The study of magneto hydrodynamic (MHD) flow and heat transfer has attracted numerous researchers due to its application to many technological and industrial processes, such as magnetic materials processing, purification of crude oil, magneto hydrodynamic electrical power generation, glass manufacturing, geophysics, and paper production, etc. Pavlov [1] used boundary-layer approximation theory to solve the problem of the flow of an electrically conducting fluid caused by a stretching elastic surface in the presence of a uniform magnetic field. Chakrabarti and Gupta [2] extended the work of Pavlov [1] and studied the flow evolution and heat transfer characteristics in flow over a stretching sheet with uniform suction. Applications of these results can be found in polymer technology and metallurgy. Watanabe [3] studied the characteristics of MHD boundary layer flow past a flat plate with a pressure gradient. Andersson [4] extended the work of Chakrabarti and Gupta [2] to a power law fluid while Chiam [5] obtained an accurate expression for the skin friction coefficient using Crocco’s transformation for a power-law velocity distribution in a conducting fluid. Chamkha [6] considered the problem of hydro-magnetic three-dimensional free convection flow on a vertical porous stretching surface. Abel et al. [7] investigated the effect of a magnetic field on a non-Newtonian fluid flow and obtained solutions for different profiles and their asymptotic limits for large and small Prandtl numbers. Recently, Sheikholeslami et al [8] used a semi analytical method to obtain solutions for nanofluid flow and heat transfer between parallel plates subject to a time-dependent magnetic field. Watanabe [3] presented a theoretical study that sought to describe the behavior of an electrically conducting fluid past a semi-infinite flat plate subject to a transverse magnetic field. In many practical situations, the material moves in a quiescent fluid with the fluid flow induced by the motion of the solid material and by the thermal buoyancy. Therefore the resulting flow and the thermal fields are determined by these two mechanisms. It is well known that the buoyancy force stemming from the heating or cooling of the continuous stretching sheet alter the flow and the thermal fields and thereby heat transfer characteristics of the manufacturing processes. However, the significance and impact of the buoyancy force were not assessed in the studies reviewed above. Furthermore, the study of convection heat transfer around or past a sphere, a cone, and a wedge has practical applications. The heat flow around these objects has applications in fields that include spacecraft design and nuclear reactors, Ostrach [9] studied free-convection flow about a flat plate and obtained theoretical and experimental results for velocity and temperature distributions. (See for details Kothandaraman and Subramanyan [10]). A study of mixed convection along a moving surface was carried out by Moutsoglou and Chen [11]. Mixed convection heat transfer at a stretching sheet with variable temperature was investigated by Vajravelu [12]. Ishak et al. [13] analyzed the hydromagnetic effects to mixed convection flow near a vertical stretching sheet. Some relevant studies in this area have been reported by researchers including Ali and Al-Youself [14], Nandkeolyar et al. [15], Mastroberardino [16], Srinivasacharya and Ram Reddy [17].
Renewed interest in the stretching sheet problem was sparked by a realization that some physical problems may be better modeled by a nonlinearly stretching sheet. Researchers who have reported the behaviour of fluid flow due to a nonlinear stretching sheet are Chaim [6], Prasad et al. [18], Ahmad et al. [19], Akyildiz et al. [20] and Kameswaran et al. [21]. Variable thickness of the sheet is useful in the mechanical, civil, marine and aeronautical structures and designs. The use of variable thickness helps to reduce the weight of structural elements and improve the utilization of the material. Sheets with variable thickness are often used in machine design, architecture, nuclear reactor technology, naval structures and acoustical components. With these industrial applications in mind, Lee [22] introduced the idea of variable thickness in theoretical studies. Fang et al. [23] studied the behaviour of boundary layer flow over a stretching sheet with variable thickness and explained the significant effects of the non-flatness of the sheet on the velocity and shear stress profiles by considering a special type of non-linear stretching $u_{w}(x)=U_{0}(x+b)^{m}$ for different values of $m$ being $b$ and $U_{0}$ constants. Khader et al. [24] extended the work of Fang et al. [23] and obtained the numerical solution for the slip velocity effect. Recently, Prasad et al. [25] and Vajravelu et al. [26], focused on heat transfer characteristics of fluid flow over a stretching sheet with variable thickness and power law velocity in the presence of a variable magnetic field.
Most of the studies above restricted their analysis to the hydromagnetic flow and heat transfer over a horizontal or a vertical plate and assumed the thermo-physical properties of the ambient fluid to be constant. However, it is known that these physical properties may change with temperature, especially the fluid viscosity and fluid thermal conductivity (Prasad et al. [18], Vajravelu et al. [27], Prasad et al. [28], and Hassanien [29]). For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the fluid, and the properties of the fluid can no longer be assumed to be constant. The increase in temperature leads to an increase in the transport processes including heat transfer at the wall. Therefore, to predict the flow and heat transfer rates, it is necessary to take into account the variable fluid properties. From the literature, we find no evidence of previous studies on the combined effects of variable fluid properties and mixed convection in flow over a slender stretching sheet with variable thickness.
The problem studied here extends the work of Prasad et al. [18] to the mixed convection flow with variable thickness. The coupled non-linear partial differential equations modeling the flow problem have been transformed to a system of coupled non-linear ordinary differential equations. These equations have been solved numerically using the Keller-box method, which is essentially a second order finite difference method. Computed numerical results for the flow and heat transfer characteristics are found to be in good agreement with experimental results in the literature (Fang et al. [23] and Khader et al. [24]. It is expected that the obtained results will not only provide useful information for industrial applications but would also serve to compliment and validate previous works.
Consider a mixed convection boundary layer flow of a viscous incompressible electrically conducting fluid in the presence of a transverse magnetic field $B(x)$ past an impermeable stretching vertical heated sheet with variable thickness. The origin is located at the slit, through which the sheet (see Fig. 1) is drawn in the fluid.
Two equal and opposite forces are applied impulsively along the
where
where
where
Substituting Eqs. (3) and (4) into Eqs. (1) and (2), we obtain
The appropriate boundary conditions for the problem are
Now we transform the system of Eqs. (2.1)-(2.3) into a dimensionless form. To this end, let the dimensionless similarity variable be
the stream function
Using Eq. (9), the velocity components can be written as
Here the prime denotes differentiation with respect to . In the present work, it is assumed
The non-dimensional parameters
The mixed convection parameter
and the corresponding boundary conditions are (
The value of
and the corresponding boundary conditions are (
where the prime denotes the differentiation with respect to
For practical purposes, the important physical quantities of interest are the local skin friction
where
Here we present some exact solutions for certain special cases. Such solutions are useful and serve as a baseline for comparison with the solutions obtained via other numerical / analytical schemes. In the case of constant fluid properties, the absence of a magnetic field and in the presence of buoyancy parameter with flat plate (
In the limiting case,
When
When
where
Eqs. (12) and (13) are highly non-linear, coupled ordinary differential equations of third-order in
Reduce equations (12) and (13) to a system of first-order equations.
Write the difference equations using central differences.
Linearize the algebraic equations by Newton’s method, and write them in matrix-vector form.
Solve the linear system by the block tri-diagonal elimination technique.
For numerical calculations, a uniform step size of Δ
Comparison of skin friction –
Fang et al. [23] By shooting method | Khader and Megahed [24] when Stip velocity parameter | Present Results | ||
---|---|---|---|---|
0.5 | 10 | 1.0603 | 1.0603 | 1.060309 |
9 | 1.0589 | 1.0588 | 1.058812 | |
7 | 1.0550 | 1.0551 | 1.055122 | |
5 | 1.0486 | 1.0486 | 1.048615 | |
3 | 1.0359 | 1.0358 | 1.035805 | |
2 | 1.0234 | 1.0234 | 1.023424 | |
1 | 1.0 | 1.0 | 1.0 | |
0.5 | 0.9799 | 0.9798 | 0.979848 | |
0 | 0.9576 | 0.9577 | 0.957727 | |
-1/2 | 1.1667 | 1.1666 | 1.166666 | |
0.25 | 10 | 1.1433 | 1.1433 | 1.143309 |
9 | 1.1404 | 1.1404 | 1.140481 | |
7 | 1.1323 | 1.1323 | 1.132332 | |
5 | 1.1186 | 1.1186 | 1.118662 | |
3 | 1.0905 | 1.0904 | 1.090409 | |
1 | 1.0 | 1.0 | 1.0 | |
0.5 | 0.9338 | 0.9337 | 0.933770 | |
0 | 0.7843 | 0.7843 | 0.784309 | |
-1/3 | 0.5000 | 0.5000 | 0.500000 | |
-1/2 | 0.0833 | 0.08322 | 0.0832227 |
Comparison of wall temperature gradient
Pr | Present results | Gmbka & Bobba [32] | Chen [33] | Ali [34] |
---|---|---|---|---|
0.01 | -0.01017936 | -0.0099 | 0.0091 | - |
0.72 | -0.4631462 | -0.4631 | -0.46315 | -0.4617 |
1.0 | -0.5826707 | -0.5820 | -0.58199 | -0.5 801 |
3.0 | -1.16517091 | -1.1652 | -1.16523 | -1.1599 |
5.0 | -1.56800866 | - | - | - |
-2.308029 | -2.3080 | -2.30796 | -2.2960 | |
100.0 | -7.769667 | -7.7657 | - | - |
In order to get a clear insight of the physical problem, numerical computations have been carried out using the Keller-box method for different values of pertinent parameters such as the fluid viscosity parameter
Variation of skin fricrion
Pr | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
2.0 | 0.1 | -5.0 | 1.0 | 0.1 | -5.0 | -1.545068 | -1.080793 | -1.522204 | -1.068792 | -1.499637 | -1.057054 |
0.0 | -1335278 | -1.146790 | -1.331519 | -1.134678 | -1.310604 | -1.122755 | |||||
0.5 | -1.185840 | -1.189561 | -1.165799 | -1.177318 | -1.146117 | -1.165239 | |||||
1.0 | -1.031486 | -1.222858 | -1.012449 | -1-210512 | -0.993795 | -1.198313 | |||||
5.0 | 0.5 | -5.0 | 1.0 | 0.0 | 0.1 | -1353883 | -1.530281 | -1.322420 | -1.502237 | -1.291802 | -1.474429 |
0.1 | -1352389 | -1380212 | -1320975 | -1356161 | -1290407 | -1332783 | |||||
0.2 | -1.351008 | -1.262758 | -1.319643 | -1.242086 | -1.289123 | -1.222008 | |||||
5.0 | 0.5 | -5.0 | 1.0 | 0.1 | 0.1 | -1.353871 | -1.383738 | -1.320976 | -1.135616 | -1.290407 | -1.332783 |
2.0 | -1.361864 | -2.092309 | -1.330033 | -2.037640 | -1.290407 | -1.984734 | |||||
5.0 | -1.371232 | -3.507749 | -1.334456 | -3.362246 | -1.307821 | -3.223160 | |||||
10.0 | -1.376316 | -5.101436 | -1.344081 | -4.806734 | -1.312650 | -4.529388 | |||||
Pr | |||||||||||
0.1 | 0.0 | -5.0 | 1.0 | 0.1 | 0.0 | -0.227799 | 5.478141 | -0.631212 | 0.880672 | -1.376995 | -1.399540 |
0.1 | -0.409797 | 3.568688 | -0.718014 | 0.679241 | -1.336629 | -1.376940 | |||||
0.2 | -0.593180 | 2.335793 | -0 808435 | 0.50008 | -1.297614 | -1.354864 | |||||
0.3 | -0.775887 | 1.545657 | -0.902260 | 0.337709 | -1259952 | -1333293 | |||||
0.5 | -1.141195 | 0.598557 | -1.099277 | 0.049424 | -1.188558 | -1.291576 | |||||
0.75 | -1.603216 | -0.121641 | -1.360758 | -0.266291 | -1.106376 | -1.241904 | |||||
Pr | |||||||||||
0.1 | 0.0 | -5.0 | 1.0 | 0.1 | -0.2 | -0.248745 | 4.692819 | -0.548033 | 3.405338 | ||
-0.1 | -0.484413 | 1.895248 | -0.725686 | 1.517615 | |||||||
0.0 | -0.640512 | 0.806492 | -0.849481 | 0.688922 | |||||||
1.0 | -1.134166 | -0.889315 | -1.275859 | -0.858419 | |||||||
5.0 | -1.375488 | -1.399645 | -1.496541 | -1.367532 | |||||||
10.0 | -1.426228 | -1.494909 | -1.543140 | -1.464027 | |||||||
Pr | |||||||||||
0.1 | 0.0 | 2.0 | 1.0 | 0.1 | -10.0 | -1212481 | -1.169201 | -1.123290 | -1.109093 | ||
-5.0 | -1.312789 | -1.152769 | -1.211471 | -1.094139 | |||||||
-3.0 | -1.435761 | -1.132209 | -1.317998 | -1.075621 | |||||||
-2.0 | -1.576301 | -1.108233 | -1.437581 | -1.054299 |
Figs. 2(
In Figs. 3(a)-3(e) the numerical results for the temperature distribution
The effects of the physical parameters on the skin friction
Constant in Eq. (3)
Temperature (
Constant in Eq. (3)
Temperature of the plate (
Ambient temperature (
Temperature difference (
Constant in Eq. (7) known as stretching rate
Uniform magnetic field (Tesla)
Wall temperature parameter
Specific heat at constant pressure (
Skin friction
Dimensionless stream function
Temperature dependent thermal conductivity (
Thermal conductivity of the fluid far away from the sheet (
Velocity exponent parameter
Magnetic parameter
Nusselt number
Prandtl number
Local Reynolds number
Sherwood number
Velocity components in the
Stretching velocity (
Reference velocity (
Cartesian coordinates (
Wall thickness parameter
Buoyancy or mixed convection parameter
Electric conductivity
Constant in (2.5) known a variable thermal conductivity parameter
Thermal expansion coefficient
Constant defined in equation (2.4)
Kinematic viscosity away from the sheet (kg/m3)
Density (kg/m3)
Thermal conductivity of the fluid far away from the sheet (W/m K)
Similarity variables
Dimensionless temperature
Fluid viscosity parameter, constant in equation (2.12)
Dynamic Viscosity (Pa s)
Constant value of dynamic viscosity (Pa s)
Stream function
Kummers' function
Condition at infinity
Condition at the wall
The numerical results indicate that the effect of increasing the mixed convection parameter is to increase the momentum boundary layer thickness whereas to decreases the thermal boundary layer thickness. Meanwhile, the dimensionless velocity distribution at a point near the plate decreases as the wall thickness parameter increases and hence the thickness of the boundary layer becomes thinner when
The rate of heat transfer increases with increasing magnetic parameter and the Prandtl numbers. Hence, the effect of Prandtl number is to decrease the thermal boundary layer thickness and the wall-temperature gradient. In addition to this, the effect of the variable thermal conductivity parameter is to enhance the temperature field.