Stability of pressure-driven flow in a deformable neo-Hookean channel

Abstract

The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (<1) for highly deformable walls. There are other types of instabilities that exist only due to wall deformability. In the absence of inertia (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

A multifold reduction in the transition Reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall

“…We used a two values of N = 50 and N = 60 to get the convergent results discussed in § 5. The linear stability calculations were first validated with the results of Gaurav & Shankar (2010) for the parabolic flow in a channel with flat walls, and found quantitative agreement for the eigenvalues for that problem. The same eigenvalue search procedure was used for the velocity profiles predicted by the CFD simulations in our linear stability analysis.…”

“…The method of analysis is identical to that of Gaurav & Shankar (2010), with two important differences. First, we have a rigid wall at the boundary at y = h 0 + h , in contrast to Gaurav & Shankar (2010) who had a flexible surface.…”

“…First, we have a rigid wall at the boundary at y = h 0 + h , in contrast to Gaurav & Shankar (2010) who had a flexible surface. Secondly, we do not consider a parabolic flow, as was done by Gaurav and Shankar, but we derive equations for a general flow profile and pressure variation, assuming only that the velocity profile is locally parallel.…”

“…Our equations are restricted to the simple case where there is no solid dissipation, in order to obtain the lower bound for the transition Reynolds number. The extension to include solid dissipation is discussed by Gaurav & Shankar (2010).…”

Manipulation of interfacial instabilities by using a soft, deformable solid layer

“…In this section, we extend the above low-k results to arbitrary wavenumbers to ensure whether the predicted suppression in long wave limit holds for finite and high wavenumber perturbations as well. Furthermore, flow past deformable solid surface (involving only a fluid-solid interface) could become unstable on increasing the deformability in the absence of inertia (Kumaran et al 1994;Gkanis & Kumar 2003), and several additional unstable fluid-solid modes proliferate when inertia is present (Chokshi & Kumaran 2008;Gaurav & Shankar 2009, 2010b. All these fluid-solid unstable modes are not captured by the low-k analysis presented in the previous section.…”