Stability Analysis of a Gravity Driven Miscible Two-fluid Flow: Role of Wall Slip
A linear stability analysis of a gravity-driven miscible two-fluid flow with matched density and different viscosities, containing a thin mixed layer between the two fluids, down a slippery inclined plane is considered. Results reveal the stabilizing effects on the surface (S), the overlap (O1 and O2) and the TS (not shown) modes that exist for a configuration with more viscous fluid adjacent to the free surface under overlap conditions. There is a possibility of destabilizing the overlap mode (O1) when the mixed layer is very close to the slippery wall, indicating the dual role played by the wall slip. The stabilizing characteristics of slip can be favourably used to suppress the nonlinear breakdown that may happen due to the coexistence of the unstable modes in a flow over a substrate with no slip.
Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane
A thin film of a power-law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy's law together with Beavers-Joseph boundary condition for a general power-law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as 'permanent waves'. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology. © 2010 Elsevier B.V.
Interfacial instability in pressure-driven core-annular pipe flow of a Newtonian and a Herschel–Bulkley fluid
The linear stability characteristics of pressure-driven core-annular flow of a Newtonian core fluid and a Herschel–Bulkley annular fluid is investigated. The fluids are assumed to have the same density and separated by a sharp interface. The modified Orr–Sommerfeld equations for each layer are derived and solved using an efficient spectral collocation method considering a configuration without any unyielded region. The effect of various dimensionless parameters, such as the Bingham number (Bn), the flow index (n), the interface radius (R0) and the inverse capillary number (Γ) on the instability characteristics of the flow is investigated, and an energy budget analysis is conducted to explain the physical mechanism of the instability observed. We found that axisymmetric mode is the most dominant unstable mode for the interfacial flow configuration considered in the present work, which is in contrast to miscible core-annular flows. It is observed that increasing Bn has a non-monotonic effect on the growth rate of the axisymmetric mode, and two dominant modes appear at high Bn. We found that increasing the thickness of the core fluid increases the bandwidth of the unstable wavenumbers and destabilises the short waves; however, displays a non-monotonic trend in the growth rate curves. The instability behaviour observed for different sets of parameters are investigated by conducting an energy budget analysis and analysing the disturbance eigenfunctions and the basic velocity profiles.