Journal of Fluid Mechanics

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Linear and nonlinear spatio-temporal instability in laminar two-layer flows

P. VALLURIa1 p1, L. Ó NÁRAIGHa1 p2, H. DINGa1 p3 and P. D. M. SPELTa1 c1

a1 Department of Chemical Engineering, Imperial College London SW7 2AZ, UK

Abstract

The linear and nonlinear spatio-temporal stability of an interface separating two Newtonian fluids in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. A linear, Orr–Sommerfeld-type analysis shows that most of such systems are unstable. The transition to an absolutely unstable regime is investigated, and is shown to occur in an intermediate range of Reynolds numbers and ratios of the thicknesses of the two layers, for near-density matched fluids with a viscosity contrast. A critical Reynolds number is found for transition from convective to absolute instability of relatively thin films. Results obtained from direct numerical simulations (DNSs) of the Navier–Stokes equations for long channels using a diffuse-interface method elucidate that waves generated by random noise at the inlet show that, near the inlet, waves are formed and amplified strongly, leading to ligament formation. Successive waves coalesce with each other further downstream, resulting in longer larger-amplitude waves further downstream. In the linearly absolute regime, the characteristics of the spatially growing wave near the inlet agree with that of the saddle point as predicted by the linear theory. The transition point from a convective to an absolute regime predicted by linear theory is also in agreement with a sharp change in the value of a healing length obtained from the DNSs.

(Received May 09 2009)

(Revised March 08 2010)

(Accepted March 08 2010)

(Online publication May 21 2010)

Correspondence:

c1 Email address for correspondence: p.spelt@imperial.ac.uk

p1 Present address: School of Engineering, The University of Edinburgh, Kings Buildings, Edinburgh EH9 3JL, UK

p2 Present address: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

p3 Present address: Department of Chemical Engineering, University of California, Santa Barbara, CA 93106-5080, USA

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