Biorthogonal stretching and shearing of an impermeable surface in a uniformly rotating fluid system

Abstract

The flow induced by an impermeable flat surface executing orthogonal stretching and orthogonal shearing in a rotating fluid system is investigated. Both the stretching and shearing are linear in the coordinates. An exact similarity reduction of the Navier–Stokes equations gives rise to a pair of nonlinearly-coupled ordinary differential equations governed by three parameters. In this study we set one parameter and analyze the problem which leads to flow for an impermeable surface with shearing and stretching due to velocity u along the x-axis of equal strength a while the shearing and stretching due to velocity v along the y-axis of equal strength b. These solutions depend on two parameters—a Coriolis (rotation) parameter $\sigma = \Omega /a$ and a stretching/shearing ratio $\lambda =b/a$. A symmetry in solutions is found for $\lambda = 1$. The exact solution for $\sigma = 0$ and the asymptotic behavior of solutions for $|\sigma | \rightarrow \infty$ are determined and compared with numerical results. Oscillatory solutions are found whose strength increases with increasing values of $|\sigma |$. It is shown that these solutions tend to the well-known Ekman solution as $|\sigma | \rightarrow \infty$.