ABSTRACT

Effects of thermal and species diffusion with one relaxation time on the boundary layer flow of a viscoelastic fluid bounded by a vertical surface in the presence of transverse magnetic field have been studied. The state space approach developed by Ezzat [1] is adopted for the solution of one-dimensional problem for any set of boundary conditions. The resulting formulation together with the Laplace transform techniques are applied to a thermal shock-chemical reactive problem. The inversion of the Laplace transforms is carried out using a numerical approach. The numerical results of dimensionless temperature, concentration, velocity, and induced magnetic and electric fields distributions are given and illustrated graphically for the problem.

1. Introduction

Viscoelastic flows are encountered in numerous areas of petrochemical, biomedical and environmental engineering including polypropylene coalescence sintering [2] and geological flows [3]. A wide range of mathematical models have been developed to simulate the nonlinear stress-strain characteristics of such fluids which exhibit both viscous and elastic properties [4].

In nature and many industrial applications, there are plenty of transport processes where simultaneous heat and mass transfer is a common phenomenon. Its application is found in many diverse fields but not limited to cleaning operations, curing of plastics, manufacturing of pulp-insulated cables, many chemical processes such as analysis of polymers in chemical engineering, condensation and frosting of heat exchangers [5]. The study of convection reduces to the determination of convective heat and mass transfer coefficients. Convective heat and mass transfer coefficients are important parameters, which are a measure of the resistance to heat and mass transfer between a surface and the fluid flowing over that surface. The convective coefficients depend on the hydrodynamic, thermal and concentration boundary layers. In many of the internal flows, both forced and natural convection play major roles in the heat and mass transfer processes. Whereas in the entrance section of a duct, forced convection becomes dominant, as the flow moves towards the downstream section, natural convection could dominate over forced convection and finally in the thermally developed region natural convection becomes negligible. Natural convection may be due to a temperature or concentration gradient or both. If the buoyancy forces are due to temperature and concentration gradients that act in the same direction, both the heat and mass transfer will increase. However, if the temperature and concentration gradients act in the opposite direction, both heat and mass transfer reduce [6].

In recent years, the study of viscoelastic fluid flow is an important type of flow occurring in several engineering processes. Such processes are wire drawing, glass fiber and paper production, crystal growing, drawing of plastic sheets, among which we also cite many applications in petroleum in drilling, manufacturing of foods and slurry transporting. The boundary layer concept of such fluids is of special importance due to its applications to many engineering problems among which we cite the possibility of reducing frictional drag on the hulls of ships and submarines.

A great deal of works has been carried out on various aspects of momentum and heat transfer characteristics in a viscoelastic boundary layer fluid flow over a stretching plastic boundary [7] since the pioneering work of Sakiadis [8]. Ezzat and Zakaria [9] studied the effects of free convection currents with one relaxation time on the flow of a viscoelastic fluid through a porous medium. Khan and Sanjayanand [10] studied heat and mass transfer in a viscoelastic boundary layer flow over exponentially stretching sheet.

In this work, we use a more general model of MHD mixed convection flow of conducting viscoelastic fluid which also includes both the relaxation time in the heat and concentration equation and the electric permeability of the electromagnetic field. The unsteady free convection heat and mass transfer flow of electrically conducting incompressible viscoelastic fluid past an infinite vertical plate in the presence of a transverse magnetic field and chemical reaction using the state space approach and Laplace transforms technique. The inversion of the Laplace transform is carried out using a numerical technique [11].

2. Formulation of the Problem

The electro-magnetic quantities satisfy Maxwell’s equations [12]:

(1)

(2)

, (3)

, (4)

These equations are supplemented by Ohm’s law

(5)

Consider an unsteady free convection flow of electrically conducting incompressible, viscoelastic fluid past an infinite vertical plate. The x-axis is taken in the vertical direction along the plate and y-axis normal to it. Let u be the component of the velocity of the fluid in the x direction and a constant magnetic field acts in the y direction of strength. This produces an induced magnetic field and an induced electric field as well as a conduction current density. All the considered functions will depend on y and the time t only.

Equation (5) reduces to

. (6)

The vector Equations (1) and (2) reduced to the following scalar equation

(7)

(8)

Eliminating J between Equations (6) and (7) we obtain

(9)

Eliminating E between Equations (8) and (9) we obtain

(10)

The Lorentz force has a non-vanishing component in the x-direction, given by:

(11)

Assume that the viscoelastic fluid contains some chemically reactive diffusive species then the equations describing the flow in the boundary layer reduce to:

(12)

(13)

(14)

(15)

(16)

Introduce the non-dimensional quantities.

(17)

With the help of the non-dimensional quantities above Equations (12)-(16) reduced to the non-dimensional equations

(18)

(19)

(20)

(21)

(22)

where,.

To simplify the algebra, only problems with zero initial conditions are considered. Taking Laplace transform of Equations (18)-(22) and writing the resulting equations in matrix form results in (23).

where

and

.

In Equation (23) the overbar denotes the Laplace transform and the prime indicates differentiations with respect to y.

(23)

Equation (23) can be written in constracted form as

(24)

The formal solution can be expressed as:

(25)

The characteristic equation of the matrix is

(26)

where

(27)

The roots, , and of Equation (25) satisfy the relations:

(28)

Two of the roots, say and have simple expression given by

(29a)

The other two roots and satisfy the relation

(29b)

The Maclaurin series expansion of is given by

.

Using the Cayley-Hamilton theorem, the infinite series can be truncated to the following form

(30)

where I is the unit matrix of order 8 and a₀ – a₇ are some parameters depending on s and y.

The characteristic roots, , and of the matrix A must satisfy the equations.

The solution of this system of linear equations is given in Appendix A:

Substituting for the parameters a₀ - a₇ into Equation (30) and computing A², A³, A⁴, A⁵, A⁶ and A⁷, we get, the elements (ℓ_ij i, j = 1, 2, 3, 4, 5, 6, 7, 8) of the matrix L(y, s) which listed in Appendix B.

It should be noted here that, we have used Equation (29) in order to write these entries in the simplest possible form. It should also be noted that this is a formal expression for the matrix exponential. In the physical problem, we should suppress the positive exponential which are unbounded at infinity. Thus we should replace each by and each by.

It is now possible to solve broad class problems in the Laplace transform domain.

3. Thermal Shock-Chemical Reactive Problem

Consider the free convection flow of an incompressible viscoelastic fluid in the presence of magnetic field occupying a semi-infinite region y ³ 0 of the space bounded by an infinite vertical plate y = 0 with quiescent initial state. A thermal-concentration shock is applied to the boundary plane y = 0 in the form

(31)

and the mechanical boundary conditions on the plate is taken as

(32)

where and are constant and H(t) is Heaviside unit step function.

Now we apply the state space approach described above to this problem.

Since the solution is bounded at infinity, then the expressions for can be obtained by suppressing the positive exponential terms in Equation (30) which are not bounded at infinity. Thus for, we should replace each by and each by.

The components of the transformed initial state vector are known name

(33)

(34)

In order to obtain the remaining four components, , and we substitute y = 0 into Equations (31) and (32) to obtain the following linear system of equations:

(35)

By solving this system, we arrive at

(36)

Finally substituting the above value into (25), we obtain the solution of the problem in the transformed domain as:

(37)

(38)

(39)

(40)

where the constants A_i, i = 1, 2, 3, 4 are listed in Appendix C.

The induced electric field and current density take the following forms

(41)

(42)

The shearing stress at the wall is given by

(43)

4. Inversion of the Laplace Transforms

In order to invert the Laplace transform in the above equations, we adopt a numerical inversion method based on a Fourier series expansion [11]. In this method, the inverse g(t) of the Laplace transform is approximated by the relation

(44)

where N is a sufficiently large integer representing the number of terms in the truncated infinite Fourier series. N must chosen such that

where is a persecuted small positive number that corresponds to the degree of accuracy to be achieved. The parameter c is a positive free parameter that must be greater than the real parts of all singularities of. The optimal choice of c was obtained according to the criteria described in [11].

5. Numerical Results and Discussion

The problem of free convective flow with heat and mass transfer of a viscous incompressible viscoelastic electrically conducting fluid past a vertical plate in presence of a transverse magnetic field has been considered. The solutions for velocity, temperature and concentration fields as well as the induced magnetic and electric fields are obtained by using the state space approach. The technique is applied to a thermal shock-chemical reactive problem without heat sources. The effects of flow parameters such as Grashof number for heat and mass transfer, , Prandtl number, Schmidt number, chemical reaction parameter K, viscoelastic parameter and relaxation time have been studied analytically and presented with the help of Figures 1-8 for the considered problem.

5.1. Velocity Field (u)

The velocity of the flow field varies vastly with the variation of the flow parameters such as Grashof number for heat and mass transfer, , viscoelastic parameter, Prandtl number, Schmidt number, chemical reaction parameter K. The effects of these parameters on the velocity fluid of flow field have been presented in Figures 1-3.

5.1.1. Effect of Viscoelastic Parameter (k_o)

Figure 1 depicts the effect of viscoelastic parameter on the velocity profiles of the flow field keeping other parameters of the flow field constant. The curve with viscoelastic parameter, corresponds to Newtonian flow and in other two curves the viscoelastic parameter is taken in increasing order. The viscoelastic parameter is found to decelerate the velocity of the flow field. The above parameter is in good agreement with the result obtained in cases of Khan and Sanjayanand [10].

5.1.2. Effect of Grashof Number for Heat (G_T)

The values of Grashof number for heat have been chosen as they are interesting from physical point of view. The free convection of heat is due to the temperature difference and hence when which physically corresponds to cooling of the surface by free convection currents. Then correspond to heating of the surface by free convection currents. In Figure 2, we observe that the effect of cooling and heating by free convection currents when and are in agreement with physical observations that cooling of the surface by free convection currents occurs for positive values of while heating corresponds to negative values of. It was also noticed that the velocity increase with the increase of.

5.1.3. Effect of Different Parameters (G_c, P_r, S_c, K)

Figure 3 present the effect of Grashof number for mass transfer, Prandtl number, Schmidt number and chemical reaction parameter K on the velocity profiles of the flow fluid. Comparing the curve (1) and (2) of the figure, it is observed that the Grashof number for mass transfer is to enhance the velocity of the flow field at all points. The effect of both Prandtl number and chemical reaction parameter K on the velocity field is shown by the curves (1), (3) and (5). It was found that the increasing of and K lead to decelerate the velocity of the flow field. Curves (1) and (4) describe the effect of Schmidt number on the velocity profiles of the flow field which reveal that the presence of heavier diffusing species has a retarding effect on the velocity of the flow field. The effect of the above parameters has the same behavior as in case of Das et al. [13].