Medical, Pharma, Engineering, Science, Technology and Business

American International University-Bangladesh, Kamal Ataturk avenue, Banani, Dhaka, Bangladesh

- *Corresponding Author:
- Maleque A

Department of Mathematics

American International University-Bangladesh

House - 23, Road - 17, Kemal Ataturk avenue

Banani, Dhaka-1213, Bangladesh

**Tel:**88-02-8813233

**E-mail:**[email protected], [email protected]

**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.

**Visit for more related articles at** Fluid Mechanics: Open Access

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 [1] studied the low Reynolds number flow of a variable property gas past an infinite heated circular cylinder. Bhat and Bose [2] examined the fluid flow with variable properties in a two dimensional channel. Zakerullah and Ackroyd [3] investigated the free convection flow above a horizontal circular disc for variable fluid properties. Correlations have been developed by Gokoglu and Rosner [4] 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 [5] and Herwig and Wickern [6], Herwig and Klemp [7]. Herwing [8] 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 [12] studied the effect of variable fluid properties on the classical Blasius and Sakiadis flow taking into account temperature dependent physical properties. More recently Maleque [13] 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 *U _{0}*. We take the plate in upward direction along the

We assume that the dependency of the fluid properties, viscosity coefficient (*μ*) and thermal **conductivity** coefficient (*k*) are function of temperature alone and obey the following laws [10].

(1)

Where A and C are arbitrary exponents, *μ _{∞}*is the uniform viscosity of the fluid and

Thus according to the above assumptions and Boussinesq’s approximation, the continuity equation, the momentum equation and the heat equation of the problem are:

(2)

(3)

(4)

The boundary conditions are:

(5)

where u and v are components of velocity along x-axis and y-axis respectively; *ρ* is the fluid density, *B _{0}* is the constant applied transverse magnetic field,

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:

(6)

From the continuity equation (2) we have

(7)

*v _{0}* is the dimensionless suction/injection velocity.

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:

(8)

(9)

Where,

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 [15] can try a one class of solutions of the equations (8)-(9) by taking that

(10)

From equation (10) we have,

(11)

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 [18], 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

(12)

Using equation (12), the equations (8)-(9) yield

(13)

(14)

The corresponding boundary conditions are obtained from equation (5) as,

(15)

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:

(16)

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

(17)

Where,

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 [23] 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 *G _{r}*,

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** _{∞}* =

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 *G _{r}*,

**Figure 3** represents the effects of γ on the velocity and temperature profiles for *G _{r}*=0, M=0.5,

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 (*E _{c}*) on velocity profiles and temperature profiles for

It has also been observed from this figure that the parameter *E _{c}* has remarkable effect on the temperature profiles. The increasing values of

**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**).

γ |
f′(0) |
–θ′(0) |
---|---|---|

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 |
6.4013 3.2132 1.9042 1.3142 1.0245 0.8638 0.7573 |
18.9261 9.1721 5.1378 3.2430 2.2272 1.6206 1.2322 |

**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 (*G _{r}*) has marked effects on velocity and temperature fields. Both velocity and temperature profiles increase as

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 *E _{c}* 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

5. It is observed that the temperature profiles decrease for increasing values of *v _{0}* and increase for negatively increasing values of

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*, *E _{c}* and

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