Abstract
The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.
Highlights
Theoretical study of a fluid flow problem with variable viscosity is a system containing equation of momentum, energy and continuity
The momentum equations for the motion of a fluid element are the Navier-Stokes equations (NSE) having capacity to incorporate all forces in the righthand side
Letting ξ be the angle between the tangents to the curves ψ = const. and φ = const. at a point P(x, y), streamline pattern equation (11) and applying differential geometric technique of [19] it is easy to show that the fundamental equations are following
Summary
Theoretical study of a fluid flow problem with variable viscosity is a system containing equation of momentum, energy and continuity. Mushtaq Ahmed: A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number Via Von-Mises Coordinates i , j , k∈{1, 2} , x1 = x , x2 = y , F = ( F1(x, y), F2 (x, y)) , v1 = u v2 = v , the equations (1-3) reduce to following system of equations ux + vy = 0. The solution of the plane equation of continuity (5) provides astream functionψ =ψ (x, y) , such thatψ y x =ψ x y , and This discourse applies successive coordinate transformations technique for solution of plane momentum and energy equations (5-7). It transforms equations firstly into a curvilinear net (φ,ψ ) where the coordinate curves ψ = const.
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