Approximately invariant solutions of creeping flow equations

TL;DR

This paper develops a method to find approximately invariant solutions of creeping flow equations for second grade fluids, demonstrating its effectiveness through a boundary value problem in a physically relevant context.

Contribution

It introduces a new approach to compute first order approximate Lie symmetries for non-Newtonian fluid flow equations and explicitly constructs invariant solutions.

Findings

Successfully computed approximate Lie symmetries

Derived explicit invariant solutions for the flow equations

Validated the method with a boundary value problem

Abstract

In this paper, the steady creeping flow equations of a second grade fluid in cartesian coordinates are considered; the equations involve a small parameter related to the dimensionless non--Newtonian coefficient. According to a recently introduced approach, the first order approximate Lie symmetries of the equations are computed, some classes of approximately invariant solutions are explicitly determined, and a boundary value problem is analyzed. The main aim of the paper is methodological, and the considered mechanical model is used to test the reliability of the procedure in a physically important application.