MHD mixed convective stagnation point flow along a vertical stretching sheet with heat source/sink

doi:10.1016/j.ijheatmasstransfer.2017.10.026

International Journal of Heat and Mass Transfer

Volume 117, February 2018, Pages 780-786

https://doi.org/10.1016/j.ijheatmasstransfer.2017.10.026 Get rights and content

Highlights

•
Effect of mixed convection parameter ( $λ$ ) with heat source/sink parameter ( $δ$ ) is studied.
•
Nusselt number increases with increasing mixed convection parameter.
•
Skin friction coefficient increases with increasing mixed convection parameter.
•
Pr shows dual effects on velocity for assisting and opposing flows respectively.
•
$δ$ shows dual effects on velocity for assisting and opposing flows respectively.

Abstract

Aim of the paper is to investigate the effects of heat generation/absorption on MHD mixed convective stagnation point flow along a vertical stretching sheet in the presence of external magnetic field. The governing boundary layer equations are formulated and transformed into nonlinear ordinary coupled differential equations using similarity transformation and numerical solution is obtained by using Runge-Kutta fourth order scheme with shooting technique. The effects of various physical parameters such as velocity ratio parameter, mixed convection parameter, Hartmann number, Prandtl number and heat source/sink on velocity and temperature distributions are presented through graphs and discussed numerically. The skin friction coefficientand Nusselt number at the sheet are derived, discussed numerically and their numerical values are presented through tables.

Introduction

MHD flow and heat transfer of viscous fluids over a continuous stretching surface has great importance because of its many applications in engineering processes such as geothermal energy extraction, purification of metal from non-metal enclosures, plasma studies, aerodynamic extrusions of plastic sheets etc. Stagnation point flow is relevant to the bodies in high speed flow. It is used to reduce drag, in designing of thrust bearings and transpiration cooling etc. Stagnation point flow was first analyzed by Hiemanz [1]. Hiemanz used similarity transformation to reduce the governing Navier-Stokes equation into ordinary differential equations of third order subject to two point boundary conditions. Later many researches considered magnetic field, heat source/sink, suction/injection etc. to enhance the properties of fluids in stagnation point flow. Gupta and Gupta [2] studied heat and mass transfer on a stretching sheet with suction or blowing. Mixed convection in stagnation flows adjacent to vertical surfaces was analyzed by Ramchandran et al. [3]. Hassanien and Gorla [4] investigated combined forced and free convection in stagnation flow of micropolar fluids. Andersson [5] discussed MHD viscoelastic fluid flow past a stretching sheet. Chiam [6] presented heat transfer in a variable conductivity in a stagnation point flow towards a stretching sheet. Stagnation point flow of a viscoelastic fluid towards a stretching surface was studied by Mahapatra and Gupta [7]. Abel et al. [8] discussed buoyancy force and thermal radiation effects in MHD boundary layer flow over continuously moving stretching surface. Ishak et al. [9] analyzed mixed convection boundary layers in the stagnation point flow towards a stretching vertical sheet. Unsteady mixed convection flow of a micropolar fluid near the stagnation point was discussed by Lok et al. [10]. Layek et al. [11] presented heat and mass transfer analysis for boundary layer stagnation point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing. Ishak et al. [12] discussed dual solutions in mixed convection flow near a stagnation point on a vertical porous plate. An effect of variable thermal conductivity and heat source/sink on MHD flow near a stagnation point on a linearly stretching sheet was studied by Sharma and Singh [13]. Pal [14] analyzed heat and mass transfer in stagnation point flow towards a stretching surface in the presence of buoyancy force and thermal radiation. Alharbi et al. [15] investigated heat and mass transfer in MHD viscoelastic fluid flow through a porous medium over a stretching sheet with chemical reaction. Bhattacharyya and Layek [16] presented effects of suction/blowing on steady boundary layer stagnation point flow and heat transfer. Makinde et al. [17] discussed buoyancy effects on MHD stagnation point flow and heat transfer along heated stretching/shrinking sheet. Singh and Sharma [18] investigated dual solution for heat and mass transfer in the boundary layer flow along a vertical isothermal reactive plate near stagnation point. Mixed convection stagnation point flow on vertical stretching sheet with external magnetic field was studied by Ali et al. [19]. Shen et al. [20] presented MHD mixed convection slip flow near a stagnation point on a nonlinearly vertical stretching sheet.

Section snippets

Mathematical formulation

Consider a two-dimensional steady laminar flow of a viscous incompressible fluid along a vertical stretching sheet placed in $x$ direction and $y$ axis is normal to the sheet. $u$ and $v$ are the velocity components in $x$ and $y$ directions respectively. $u = u_{e} (x) = ax$ is the free stream velocity and the velocity by which the sheet is stretching is $u = u_{w} (x) = cx$ where both $a$ and $c$ are positive constants. An external magnetic field $H_{0}$ is applied normal to the sheet in the presence of heat source/sink. The

Method of solution

In order to get solution of Eqs. (1), (2), (3) with boundary conditions (4), we use the following transformations and dimensionless quantities $η = \sqrt{\frac{a}{υ}} y, ψ = x \sqrt{a υ} f (η), θ (η) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}} and u = \frac{\partial ψ}{\partial y}, v = - \frac{\partial ψ}{\partial x}$ into equations, so that equation of continuity is automatically satisfied. The equation of momentum and energy become $f^{‴} + {ff}^{″} - {(f^{'})}^{2} + 1 + {Ha}^{2} (1 - f^{'}) + λ θ = 0,$ $θ^{″} + \Pr (f θ^{'} - f^{'} θ + δ θ) = 0,$ where prime denotes the derivative with respect to $η$ , $Ha (= μ_{e} H_{0} \sqrt{\frac{σ}{ρ a}})$ is the Hartmann number, $λ (= \frac{{Gr}_{x}}{{Re}_{x}^{2}})$ is mixed convection parameter, ${Gr}_{x} (= g β (T_{w}$

Results and discussion

Fig. 1 shows the influence of Hartmann number on velocity profiles for both buoyancy added ( $λ > 0$ , assisting) and buoyancy opposed ( $λ < 0$ , opposing) flow. Fluid velocity decreases with increase in Hartmann number for $A > 1$ , because transverse magnetic field to an electrically conducting fluid gives rise to a resistive force called the Lorentz force, this force slow down the motion of the fluid in boundary layer, but in case when free stream velocity dominates stretching velocity of the surface there

Conclusions

The influence of heat source/sink on MHD mixed convective stagnation point flow along a vertical stretching sheet is investigated. Effects of various physical parameters on velocity profile, temperature profile, skin friction coefficient and Nusselt number are observed and concluded as follows:

1.
Fluid velocity increases with increase in mixed convection parameter $λ$ , velocity ratio parameter $A$ , Hartmann number $Ha$ , heat source/sink parameter $δ$ (for assisting flow) or Prandtl number $\Pr$ (for opposing

Conflicts of interest

None.

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Buoyancy force and thermal radiation effects in MHD boundary layer viscoelastic fluid flow over continuously moving stretching surface
Int. J. Therm. Sci.
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Effects of suction/blowing on steady boundary layer stagnation point flow and heat transfer towards a shrinking sheet with thermal radiation
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Buoyancy effects on MHD stagnation point flow and heat transfer of a nano fluid past a convectively heated stretching/shrinking sheet
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View full text

MHD mixed convective stagnation point flow along a vertical stretching sheet with heat source/sink

Highlights

Abstract

Introduction

Section snippets

Mathematical formulation

Method of solution

Results and discussion

Conclusions

Conflicts of interest

Int. J. Eng. Sci.

Int. Commun. Heat Mass Trans.

Int. J. Non-linear Mech.

Int. J. Therm. Sci.

Int. Commun. Heat Mass Transf.

Int. J. Therm. Sci.

Int. J. Heat Mass Transf.

Int. J. Heat Mass Trans.

Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder

Dingler’s Polytech. J.

Heat and mass transfer on a stretching sheet with suction or blowing

Can. J. Chem. Eng.