A thin film on a porous substrate: A two-sided model, dynamics and stability
A two-sided model (TLM) is employed to investigate the dynamics and stability of a thin film of Newtonian fluid overlying a porous substrate; the model consists of a free fluid interfacing a Brinkman-type porous transition layer, which overlies a porous medium described by the Darcy equation. The model explicitly describes the transition flow at the top of the porous medium. A nonlinear evolution equation for the free surface of the film is derived through long-wave approximation. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. It is observed that the stability characteristics of the film are influenced by the permeability, the porosity of the porous medium and the ratio of the porous to liquid layer thickness d̂. Further, the conditions under which the two-sided model (TLM) can be replaced by an effective one-sided slip model (SM) is analyzed and the corresponding slip length is computed in terms of the porous layer characteristics. A weakly nonlinear stability analysis is performed and the range of preferred wave numbers for which the disturbances reach finite equilibrium amplitude or an explosive state is obtained. The nonlinear equation is then numerically solved as an initial value problem on a periodic domain and different scenarios of surface structures are captured. The long-time wave forms are shown to agree very well with the corresponding stationary solutions of the evolution equation. The results show that the long-time wave forms are either time-independent waves that propagate or time-dependent modes that oscillate slightly in amplitude. The fundamental modes dominate the stationary solution for shorter periods and the higher modes dominate as the period increases. Further, for certain bands of the period, the steady state is observed to lose its stability to oscillations. © 2012 Elsevier Ltd.
Instabilities in a liquid film flow over an inclined heated porous substrate
Stability of a thin viscous Newtonian fluid draining down a uniformly heated porous inclined plane is examined. The long-wave linear stability analysis is performed within the generic Orr-Sommerfeld framework both theoretically and numerically. An evolution equation for the local film thickness for two-dimensional disturbances is derived to analyze the effect of long-wave instabilities. The parameters governing the film flow system and the porous substrate strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The long-time wave forms are either time-independent wave forms that propagate or time-dependent modes that oscillate slightly in the amplitude. The role of permeability and Marangoni number is to increase the amplitude of the disturbance leading to the destabilization state of the film flow system. The permeability of the porous medium promotes the oscillatory behavior. © 2010 Elsevier Ltd. All rights reserved.
Instabilities of a confined two-layer flow on a porous medium: An Orr-Sommerfeld analysis
Instabilities of a pressure driven two-layer Poiseuille flow confined between a rigid wall and a Darcy-Brinkman porous layer are explored. A linear stability analysis of the conservation laws leads to an Orr-Sommerfeld system, which is solved numerically with appropriate boundary conditions to identify the time and length scales of the instabilities.fde The study uncovers the coexistence of twin instability modes, (i) long-wave interfacial mode-engendered by the viscosity stratification across the interface and (ii) finite wave number shear mode-originating from the inertial stresses, for almost all combinations of the viscosity (μr) and thickness (h r) ratios of the liquid layers. The presence of the porous layer reduces the frictional influence on the films, which significantly alters the length and time scales of the shear mode while the interfacial mode remains dormant to this effect. This is in stark contrast to the two-layer flow confined between non-porous plates where an unstable interfacial (shear) mode is observed when μr>hr2(μr