Flows between orthogonally stretching parallel plates

Abstract

Navier-Stokes equilibrium solutions of a viscous fluid confined between two infinite parallel plates that can independently stretch or shrink in orthogonal directions are studied. It is assumed that the admissible solutions satisfy spatial self-similarity in the stretching or shrinking perpendicular coordinates. The nonlinear steady boundary-value problem is discretized using a spectral Legendre method, and equilibrium solutions are found and tracked in the two-dimensional parameter space by means of pseudoarclength continuation Newton-Krylov schemes. Different families of solutions have been identified, some of which are two-dimensional and correspond to the classical Wang & Wu self-similar flows arising in a plane channel with one stretching-shrinking wall; C. A. Wang and T. C. Wu, Comput. Math. Applic., 30, 1-16 (1995). However, a large variety of three-dimensional solutions have also been found, even for low stretching or shrinking rates. When slightly increasing those rates, some of these solutions disappear at saddle-node bifurcations. By contrast, when both plates are simultaneously stretching or shrinking at higher rates, a wide variety of new families of equilibria are created-annihilated in the neighbourhood of cuspidal codimension-2 bifurcation points. This behaviour has similarities with the one observed in other planar and cylindrical self-similar flows.

Stability of the flow due to a linear stretching sheet

“…Having validated our numerical scheme we begin by solving (4) subject to (6) for fixed values of β whilst cycling through a range of values of ω and R in order to determine points where α i ≤ 0. A point is deemed to be neutrally stable if α i = 0.…”

“…Given that these flows exhibit fluid entrainment at the boundary-layer edge, and that the surface stretching is linear in nature, an association can be made between these flows and the three-dimensional flow induced by a rotating disk. Very recently the ideas of Crane and Wang have been extended by [6] to include the family of flows associated with the independent stretching or shrinking of two infinite parallel plates.…”

Phase-locking flows between orthogonally stretching parallel plates

“…In this section we closely follow Ayats et al (2021), summarizing the formulation and adding explicitly the time dependence, necessary for the linear stability analysis and computation of time dependent solutions.…”

“…becoming a good approximation of the exact solution for small Reynolds numbers, as seen in Ayats et al (2021). Henceforth in this study, we will characterize the dynamical properties of the flow by monitoring the time evolution of the three quantities…”

“…In Ayats et al (2021) the generalized problem of a viscous fluid confined between two orthogonally stretching-shrinking parallel plates was studied numerically. In that work, five different families of steady self-similar flows were identified and explored for a wide range of independent orthogonal acceleration rates of the two walls.…”