Group theoretical analysis and invariant solutions for unsteady flow of a fourth-grade fluid over an infinite plate undergoing impulsive motion in a darcy porous medium

In this study, an incompressible time-dependent flow of a fourth-grade fluid in a porous half space is investigated. The flow is generated due to the motion of the flat rigid plate in its own plane with an impulsive velocity. The partial differential equation governing the motion is reduced to ordinary differential equations by means of the Lie group theoretic analysis. A complete group analysis is performed for the governing nonlinear partial differential equation to deduce all possible Lie point symmetries. One-dimensional optimal systems of subalgebras are also obtained, which give all possibilities for classifying meaningful solutions in using the Lie group analysis. The conditional symmetry approach is also utilised to solve the governing model. Various new classes of group-invariant solutions are developed for the model problem. Travelling wave solutions, steady-state solution, and conditional symmetry solutions are obtained as closed-form exponential functions. The influence of pertinent parameters on the fluid motion is graphically underlined and discussed.