Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability

Abstract

The non-linear equations of motion describing the laminar, isothermal and incompressible flow in a rectangular domain bounded by two weakly permeable, moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions, are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which given partial differential equations are invariant, then, the determining equations are derived. The determining equations are a set of linear differential equations, the solution of which gives the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equation may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables in the system. Effect of the permeation Reynolds number R e and the dimensionless wall dilation rate a on self-axial velocity have been studied both analytically and numerically and the results are plotted.

Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall

“…Similar to Dauenhauer and Majdalani [11], Uchida and Aoki [20], and Boutros et al [13,14], we substitute Eqs. (6) and (7) into governing equations and consider the similarity solutions with respect to space and time, then the following ordinary equations can be obtained:…”

“…Furthermore, Majdalani, Zhou and Dawson [12] also obtained an asymptotic solution for the flow in a porous channel, with slowly expanding or contracting walls, by considering the permeation Reynolds number and expansion ratio as two small parameters. Boutros et al [13,14] also discussed the flow through an expanding porous channel or pipe using the Lie group method and obtained the analytical solution with the perturbation method. Recently Si et al [15] also investigated the micropolar fluid in a porous deforming channel and discussed the effects of the micropolar parameter and the expansion ratio on the velocity and microrotation distribution.…”

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

“…In the interim, Dauenhauer and Majdalani [56][57][58] formulated the corresponding equation and numerical solution for the planar flow analog. The analytical solutions promoted through these efforts would later receive attention in follow-up studies aimed at devising asymptotic approximations over various ranges of the control parameters or at reconstructing the solution using alternative techniques such as the Lie-group theory [59,60] or the HomotopyAnalysis Method [61][62][63].…”

A Note on Some Solutions of Copper-Water (Cu-Water) Nanofluids in a Channel with Slowly Expanding or Contracting Walls with Heat Transfer

“…Xinhui et al [12,13] analyzed the flow of non-Newtonian fluid in a porous channel with expanding or contracting walls. Many researchers have investigated the fluid flow behavior between expanding or contracting walls analytically as well as numerically under the various fluid flow conditions [14][15][16][17][18][19][20].…”