A Thermal Shock-Chemical Reactive Problem in Flow of Viscoelastic Fluid with Thermal Relaxation

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.


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 Sa-kiadis [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].

Formulation of the Problem
The electro-magnetic quantities satisfy Maxwell's equations [12]: These equations are supplemented by Ohm's law 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 0, , 0 . All the considered functions will depend on y and the time t only.
Equation ( 5) reduces to The vector Equations (1) and (2) reduced to the following scalar equation Eliminating J between Equations ( 6) and ( 7) we obtain Eliminating E between Equations ( 8) and ( 9) we obtain 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: Introduce the non-dimensional quantities.
In Equation (23) the overbar denotes the Laplace transform and the prime indicates differentiations with respect to y.
With the help of the non-dimensional quantities above Equations ( 12)-( 16) reduced to the non-dimensional equations Equation ( 23) can be written in constracted form as The formal solution can be expressed as: The characteristic equation of the matrix 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 The roots of Equation (25) satisfy the relations: The other two roots 3 k  and satisfy the relation The Maclaurin series expansion of Using the Cayley-Hamilton theorem, the infinite series can be truncated to the following form where I is the unit matrix of order 8 and a 0 -a 7 are some parameters depending on s and y.
The characteristic roots 1 , 2 , and The solution of this system of linear equations is given in Appendix A: Substituting for the parameters a 0 -a 7 into Equation (30) and computing A 2 , A 3 , A 4 , A 5 , A 6 and A 7 , 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 It is now possible to solve broad class problems in the Laplace transform domain.

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  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 and the mechanical boundary conditions on the plate is taken as where and are constant and Now we apply the state space approach described above to this problem.
Since the solution is bounded at infinity, then the expressions for 1, 2, ,8 ij    can be obtained by suppressing the positive exponential terms in Equation (30) which are not bounded at infinity.Thus for , we should replace each In order to obtain the remaining four components 31) and (32) to obtain the following linear system of equations: By solving this system, we arrive at Finally substituting the above value into (25), we obtain the solution of the problem in the transformed domain as: , e x p e x p exp exp 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 The shearing stress at the wall is given by

Inversion of the Laplace Transforms
above In order to invert the Laplace transform in the 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   g s is approximated by the relation where N is a sufficiently large integer representing the number of terms in the truncated infinite Fourier series.N must chosen such that where 1  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   g s .The optimal choice of c was obtained according e criteria described in [11].to th  3) and ( 5).It was found that the increasing of r P and K lead to decelerate the velocity of the flow fie Curves (1) and ( 4) describe the effect of Schmidt number c S on the velocity profiles of the flow field which reve 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].S .This shows that the heavier the diffusing species h e a greater retarding effect on the concentration distribution of the flow field.The concentration profiles are in good agreement with the results obtained in case of Hsiao [14].av  S and chemical reaction parameter K on the induced electric field E nd current density rofiles of the flow id, respectively.It was found that the increasing of these parameters lead to decelerate the magnitude of both the induced magnetic field and current density but Grashof number for heat transfer is to enhance them.

Skin Friction (τ)
, ff friction coe wh 1) Owing to the complicated equations for the unsteady MH tempts have been made to solve problems in this field.These attempts utilized approximate methods valid for only a specific range of some parameters.In this work, the method of direct integration by means of the matrix exponential, which is a standard approach in modern control theory and is developed in detail in many texts such as Ezzat [1], is introduced in the field of MHD and is applied to specific problems in which the temperature, velocity, concentration and magnetic field are coupled.This method gives exact solutions in the Laplace transform omain without any assumed restrictions on the applied n this work, we use a more general model of equatio d magnetic field or viscoelastic parameters.The same approach was used quite successfully in dealing with problems in generalized thermoelasticity theory by Ezzat et al. [15].
2) I ns, which includes the relaxation time of heat conduction o  and the electric permeability of the electromagnetic field o  .The inclusion of the relaxation time and electric permeability modifies the governing thermal, concentration and electromagnetic equations, changing them from parabolic to hyperbolic type, and thereby eliminating the unrealistic result that thermal and chemical reactive disturbance is realized instantaneously everywhere within a fluid.
3) The work considered here in reflects the effects tra over the boundaries has many ap agriculture, engineering, petroleum industries, and heat

REFERENCES
[1] M. A. Ezzat, to Solids and Fluids," Canadia view, Vol.86, No. of nsverse magnetic field and chemical reaction on an unsteady laminar incompressible free convection heat and mass transfer flow of a viscoelastic fluid past an infinite vertical plate.A set of linear differential equations governing the fluid velocity, temperature, species concentration and induced magnetic field is solved by the method of state space approach.The parameters that arise in this analysis are prandtl number r P (thermal diffusivity), Schmidt number c S (mass diffu ity), thermal Grashof number T G ( ee convection), Solutal Grashof number c G , che cal reaction parameter K and viscoelastic param er o k .A comprehensive set of graphical results for the velo ity, temperature, concentration, induced magnetic and electric field and current density is presented and discussed.
4) The flow of fluids siv fr mi c et plications such as boundary-layer control.The study of unsteady boundary layers owes its importance to the fact that all boundary layers that occur in real life are, in a sense, unsteady.In recent years, the requirements of modern technology have stimulated interest in fluid-flow studies, which involve the intersection of several phenomena.One such study is related to the effects of free convection flow through, which play an important role in transfer.
Appendix B: Elements of the Matrix L(s, y)

T G  and 0 TGFigure 3
Figure 3 present the effect of Grashof number for mass transfer c G , Prandtl number r P , Schmidt number c S and chem l reaction paramete K on the velocity pr files 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 r P 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 r P and K lead to decelerate the velocity of the flow fie Curves (1) and (4) describe the effect of Schmidt number c S on the velocity profiles of the flow field which reve 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].

Figure 1 .
Figure 1.Velocity distribution for different values of viscoelastic parameter k o .

Figure 2 .Figure 3 .
Figure 2. Velocity distribution for different values of G T .

Figure 4 .
Figure 4. Temperature distribution for different values of P r .

Figure 5 .
Figure 5. Concentration distribution for different values of S c .

Figure 6 .
Figure 6.Induced magnetic field distribution for different values of G T , G c .

Figure 7 .
Figure 7. Induced electric field distribution for different values of G T , P r , S c and K.

Figure 8 . 5 . 5 .
Figure 8.Current density distribution for different values of G T , P r , S c and K.f the flow field diminishes as the Prandtl number in-

Figure 6 TG  and heating 0 TG
Figure 6 concentrate on variations in the netic field profiles h for cooling 0 T G  and heating 0 T G  of the plate due to chang he values of number for mass transfer c G .It is observed that for cooling (heating) of the plate the induced magnetic field increases (decreases) rapidly in the vicinity of the plate and decreases (increases) asymptotically for higher values of y.It is also observed that an increase in Grashof number for mass transfer c G increases (decreases) the induced magnetic field.e in t Grashof

Table 1 .
It T G  are entered in S , k