| dc.description.abstract | Fluid flow adjacent to a flexible surface is often encountered in biological systems and bio-technological processes. These flows have been analyzed using models similar to those for the flow past a rigid surface, but the characteristics of the fluid flow could be very different from that past a rigid surface due to the effect of the elasticity of the gel.
The earlier studies on the stability of the flow of a fluid adjacent to a gel at low Reynolds number showed that there could be instability, but at high Reynolds number, the flow was found to be neutrally stable. The objective of this study is to extend the study to finite Reynolds number using numerical techniques in order to resolve these paradoxical results. We consider two systems for this purpose-the gel-fluid system and the fluid-membrane-fluid system.
The gel-fluid system consists of a Newtonian fluid of density ?, viscosity ? and thickness R flowing adjacent to a polymer gel of density ?, modulus of elasticity E, viscosity ?_r and thickness H R. The base flow in the fluid is a plane Couette flow. The fluid-membrane-fluid system consists of two Newtonian fluids of viscosity ? and density ?, one of thickness R and the other of thickness H R, separated by a membrane of infinitesimal thickness along z = 0. The membrane has a surface tension T. The base flow of the fluid above the membrane is a plane Couette flow, and the fluid below it is at rest.
The characteristic equations for the gel-fluid system and fluid-membrane-fluid system are obtained using the boundary conditions at the interface. The characteristic equation for the growth rate is a non-linear equation, so analytical solutions cannot be obtained in general. Numerical solutions are obtained by analytic continuation using the exact solutions at zero Reynolds number as the initial guess. The growth rate depends on the ratio of Reynolds number to dimensionless strain rate Z, the ratio of thickness H and the wave number k. In the case of gel-fluid system, it depends on the ratio of viscosities ?_r = ?_g/?_f as well.
The important results obtained for the gel-fluid system are as follows. For ?_r = 0, it is found that the perturbations become unstable when the Reynolds number is increased beyond a transition value Re, for all Z and k. The critical Reynolds number, which is the minimum of the Re-k curve, increases proportional to E for Z < 1, and it shows a scaling behavior ? Z^p for Z > 1, where 0.75 < p < 0.8. Rec decreases with increase in the ratio of thickness of gel to fluid H, but the scaling behavior remains unchanged. A variation in the ratio of viscosities ?_r qualitatively changes the stability characteristics. For relatively low values of 1 < Z < 10^2, it is found that the transition Reynolds number decreases as ?_r is increased, indicating that an increase in the gel viscosity has a destabilizing effect. For relatively higher values of 10^2 < Z < 10^4, the transition Reynolds number increases as ?_r is increased and goes through a turning point. In this case, perturbations are unstable only when ?_r is less than a maximum value ?_r(max), and there is no instability for ?_r > ?_r(max). A boundary layer is observed above the gel surface whose thickness is O(Re^(-1)) for Z > 10.
The results obtained for the fluid-membrane-fluid system are as follows. An asymptotic analysis based on the wave number k shows a linear dependence of Re_c with k for small k. The result obtained numerically is in agreement with this analysis. The variation of Re_c with Z shows that there is no instability for Z > Z_max. As Z ? 0, Re_c tends to two values: 0 and Re_max. This means that the system cannot be unstable for Re > Re_max. The study of the effect of H on the stability shows that there is no instability for H > 2.748, and unstable modes exist only for H < 2.748. | |
