Numerical solution of a viscous incompressible fluid along a semi-infinite vertical plate with large suction

Abstract:

Velocity profile and temperature distribution of a viscous incompressible fluid along a semi-infinite vertical plate with large suction is investigated in this work. The governing equations are solved numerically using perturbation technique. The flow phenomenon has been characterized with the help of flow parameters such as Suction parameterAbbildung in dieser Leseprobe nicht enthalten, Prandtl numberAbbildung in dieser Leseprobe nicht enthaltenand Eckart numberAbbildung in dieser Leseprobe nicht enthalten.The effects of these parameters on the velocity field and temperature distribution have been studied and the results are presented graphically and discussed quantitatively. This type of problem is significantly relevant geophysical and astrophysical studies.

Keywords: Heat transfer, mass transfer, accelerated plate, suction, perturbation technique

C hapter One

Introduction

Boundary layer flows on moving solid surfaces are frequently encounted in transport processes occuring both in nature and industry. The heat transfer flows play a decisive role in many engineering applications such as distillation, condensation, evaporation, rectification and absorption of a fluid as well as in fluids condensing or boiling at a solid surface. The heat transfer processes are of great interest in power engineering, metallurgy, astrophysics and geophysics. Many researchers have studied the problems of free convection boundary layer flow over or on a various types of shapes. Pop and Soundalgekar (1980) have investigated the free convection flow past an accelerated infinite plate. Raptis et al. (1987) have studied the unsteady free convective flow through a porous medium adjacent to a semi-infinite vertical plate using finite difference scheme. Singh and Soundalgekar (1990) have investigated the problem of transient free convection in cold water past an infinite vertical porous plate. Vajravelu and Sastry (1978) studied about free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. Free convection boundary layer flow of a non-Newtonian fluid along a vertical wavy wall was considered by Kumari et al. (1997). Chandran et al. (1998) have discussed the unsteady free convection flow with heat flux and accelerated boundary motion. The significance of suction for the boundary layer control has been well recognized. It is often necessary to prevent (or postpone) separation of the boundary layer to reduce drag and attain high lift values. Hence Pop and Watanabe (1992) studied the effect of suction or injectionon boundary layer flow and heat transfer. Hence the main aim of this paper is to find the transfer of heat and velocity of a viscous incompressible fluid along a thermal boundary layer with large suction

Chapter Two

Numerical Solution of a viscous incompressible fluid along a semi-infinite vertical plate with Large Suction

2.1 Mathematical Model of Flow

By introducing Cartesian Co-ordinate system, Abbildung in dieser Leseprobe nicht enthalten-axis is chosen along the plate in the direction of the flow and Abbildung in dieser Leseprobe nicht enthalten-axis –is normal to it. Initially it is considered that the plate as well as the fluid is remained at same temperatureAbbildung in dieser Leseprobe nicht enthalten. Also it is considered that the fluid and the plate is at rest after that the plate is to be moving with a constant velocity Abbildung in dieser Leseprobe nicht enthalten is its own plate instantaneously at timeAbbildung in dieser Leseprobe nicht enthalten, the species temperature of the plate is raised to Abbildung in dieser Leseprobe nicht enthalten.Where Abbildung in dieser Leseprobe nicht enthaltenis species temperature at the wall of plate and Abbildung in dieser Leseprobe nicht enthalten be the temperature species far away from the plate. The physical model of the study is shown in Figure 2.1.1 .

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Figure 2.1.1: The physical model and coordinate system

Within the framework of the above stated assumptions with references to the generalized equations described before the equation relevant to the transient two dimensional problem are governed by the following system of coupled non-linear differential equations.

Continuity Equation

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(2.1.1)

Momentum Equation

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(2.1.2)

Energy Equation

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(2.1.3)

and the boundary conditions are

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where Abbildung in dieser Leseprobe nicht enthalten be the Cartesian coordinate system. Abbildung in dieser Leseprobe nicht enthaltenare illustration not visible in this excerpt component of flow velocity respectively is the local acceleration due to gravity, illustration not visible in this excerpt is the kinematic viscosity, Abbildung in dieser Leseprobe nicht enthalten be the density of the fluid,illustration not visible in this excerpt be the thermal conductivity, illustration not visible in this excerpt be the specific heat at the constant pressure (Kishore et al., 2010).

2.2 Mathematical Formulation

Introducing the stream functionAbbildung in dieser Leseprobe nicht enthalten. Such that,

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(2.2.1)

where illustration not visible in this excerpt and illustration not visible in this excerpt

Againillustration not visible in this excerpt

(2.2.2)

and illustration not visible in this excerpt

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(2.2.3)

Again

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(2.2.4)

Now differentiating (2.2.3) with respect to x

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(2.2.5)

Again, differentiating (2.2.3) with respect to y

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(2.2.6)

Differentiating (2.2.6) with respect to y

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(2.2.7)

Now, multiplying (2.2.3) and (2.2.5)

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(2.2.8)

Again multiplying (2.2.4) and (2.2.6)

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(2.2.9)

But,

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(2.2.10)

Differentiating this with respect to x

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(2.2.11)

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