Benchmark solution for the stability of plane Couette flow with net throughflow

I. S. Shivakumara,B. M. Shankar

doi:10.1038/s41598-021-90552-5

I. S. Shivakumara, B. M. Shankar

Open Access

https://doi.org/10.1038/s41598-021-90552-5

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Journal: Scientific Reports	Publication Date: May 25, 2021
Citations: 2	License type: open-access

Affiliation: Bangalore University, PES University

Abstract

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

Highlights

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow
The stability of plane Poiseuille flow (PPF) where both the plates are at rest, plane Couette flow (PCF) where both upper and lower plates are moving in opposite directions and plane Couette-Poiseuille flow (PCPF) where the lower plate is at rest and the upper plate is in motion in its own plane have been the subject of many investigations from the past many decades
Attempts to solve the Orr-Sommerfeld equation for the flow associated with PCF led to the prediction of stability for all values of Reynolds numbers, in apparent disagreement with the experiment, the prediction is correct for infinitesimal disturbances

Summary

Introduction

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. It is natural to assume that the PCF in the presence of throughflow will be stable against small disturbances for all the values of Reynolds number.

Results

Conclusion