The problem of unsteady free convective boundary layer flow has attracted the interest of many investigators in view of its application in many engineering and geophysical problems. Over the years studies involving the boundary layer flow along reactive surface have grown to researchers due to its implicit occurrence in different engineering and natural applications like melt spinning, contamination transport from surfaces, acid mine drainage remediation, waste disposal, reactive flow of gases adjacent to the surface and so on. Fluid is also sprayed for cleaning the surface; the contamination at the surface reacts with the plate and washes away. In addition the processes of Micro Electro Mechanical System (MEMS) require deposition or selective removal of specie from the surface. Thus it has several utilities in micro or macro engineering, material science and environmental science, etc. In this context, Soundalgekar [1] studied the effect of mass transfer and free convection on the flow past an impulsively started vertical flat plate. Raptis et al. [2] constructed similarity solutions for boundary layer near a vertical surface in a porous medium with constant temperature and concentration. Many transport pro- cesses exist in nature and in industrial applications in which the simultaneous heat and mass transfer occur as a result of combined buoyancy effects of thermal diffusion and diffusion of chemical species. Bejan and khair [3] reported on the natural convection boundary layer flow in a saturated porous medium with combined heat and mass transfer. Williams et al. [4] studied the unsteady free convection flow over a vertical flat plate under the assumption of variations of the wall temperature with time and distance. They found possible solution for a variety of classes of wall temperature distribution. The similarity solutions of free, mixed and forced convection problems in a saturated porous media were discussed by Ramanaiah et al. [5] . Watanabe [6] considered the mixed convection boundary layer flow past an isothermal vertical porous flat plate with uniform suction or injection. Kafoussias and Williams [7] investigated the effect of temperature-dependent viscosity on free-forced convective laminar boundary layer flow past a vertical isothermal plate. Erikson et al. [8] examined heat and mass transfer on a moving continuous flat plate with suction or injection on the flat plate. Slaouti et al. [9] studied the unsteady free convection flow in the stagnation point region of a three-dimensional body. It was seen that temperature and surface heat transfer were changed in a small interval of time. The surface heat transfer parameter increased with the increase of Prandtl number while the surface skin friction parameter decreased with the increase of Prandtl number. Healmy K. A. [10] studied MHD unsteady free convective boundary layer flow past a vertical plate embedded in a porous medium. Pop and Postelnicu [11] obtained similarity solutions of free convection boundary layers over vertical and horizontal surface porous media with internal heat generation. Chamkha and Khaled [12] obtained similarity solutions for hydro magnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Hossin and Monir [13] analyzed a two dimensional mixed convection flow of a viscous incompressible fluid of temperature dependent viscosity past a vertical plate. Seddek and Salem [14] studied the laminar mixed convection flow adjacent to vertical continuously stretching sheets with variable viscosity and thermal diffusivity. Aydin and Kaya [15] studied mixed convection of a viscous dissipating fluid about a vertical flat plate. They assumed that the fluid thermal conductivity was supposed to vary as a linear function of temperature. Aziz [16] investigated a similarity solution for laminar thermal boundary layer over a flat plate with a convection surface boundary condition. Ali and Magyari [17] studied the problem for unsteady stretching surface under different conditions by using a similarity method to transform governing time-dependent boundary layer equations into a set of ordinary differential equations. Mukhopadhyay [18] obtained similarity solution for unsteady mixed convection flow past a stretching sheet considering the combined effects of porous medium and the thermal radiation. Vajravelu [19] studied unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties. Singh and Sharma [20] analyzed heat and mass transfer in the boundary layer flow along a vertical isothermal reactive plate near stagnation point.

The present problem is to study the similarity solutions of unsteady convective boundary layer flow along isothermal vertical plate with porous medium. Mentioned fluid flow, heat and mass transfer models are parameterized by dimensionless quantities such as Prandtl number, Schmidt number, unsteadiness and Buoyancy parameters, etc. The effect of governing parameters on the velocity, temperature, concentration profile are investigated and analyzed. The local Skin-friction coefficient, Nusselt number and Sherwood number at the plate for physical interest are also illustrated for required parameters.

We consider an unsteady two dimensional flow of a viscous incompressible fluid through porous media along vertical plate. The flux of the specie at the plate is considered proportional to specie concentration at the plate. All the fluid properties assumed to be constant except the effect of buoyancy and the density variation. In addition the electric field, pressure gradient, Hall effects, joule heating terms and induced magnetic field are also neglected. Let the x-axis be taken along the direction of the plate and y-axis normal to it. If the velocity distribution in the potential flow is given by and, where a is a positive constant. Under assumptions mentioned above, the governing boundary layer equations of continuity, momentum, energy and specie concentration can be written as:

The boundary condition for velocity, temperature and concentration fields are:

where u and v are the fluid velocity components along x and y-axis respectively, is the kinematics viscosity, ρ is the density of the fluid, β and β* are the thermal and concentration expansion co-efficient respectively, g is the acceleration due to gravity, T is the temperature inside the boundary, is the temperature for away from the plate, C is the species concentration in the boundary layer, is the species concentration of the ambient fluid, is the specific heat, D is the molecular diffusivity of the species concentration, is the permeability of the porous medium. and is the co-efficient of thermal conductivity.

In order to simplify the mathematical analysis of the problem we introduce the flowing dimensionless variables:

Introducing dimensionless similarity variables into the system of Equation (2) to Equation (4) is reduced to system of ordinary differential equations;

and

It is observed that the Equation (1) is identically satisfied.

The Boundary condition in Equation (5) is reduced to the corresponding boundary condition for velocity, temperature and concentration fields are as:

at.

where

(Local Temperature Grashof number) =,

(Local Concentration Grashof number) =,

(Permeability parameter) =,

(Schomidt number) =,

(Prandtl number) =

and Unsteadiness parameters

The above equations with boundary condition are solved numerically by using shooting method.

From the engineering point of view, the physical quantities of interest are the local Skin-friction coefficient, local Nusselt number and local Sherwood number respectively defined as

where the wall shear stress, the wall heat flux and the quantity of mass transfer through the unit area of the surface. These are given by

Thus the values of the local skin-friction co-efficient, local Nusselt number and local Sherwood number are proportional to and respectively as Vajravelu et al. [19] .

Systems of ordinary differential equations together with boundary conditions are nonlinear and coupled. First of all, higher order non-linear differential Equations (7) to (9) are converted into simultaneous linear differential equations of first order and these are further transformed into initial value problem solved numerically by applying a standard initial value solver i.e., the Shooting method namely Nachtsheim-Swigert (1965) iteration technique (guessing the missing value) along with Runge-Kutta integration scheme.

In a shooting method, the missing (unspecified) initial condition at the initial point of the interval is assumed, and the differential equation is then integrated numerically as an initial value problem to the terminal point. The accuracy of the assumed missing initial condition is then checked by comparing the calculated value of the dependent variable at the terminal point with its given value. If a difference exists, another value of the missing initial condition must be assumed and the process is repeated. This process is continued until the agreement between the calculated and the given condition for the specified degree of accuracy.

In the process of iteration the Skin friction coefficient, the Nusselt number and the Sherwood number, are proportional to and also evaluated. The numerical result obtained for several selected values of the established parameter are displayed in figures and tables. These figures and tables show that velocity, temperature, concentration, rate of heat transfer and skin friction coefficient are affected significantly with the variations of the considered controlling parameters.

In order to get physical insight into the problem numerical solution have been carried out for different values of the considered thermo physical parameters of controlling the fluid dynamics in the flow regime. The value of the Prandtl number are taken to be Pr = 0.71, 01.00 and 07.00 which correspond to air, electrolyte solution such as salt water and fresh water at 25˚C and 1 atm pressure. The values of Schmidt number are chosen to be Sc = 00.22, 00.62 and 02.62 representing diffusing chemical species of most common interest in air like hydrogen, water vapor , and Propel Benzene respectively at 25˚C and 1 atm pressure. Our attention is based on the positive values of the buoyancy parameter (Gr, Gc) in such way that Grashof number Gr corresponds to the cooling problem and solutal Grashof number Gc indicates the chemical species concentration in the free stream region is less than the concentration at the boundary surface. Here it is considered that the value of Suction parameter is 0.06. The cooling problem is often considered in engineering applications for example in the cooling of electronic components and nuclear reactors. For the purpose of discussing the result, the numerical calculations are presented in the form of non dimensional velocity, temperature and concentration.

In this paper we have studied similarity solutions of unsteady convective boundary layer flow of viscous incompressible fluid along vertical isothermal plate. So far we know, experimental works for the problem are not evaluated ever. We are also expecting that using this solution the present study gives significant result for filling the gap in the existing literature. Numerical calculations are carried out for various values of the dimensionless parameters of the problem. It is concluded that the fluid velocity, temperature and concentration profiles decrease as the unsteadiness parameter increases. Both the Nusselt number and Sherwood number increase whereas Skin friction coefficient decreases with the increasing values of the unsteadiness parameter. The Nusselt number increases as the Prandtl number increases whereas Skin friction coefficient decreases with the increase of the Schmidt number (Sc) and Prandtl number (Pr). It is also revealed that velocity and specie concentration are higher at lower Schmidt number for low Prandtl fluid.

The authors are highly thankful to the authority of Chittagong University of Engineering & Technology (CUET) for providing technical supports during this research work at Simulation Lab, Department of Mathematics, CUET, Chittagong, Bangladesh.

Md. YeakubAli,Mohammed NasirUddin,Md. JasimUddin,N. M. RidwanZahed, (2015) Similarity Solutions of Unsteady Convective Boundary Layer Flow along Isothermal Vertical Plate with Porous Medium. Open Journal of Fluid Dynamics,05,391-406. doi: 10.4236/ojfd.2015.54038