0.7  5  0.71  3  2  2  0.94804946  1.17171216 
0.7  1  0.50  3  2  2  0.99912289  1.52868893 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  1  7.00  3  2  2  0.83093465  1.01252743 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  1  0.71  5  2  2  0.49014643  0.89196706 
0.7  1  0.71  7  2  2  0.11796087  0.44178695 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  1  0.71  3  4  2  1.32937345  2.41054975 
0.7  1  0.71  3  6  2  1.67864230  3.30040676 
0.7  1  0.71  3  2  2  0.98010460  1.52069275 
0.7  1  0.71  3  2  4  2.08812754  2.62871570 
0.7  1  0.71  3  2  6  3.19615049  3.73673864 
The Nusselt number and Sherwood number
t  ϕ  Pr  Sc  Nu  $\mathit{N}{\mathit{u}}_{\mathit{i}}$  Sh 

0.3  1  0.71  0.6  0.57134752  1.11605411  0.79788456 
0.5  1  0.71  0.6  0.77913255  0.98302070  0.61803872 
0.7  1  0.71  0.6  0.96929143  0.92531051  0.52233806 
0.7  1  0.71  0.6  0.96929143  0.92531051   
0.7  3  0.71  0.6  1.26243402  1.47003548   
0.7  5  0.71  0.6  1.50704023  1.88594507   
0.7  1  0.50  0.6  0.81341130  0.77650333   
0.7  1  0.71  0.6  0.96929143  0.92531051   
0.7  1  7.00  0.6  3.04350641  2.90540943   
0.7  1  0.71  0.40      0.42648724 
0.7  1  0.71  0.60      0.52233806 
0.7  1  0.71  0.78      0.59555702 
5 Conclusions

the mass diffusion rate and time tend to increase species concentration,

heat absorption reduces the fluid temperature, whereas thermal diffusion and time have the opposite effect,

heat absorption and the magnetic field tend to retard the fluid flow, whereas thermal diffusion, mass diffusion, thermal buoyancy force and mass buoyancy force have the opposite effect,

heat absorption and the magnetic field tend to increase the shear stress at the plate, whereas thermal diffusion, thermal buoyancy force and mass buoyancy force have the reverse effect,

heat absorption and time tend to increase the rate of heat transfer at the plate, whereas thermal diffusion has the reverse effect, and

mass diffusivity and time tend to reduce the rate of mass transfer at the plate.
Notes
Acknowledgements
The authors sincerely thank the reviewers for their valuable comments which helped to improve the quality of the paper. RN is grateful to the University of KwaZuluNatal for financial support.
Supplementary material
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