Numerical study of boundary stresses on Jeffery-Hamel flow subject to Soret/Dufour effects

Abstract

We have mathematically modeled the problem of convective Jeffery-Hamel flow in the presence of thermo-diffusion and diffusion-thermo effects. The flow is considered to be the steady, incompressible and unidirectional by assuming the velocity only in radial direction. We have also modeled the boundary conditions representing the boundary stresses and assumed that these stresses apply only radially. Then we analyzed the effects of these stresses on the problem. The boundary stresses are defined through traction boundary conditions which are of Robin type. The flow is assumed in the domain [[Formula: see text]] and the constant flow rate condition is applied instead of taking symmetry of the channel. The governing equations are nonlinear coupled partial differential equations. They become a system of nonlinear coupled ordinary differential equations by adopting suitable transformations. Therefore, they are handled numerically. For this purpose, we have developed a scheme using shooting method to solve nonlinear coupled differential equations representing the flow problem and shown the solution procedure via a flow chart. In order to obtain the numerical results, we have taken different values of parameters. Since we have considered the problem of Newtonian fluid, therefore, we considered the values [Formula: see text] for air, carbon disulfide and water respectively at 25°C. Whereas the values of Schmidt number are considered as [Formula: see text] for hydrogen, water vapor, and ammonia respectively at 25°C. The numerical solution is validated by comparing the results with those published in literature. The numerical results of present work show that there occurs no-slip for wedge angle of [Formula: see text] by fixing the other parameters whereas slip is achieved in some other situations. It is noticed that the flow reversal occurs for convergent as well as divergent flow. All boundary layers are affected greatly due to presence of the boundary stresses and the hydrodynamic boundary layer that disappears in the absence of boundary stresses. The freedom in constraint of channel symmetry leads to the occurrence of symmetric as well as asymmetric flow. An interesting theoretical result of opposite behaviors of temperature and concentration profiles for convergent and divergent flows is observed in case of variation of all of the governing parameters. It is also noticed that the simultaneous increase of Soret number and decrease of Dufour number results in occurrence of opposite behavior of temperature and concentration profiles for divergent flow. However, it results in similar behavior of these profiles. The results show that the boundary layer completely disappears in the absence of boundary stresses.

Investigation of Entropy Production with Thermal Analysis under Soret and Dufour Effects in MHD Flow between Convergent and Divergent Channels

“…With all highlighted assumption the mathematical model for the physical problem can be represented in the form of following partial differential equation: − 1 ρ f r ∂ p ∂ θ = 2 v f r 2 [ 1 + Π 2 true( 2 true( ∂ U normalr ∂ r true) 2 + 1 r 2 true( ∂ U normalr ∂ θ true) 2 + 2 U normalr 2 r 2 true) ] 0.5 false( n − 1 false) true( ∂ U normalr ∂ θ true) + v f Π 2 ( n − 1 ) 2 r …”

“…With all highlighted assumption the mathematical model for the physical problem can be represented in the form of following partial differential equation: − …”