UNSTEADY FLOW PAST AN ACCELERATED VERTICAL PLATE WITH VARIABLE TEMPERATURE IN PRESENCE OF THERMAL STRATIFICATION AND CHEMICAL REACTION

This work aims to investigate the effect of thermal stratification on fluid flow past an accelerated vertical plate in the presence of first order chemical reaction. The dimensionless unsteady coupled linear governing equations are solved by Laplace transform technique for the case when the Prandtl number is unity. The important conclusions made in this study the effect of thermal stratification is compared with the scenario in which there was no stratification. The results of numerical computations for different sets of physical parameters, such as velocity, temperature, concentration, skin-friction, Nusselt number and Sherwood number are displayed graphically. It is shown that the steady state is attained more quickly when the flow is stratified.


INTRODUCTION
Thermal stratification is a natural phenomenon that may be seen in many natural systems, such as lakes and seas.The presence of chemical reactions might further complicate the flow's dynamics.In this paper, we investigate how flow dynamics and interactions with chemical processes are impacted by thermal stratification.The applications of this study are wide.It may be used to build more efficient chemical reactors and heat exchangers.It may also be used to look at how the performance of cooling systems in electrical equipment is affected by thermal stratification.[1] investigated the influence of a chemical reaction on the behavior of an unsteady flow through an accelerating vertical plate, where the mass transfer was variable and without considering stratification.The purpose of this research is to determine how fluid flow past an accelerated vertical plate impacts the interaction between thermal stratification and chemical reaction.[2] and [3] investigated the unsteady flow of a thermally stratified fluid past a vertically accelerated plate under a variety of conditions.Researchers [4], [5], and [6] have investigated steady flows in a stable stratified fluid with a focus on infinite vertical plates.[7] and [8] both investigated buoyancy-driven flows in a stratified fluid.The interaction between thermal stratification and chemical reaction to change MHD flow for vertical stretching surfaces has been studied by researchers [9] and [10].These two phenomena were also investigated by [11], who investigated the impact of non-Newtonian fluid flow in a porous medium.The unsteady MHD flow past an accelerating vertical plate with a constant heat flux and ramped plate temperature respectively was researched by [12] and [13].
In this paper, we derived the special solutions for Sc = 1 and classical solutions for the case S = 0 (without stratification).These solutions are compared with the primary solutions, and graphs are used to demonstrate the differences.The impacts of physical parameters on velocity, temperature, and concentration profiles, including the stratification parameter (S), thermal Grashof number (Gr), mass Grashof number (Gc), Schimdt number (Sc) and Chemical Reaction Parameter (K), are explored and presented in graphs.The results of this research have a wide range of applications in a variety of industries and chemical factories.

MATHEMATICAL ANALYSIS
We consider a fluid that is stratified, viscous, and in-compressible, traveling along an accelerating vertical plate with first-order chemical reaction present.As can be seen in fig. 1, we use a coordinate system in which the y ′ axis is perpendicular to the plate and the x ′ axis is taken vertically upward along the plate to study the flow situation.The starting temperature T ′ ∞ and initial fluid concentration C ′ ∞ of the plate and fluid are the same.At time t ′ > 0, the plate is subjected to an impulsive constant acceleration u 0 , and the concentration and temperature of the plate are increased to C ′ w and T ′ w , respectively.All flow variables are independent of x ′ EEJP.3(2023)Nitul Kalita, et al. and only affected by y ′ and t ′ since the plate has an infinite length.As a result, we are left with a flow that is only one dimension and has one non-zero vertical velocity component, u ′ .The Boussinesqs' approximation is then used to represent the equations for motion, energy, and concentration as follows: with the following initial and boundary Conditions: where, α is the thermal diffusivity, β is the volumetric coefficient of thermal expansion, β * is the volumetric coefficient of expansion with concentration, η is the similarity parameter, ν is the kinematic viscosity, g is the acceleration due to gravity, D is the mass diffusion coefficient.Also, γ = Cp denotes the thermal stratification parameter and dT ′ ∞ dx ′ denotes the vertical temperature convection known as thermal stratification.In addition, g Cp represents the rate of reversible work done on fluid particles by compression, often known as work of compression.The variable (γ) will be referred to as the thermal stratification parameter in our research because the compression work is relatively minimal.For the purpose of testing computational methods, compression work is kept as an additive to thermal stratification.and we provide non-dimensional quantities in the following: where, A = is the constant.
The non-dimensional forms of the equations ( 1)-( 3) are given by Non-dimensional form of initial and boundary Conditions are:

METHOD OF SOLUTION
The non-dimensional governing equations ( 4)-( 6) with boundary conditions (7) are solved using Laplace's transform method for P r = 1.Hence, the expressions for concentration, velocity and temperature with the help of [14] and [15] are given by where, ) ) ) ) Also, f i 's are inverse Laplace's transforms given by We separate the complex arguments of the error function contained in the previous expressions into real and imaginary parts using the formulas provided by [15].EEJP.3(2023)Nitul Kalita, et al.

SPECIAL CASE [FOR SC=1]
We came up with answers for the special case where Sc = 1.Hence, the solutions for the special case are as follows:

CLASSICAL CASE (S=0)
We derived solutions for the classical case of no thermal stratification (S = 0).We want to compare the results of the fluid with thermal stratification to the case with no stratification.Hence, the corresponding solutions for the classical case is given by :

Skin-Friction
The non-dimensional Skin-Friction, which is determined as shear stress on the surface, is obtained by The solution for the Skin-Friction is calculated from the solution of Velocity profile U , represented by ( 9), as follows:  in which stratification does not take place.As S, Sc and K grow, the fluid's velocity decreases, whereas an increase in Gr, Gc increases it.This research is more practical than earlier ones because it applies thermal stratification, which lowers velocity and temperature in comparison to the classical scenario (S = 0).The temperature decreases when K and Gc decreases, and it increases when S, Gr increases.Thermal stratification increases the recurrence of oscillations in the skin friction and Nusselt number.

Figure 1 .
Figure 1.Physical Model and coordinate system