American International University-Bangladesh, Kamal Ataturk avenue, Banani, Dhaka, Bangladesh
Received Date: August 30, 2016; Accepted Date: September 19, 2016; Published Date: September 28, 2016
Citation: Maleque KA (2016) Temperature Dependent Variable Properties on Mixed Convective Unsteady MHD laminar Incompressible fluid Flow with Heat Transfer and Viscous Dissipation. Fluid Mech Open Acc 3:134.
Copyright: © 2016 Maleque KA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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In the present paper we investigated that the numerical solution of mixed convective laminar boundary layer flow with uniform transverse magnetic field for a vertical porous plate. The fluid properties are considered the nonlinear functions of temperature as μ = μ∞(T / T∞)A and κ = κ∞(T / T∞)C For flue gases the values of the exponents A and C are taken as A=0.7, and C=0.83. Also time dependent suction velocity is considered. The governing equations are reduced into ordinary differential equations by introducing the suitable similarity variables and then solved numerically by using sixth ordered Runge-Kutta method and Nachtsheim Swigert iteration technique. The numerical results of our physical interest are shown graphically and tabular form.
Mixed convective; Viscous dissipation; Fluid flow; MHD laminar; Temperature; Heat transfer
This is well known that the physical properties of the flow are functions of temperature. In light of this concept, Hodnett  studied the low Reynolds number flow of a variable property gas past an infinite heated circular cylinder. Bhat and Bose  examined the fluid flow with variable properties in a two dimensional channel. Zakerullah and Ackroyd  investigated the free convection flow above a horizontal circular disc for variable fluid properties. Correlations have been developed by Gokoglu and Rosner  for deposition rates in force convection systems with variable properties. The influence of variable properties on laminar fully developed pipe flow with constant heat flux across the wall studied by Herwing  and Herwig and Wickern , Herwig and Klemp . Herwing  examined an asymptotic analysis of laminar film boiling on a vertical plate including variable property effects. Steady and unsteady MHD laminar convective flow due to rotating disk with the effects of variable properties and Hall current investigated by Maleque and Sattar [9-11]. Recently Asterios Pantokratoras  studied the effect of variable fluid properties on the classical Blasius and Sakiadis flow taking into account temperature dependent physical properties. More recently Maleque  considered combined temperature- and depth-dependent viscosity and Hall current on MHD convective flow due to a rotating disk.
Considering the importance of MHD combined convective flow, in present study we are to investigate the variable fluid properties (viscosity (μ) and thermal conductivity (k) transverse magnetic field and time dependent suction/injection velocity on an unsteady MHD laminar incompressible fluid flow for a vertical porous plate. Using suitable dimensionless similarity variables the governing partial differential equations of the MHD combined convective boundary layer flow are reduced to nonlinear ordinary differential equations and then by using Range-Kutta six order integration scheme and Nachtsheim-Swigert iteration technique the nonlinear ordinary differential equations are solved numerically. The obtained numerical results of physical properties are shown graphically and tabular form for the different values of the dimensionless parameters. Finally the effects of the relative temperature difference parameter (γ) on the skin friction and heat transfer coefficients are also examined.
Let us consider an unsteady combined forced and free convective incompressible laminar boundary layer MHD electrically conducting viscous fluid flow past an infinite vertical porous plate y=0. Considering the flow is moving in upward direction with a constant velocity U0. We take the plate in upward direction along the x-axis and y-axis is normal to plate. A uniform non conducting magnetic field is perpe3ndicular to the plate and parallel to the y-axis. The magnetic field is assumed as the form = (0, B0 , 0). The equation of conservation of electric charge is ∇· = 0 , where = (Jx,Jy,Jz), the propagation direction is considered only along the y-axis and does not have any variation along the y-axis. Considering , which gives result Jy = constant. For an electrically non-conducting plate, this constant is zero and hence Jy = 0 everywhere in the flow. It is assumed that an infinite plate in extent along x-axis and hence all the physical quantities velocity and temperature are functions of y and t.
Where A and C are arbitrary exponents, μ∞is the uniform viscosity of the fluid and k∞ is a uniform thermal conductivity of heat. The fluid is assumed to be flue gas for our present analysis. The exponents values are considered as A=0.7, and C=0.83 for flue gases .
Thus according to the above assumptions and Boussinesq’s approximation, the continuity equation, the momentum equation and the heat equation of the problem are:
The boundary conditions are:
where u and v are components of velocity along x-axis and y-axis respectively; ρ is the fluid density, B0 is the constant applied transverse magnetic field, g is the gravitational acceleration, β is the volume expansion coefficient for temperature, T and T∞ are respectively the thermal boundary layer fluid temperature and the fluid temperature in the free stream, Tw is the plate temperature; σ ′ is the electrical conductivity; and vw(t), the normal velocity at the plate having a positive value for suction and negative value for injection.
The governing equations (2)-(4) under the boundary conditions (5) can be solve by considering the well-defined similarity technique to obtain the similarity solutions. Now we introduce the following nondimensional suitable similarity variables:
From the continuity equation (2) we have
v0 is the dimensionless suction/injection velocity. v0<0 represents suction and v0>0 represents injection.
The dimensionless quantities from equation (6) and v from equation (7) are introduced in equations (3)—(4) finally the nonlinear ordinary differential equations are obtained as:
Where, is the relative temperature difference parameter which is positive for a heated plate, negative for a cooled plate and zero for uniform properties.
The equations (8)-(9) are similar except for the term , where time t appears explicitly. Thus must be a constant quantity for the similar condition. Hence following Maleque  can try a one class of solutions of the equations (8)-(9) by taking that
From equation (10) we have,
Where L is the integrating constant which is obtained through the condition that when t = 0 then δ = L. Here K =0 that is δ = L. represents the length scale for steady flow and K ≠ 0 implies that δ denotes the length scale for unsteady flow [16,17].
It thus appear from equation (10) that by puting K=2 in equation (11), the length scale δ(t) becomes . The usual scale factor δ(t) exactly suitable for unsteady viscous boundary layer flows considered by Schlichting , Maleque [19,20]. Since δ is a similarity parameter as well as scale factor, except that the scale would be different any value of K in equation (11) would not change the nature of solutions. Finally introducing K=2 (Maleque [21,22] in equation (10) , we have
Using equation (12), the equations (8)-(9) yield
The corresponding boundary conditions are obtained from equation (5) as,
Prime denotes d/d in the above equations.
The chief physical interests are the skin friction coefficient and the rate of heat transfer to the plate, are calculated out. The wall shearing stress (τ) is:
From the plate to the fluid the rate of heat transfer (q) is computed by the application of Fourier’s law as given in the following:
Hence the Nusselt number (Nu) is determined as
is the square root of local Reynolds number.
In equations (16)-(17), the gradient values of f and θ at the plate are evaluated when the corresponding differential equations are solved satisfying the convergence criteria.
The set of coupled, nonlinear ordinary differential equations (13)- (14) are solved numerically using a standard initial value solver called the shooting method. For this purpose Nachtsheim and Swigert  iteration technique has been employed. A computer program was made for the solutions of the basic non-linear differential equations of our problem by adopting this numerical technique and six ordered Range-Kutta method of integration. In different phases various groups of parameters Gr, M, γ, Ec and v0 were considered.
The step size Δη=0.01 was selected that satisfied a convergence criterion of 10-6 in almost all phases mentioned above in all computations. By setting η∞ = η∞ + Δη the value of η∞ was found to each iteration loop. Thus (η∞)max, to each group of parameters, has been determined when the value of unknown boundary conditions at η=0 (that arise in the Shooting method) does not change to successful loop with error less than 10-6 . However, different step sizes such as Δη=0.01, Δη=0.005 and Δη=0.001 were also tried and the obtained solutions have been found to be independent of the step sizes observed in (Figure 1).
As a result of the numerical calculation, the velocity and temperature distributions for the flow are obtained from equations (13)-(14) and are shown in (Figures 2-6) for different values Gr, M, γ, Ec and v0. The fluid is considered as flue gas (Pr – 0.64) for the present study. The effects of Gr (Grashof number) on the velocity and temperature profiles are shown in (Figure 2). It appears from (Figure 1).that the temperature profile shows its usual trend of gradual decay when Gr=0 that is for forced convection. The profiles overshoot the uniform temperature close to the boundary when Grashof number Gr becomes larger. (Figure 2) also shows that the velocity profiles increase as Gr increases.
Figure 3 represents the effects of γ on the velocity and temperature profiles for Gr=0, M=0.5, Ec=0.5, v0=0.5. In this figure comparison is made between constant property and variable property solutions. In (Figure 3), it is observed that the velocity profiles decreases for increasing values of temperature relative parameter γ, this figure also shows that for increase value of γ leads to the increase the values of temperature. It is also observed that the velocity profile increases and temperature profile decreases for negatively increasing values of relative temperature different parameter γ.
The act of imposing of a magnetic field to an electrically conducting fluid creates a drag like force called Lorentz force. The force has the tendency to slow down the flow around the plate at the expense of increasing its temperature. This is depicted by decreases in velocity profiles and increases in the temperature profiles as M increases as shown in (Figure 4). In addition, the increases in the temperature profiles as M increases are accompanied by increases in the thermal boundary layer.
The effects of Eckert number (Ec) on velocity profiles and temperature profiles for Gr=10, M=0.5, γ=0.5, v0=0.5 and Pr=0.64 shown in (Figure 5). In the equation of continuity and momentum equation, the dimensionless Eckert number (Ec) does not enter directly its influence come through the energy equation. It has been found that the Eckert number (Ec) has marked effect on velocity profiles in presence of buoyancy force. The velocity profiles increase as Eckert number increases shown in (Figure 5).
It has also been observed from this figure that the parameter Ec has remarkable effect on the temperature profiles. The increasing values of Ec leads to the increasing value of temperature profile. It indicates that the thermal boundary layer thickness increases with increasing values of Eckert number. For Gr=10, M=0.5, γ=0.5, Ec=0.5, (Figure 6) shows the effects of suction/injection parameter v0 on the velocity profiles and the temperature profiles. It has been observed from this figure that the velocity profile decreases for increasing values of injection (v0>0) while the velocity profile increases for increasing value of suction parameter (v0<0). It implies that the boundary layer is increasingly blown away from the plate to form an interlayer between the suction and the outer flow regions. It has also been observed from this figure that the temperature profiles decrease for increasing values of v0 and increase for negatively increasing values of v0. The thermal boundary layer increases rapidly for strong suction while the temperature usual decays for strong injection.
Table 1 shows the effects of temperature related parameter γ on the skin-friction coefficient and the heat transfer co-efficient. From Table 1, it is investigated that the increase of relative temperature difference parameter γ leads to decrease both the shearing stress (skin-friction coefficient) and the heat flux (Nusselt number). These are found to in agreement with the effects of γ on the boundary layer (velocity profiles) and thermal boundary layer thicknesses ( temperature profiles) (Table 1).
Table 1: The skin-friction coefficient and Nusselt number for different values of γ.
In the present investigation, the effects of suction/injection along with the variable properties on MHD combined convective unsteady laminar incompressible boundary layer flow were studied. We obtained the nonlinear ordinary differential equations by introducing the similarity variables in basic equations. Using Range-Kutta six order integration schemes and Nachtsheim-Swigert iteration technique differential equations are then solved numerically.
As a result of the computations the following conclusions can be made:
1. Grashof number (Gr) has marked effects on velocity and temperature fields. Both velocity and temperature profiles increase as Gr increases.
2. Variable properties (γ) have marked effects on the velocity and temperature profiles. It is shown increasing value relative temperature different parameter γ leads the decrease of the non-dimensional velocity profile and increase of temperature profile.
3. We investigate that due to the application of a transverse magnetic field normal to the flow direction will result in a resistive force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reduces its velocity. As a result the magnetic interaction parameter M the boundary layer thickness decreases and thermal boundary layer thickness increases.
4. It has been observed that although Eckert number Ec does not involve directly in boundary layer equation but it has marked effect on velocity profile through thermal boundary layer equation by buoyancy force. Both the velocity and the temperature profiles increase with the increasing values of Ec.
5. It is observed that the temperature profiles decrease for increasing values of v0 and increase for negatively increasing values of v0. The temperature usually decays for strong injection while the thermal boundary layer increases rapidly for strong suction.
Increasing the values of γ (-1, to 1) lead to the decrease in skin friction coefficient and heat transfer coefficient for fixed values of M, Ec and v0.