Analytical Solution of a Non-isothermal Flow in Cylindrical Geometry

Abstract

This study proposes analytical solution to the problem of transport in a Newtonian fluid within a cylindrical domain. The flow is assumed to be dominated along the channel axis, and is taken to be axi-symmetric. No-slip boundary condition is considered for velocity while the temperature and concentration have Dirichlet boundary values. The resulting problem is transformed into a set of non-trivial variable coefficient differential equations in a cylindrical geometry. By adopting the series solution method of Frobenius, the closed-form analytical solutions are derived for the flow variables. We conduct an analysis of the derived model, and showed that, indeed, the flow variables are axi-symmetric. We also state and prove another theorem to show that the derived concentration model is positivity preserving – meaning that it yields positive concentration - provided the boundary value is non-negative. Finally, we present graphical results for the flow variables and discuss the effect of the relevant flow parameters. The results showed that (i) an increase in the cooling parameter, reduces the fluid velocity, (ii) the temperature decreases as the cooling parameter increases and (iii) an increase in the injection parameter, leads to increase in the concentration.

Laplace Transforms Method on a System of Differential Equations for Non-isothermal Chemically Reactive Flow

“…It was observed that an increase in pressure gradient (steady or pulsatile) results in an increase in velocity profile of the fluid in a cylindrical channel. This is in agreement with Nwaigwe and Amadi [14].…”

“…This is because as the magnetic field increases, a Lorentz force is introduced and this inhibits the flow of the fluid. This is in line with Nwaigwe and Amadi [14], Omamoke and Amos [17], Bunonyo and Amos [16], Baoku et al [18]. Fig.…”

“…With the given boundary conditions (5) (6) Equation ( 1) to (4) with equation ( 5) and ( 6) as the boundary conditions, we present the non-dimensional form by Nwaigwe and Amadi [14] to obtain the governing equation given as:…”

“…Subject to: (10) 3 Methods of Solution Equations ( 7) -( 9) are coupled nonlinear partial differential equations and are reduced to a set of ordinary differential equations with the expressions for velocity ( ), temperature ( ) and concentration ( ) in dimensionless form as follows (11) (12) (13) (14) Substituting equation ( 11) to (14) in the set of equations ( 7), ( 8) and ( 9) and equating nonharmonic and harmonic terms and neglecting the higher order terms of 0( 2 ), the following set of ordinary differential equations are obtained. (15) (16) (17) By Splitting, Equation ( 15) to ( 17) is reduced to Equations (18a) to (20b)…”

“…Section 5 gives the conclusion of the paper. This paper is aimed at extending the work of Nwaigwe and Amadi [14] by solving the system using the method of Laplace transform.…”