Studies in hydrodynamic and electrohydrodynamic instabilities of fluid-solid and fluid- fluid interfaces

Abstract

The study of hydrodynamic and electrohydrodynamic stability is an important branch of fluid mechanics and has applications in various areas of chemical engineering and biophysics. The detailed study of hydrodynamic stability dates back to the 19th-century experiments of Reynolds, who showed that laminar flow in a rigid pipe undergoes a transition to a more complicated turbulent flow when a dimensionless parameter, now called the Reynolds number, exceeds a certain critical value. Almost at the same time, Lord Rayleigh showed that a charged drop of fluid becomes unstable when the net charge on the drop exceeds a critical value, and this discovery laid the foundations of electrohydrodynamics. Since these two pioneering studies, there have been a large number of theoretical and experimental investigations devoted to understanding these subjects, and work in the last century has put the theory on a strong footing.

Hydrodynamic Stability The thesis deals with two problems in hydrodynamic stability: 1. Stability of Flow Over Flexible Surfaces The stability of flow over flexible surfaces is extremely important in technological applications like membrane reactors, aerospace industry, and is ubiquitous in biological systems. Earlier studies of fluid flow past membranes had been restricted to incompressible membranes. Here, the stability of linear shear flow of a Newtonian fluid past a flexible membrane is analyzed in the limit of low Reynolds number as well as in the intermediate Reynolds number regime for compressible viscoelastic membrane models. The objective is to examine the importance of tangential motion in the membrane on the stability characteristics of shear flow. The membrane is assumed to be a two-dimensional compressible viscoelastic sheet of infinitesimal thickness, in which the constitutive relation for shear stress contains:

An elastic part that depends on the local displacement field A viscous component that depends on the local velocity in the membrane

Key findings:

Stability characteristics at low Reynolds number depend crucially on tangential motion in the membrane. Flow is stable in the absence of tangential motion but destabilized when tangential motion is present, even without fluid inertia. The non-dimensional velocity (?t) required for unstable fluctuations scales as k² in the viscoelastic membrane type of wall (?t ~ k²) in the limit k ? 0. Numerical extension to intermediate Reynolds number shows instability only in a finite range of Reynolds number; flow becomes stable at high Reynolds number. Weakly nonlinear analysis at zero Reynolds number shows Landau constant is negative, and perturbations are supercritically stable in the k ? 0 limit.

2. Stability of Oscillatory Flows Over Compliant Surfaces Two types of compliant surfaces are studied:

Spring-backed wall model (permits tangential motion) Incompressible viscoelastic gel model

Stability is determined using Floquet analysis, examining amplitude of perturbations at intervals separated by one time period. Key findings:

Oscillatory flows past both models exhibit instability at zero Reynolds number. Transition amplitude of oscillatory velocity increases with oscillation frequency. Minimum transition amplitude occurs at zero wavenumber for spring-backed plate model, but at finite wavenumber for viscoelastic gel model. For spring-backed plate model, instability due to steady mean flow and oscillatory instability reinforce each other; regions of instability mapped on (A? – A) plane (steady strain rate amplitude vs oscillatory strain rate amplitude). For viscoelastic gel model, instability depends strongly on gel viscosity (?g). Experimental studies confirm theoretical predictions.

Electrohydrodynamic Stability Two problems are analyzed: 1. Nonlinear Interactions on Shape Fluctuations of Charged Membranes

Linear instability occurs when surface tension drops below a critical value for a given charge density. Displacement of membrane surface causes fluctuation in counterion density, resulting in additional Maxwell normal stress opposite to surface tension. Nonlinear analysis shows:

At low charge densities, nonlinear interactions saturate growth of perturbations ? new steady state. At high charge densities, nonlinear terms destabilize perturbations ? subcritical bifurcation.

Significant difference between Debye–Hückel and Poisson–Boltzmann predictions at high charge densities.

2. Stability of Interface Between Two Dielectric Fluids

Studied under normal electric field using linear and weakly nonlinear analysis, thin film analysis, and boundary integral computations at zero Reynolds number for long and short waves. Both perfect dielectrics and leaky dielectrics considered. Instability depends on ratio of dielectric constants, electrical conductivities, viscosities, fluid thicknesses, and surface tension. Long waves: instability always subcritical; nonlinear evolution depends on dielectric and conductivity ratios. Shear flow stabilizes long-wave instability and admits traveling waves. Short waves: supercritical bifurcation for small range of wavenumbers and dielectric ratios even without flow; shear flow stabilizes short-wave instability and changes subcritical bifurcation to supercritical. Fluid–viscoelastic gel systems show strong coupling between electrohydrodynamic instability and flow-induced surface instability.