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Linear stability of viscous flow induced by surface stretching

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Abstract

The stability of the two-dimensional flow in a semi-infinite fluid induced by the stretching of a planar surface is investigated by a normal-mode linear stability analysis. The base flow is described by Crane’s exact analytical solution of the Navier–Stokes equation. A finite-difference method and a shooting method are implemented to solve the relevant eigenvalue problem for three-dimensional perturbations that vary sinusoidally with arbitrary wave number in the spanwise direction normal to the plane of the flow. The Crane flow is found to be linearly stable, with the least-damped mode arising in the limit of two-dimensional perturbations. Eigenfunctions for the disturbance velocity are presented and discussed for selected values of the transverse wave number. The numerical results for the dispersion relation corresponding to the least-damped mode for each transverse wave number are described by a simple analytical expression. The results of the stability analysis are contextualized by superposing a planar stagnation-point flow toward the stretching surface. As the ratio of the rate of elongation of the stagnation-point flow to the rate of stretching of the surface tends to zero, the dispersion curves smoothly asymptote to those found for the Crane flow. The results for planar stagnation-point flow toward a stationary surface are recovered in the opposite limit. The analysis is extended to account for suction or injection through a porous surface undergoing in-plane stretching.

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  1. Department of Chemical Engineering, University of Massachusetts, Amherst, MA, 01003, USA

    J. M. Davis & C. Pozrikidis

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  1. J. M. Davis
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  2. C. Pozrikidis
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Correspondence to J. M. Davis.

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Davis, J.M., Pozrikidis, C. Linear stability of viscous flow induced by surface stretching. Arch Appl Mech 84, 985–998 (2014). https://doi.org/10.1007/s00419-014-0843-0

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