Usha Ranganathan

Bio: Usha Ranganathan is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Reynolds number & Laminar flow. The author has an hindex of 2, co-authored 2 publications receiving 15 citations.

Instabilities in viscosity-stratified two-fluid channel flow over an anisotropic-inhomogeneous porous bottom

Abstract: A linear stability analysis of a pressure driven, incompressible, fully developed laminar Poiseuille flow of immiscible two-fluids of stratified viscosity and density in a horizontal channel bounded by a porous bottom supported by a rigid wall, with anisotropic and inhomogeneous permeability, and a rigid top is examined. The generalized Darcy model is used to describe the flow in the porous medium with the Beavers-Joseph condition at the liquid-porous interface. The formulation is within the framework of modified Orr-Sommerfeld analysis, and the resulting coupled eigenvalue problem is numerically solved using a spectral collocation method. A detailed parametric study has revealed the different active and coexisting unstable modes: porous mode (manifests as a minimum in the neutral boundary in the long wave regime), interface mode (triggered by viscosity-stratification across the liquid-liquid interface), fluid layer mode [existing in moderate or O(1) wave numbers], and shear mode at high Reynolds numbers. As a result, there is not only competition for dominance among the modes but also coalescence of the modes in some parameter regimes. In this study, the features of instability due to two-dimensional disturbances of porous and interface modes in isodense fluids are explored. The stability features are highly influenced by the directional and spatial variations in permeability for different depth ratios of the porous medium, permeability and ratio of thickness of the fluid layers, and viscosity-stratification. The two layer flow in a rigid channel which is stable to long waves when a highly viscous fluid occupies a thicker lower layer can become unstable at higher permeability (porous mode) to long waves in a channel with a homogeneous and isotropic/anisotropic porous bottom and a rigid top. The critical Reynolds number for the dominant unstable mode exhibits a nonmonotonic behaviour with respect to depth ratio. However, it increases with an increase in anisotropy parameter ξ indicating its stabilizing role. Switching of dominance of modes which arises due to variations in inhomogeneity of the porous medium is dependent on the permeability and the depth ratio. Inhomogeneity arising due to an increase in vertical variations in permeability renders short wave modes to become more unstable by enlarging the unstable region. This is in contrast to the anisotropic modulations causing stabilization by both increasing the critical Reynolds number and shrinking the unstable region. A decrease in viscosity-stratification of isodense fluids makes the configuration hosting a less viscous fluid in a thinner lower layer adjacent to a homogeneous, isotropic porous bottom to be more unstable than the one hosting a highly viscous fluid in a thicker lower layer. An increase in relative volumetric flow rate results in switching the dominant mode from the interface to fluid layer mode. It is evident from the results that it is possible to exercise more control on the stability characteristics of a two-fluid system overlying a porous medium in a confined channel by manipulating the various parameters governing the flow configurations. This feature can be effectively exploited in relevant applications by enhancing/suppressing instability where it is desirable/undesirable.

Evolution of a thin film down an incline: A new perspective

Abstract: A new model which accounts for energy balance while describing the evolution of a thin viscous, Newtonian film down an incline at high Reynolds numbers and moderate Weber numbers has been derived. With a goal to improve the predictions by the model in inertia dominated regimes, the study employs the Energy Integral Method with ellipse profile EIM(E) as a weight function and is motivated by the success of EIM in effectively and accurately predicting the squeeze film force in squeeze flow problems and in predicting the inertial effects on the performance of squeeze film dampers [Y. Han and R. J. Rogers, “Squeeze film force modeling for large amplitude motion using an elliptical velocity profile,” J. Tribol. 118(3), 687–697 (1996)]. The focus in the present study is to assess the performance of the model in predicting the instability threshold, the model successfully predicts the linear instability threshold accurately, and it agrees well with the classical result [T. Benjamin, “Wave formation in laminar flow down an inclined plane,” J. Fluid Mech. 2, 554–573 (1957)] and the experiments by Liu et al. [“Measurements of the primary instabilities of film flows,” J. Fluid Mech. 250, 69–101 (1993)]. The choice of the ellipse profile allows us to have a free parameter that is related to the eccentricity of the ellipse, which helps in refining the velocity profile, and the results indicate that as this parameter is increased, there is a significant improvement in the inertia dominated regimes. Furthermore, the full numerical solutions to the coupled nonlinear evolution equations are computed through approximations using the finite element method. Based on a measure {used by Tiwari and Davis [“Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface,” Phys. Fluids 21, 102101 (2009)]} of the temporal growth rate of perturbations, a comparison of the slope of the nonlinear growth rate with the linear growth rate is obtained and the results show an excellent agreement. This confirms that the present model’s performance is as good as the other existing models, weighted residual integral boundary layer (WRIBL) by Ruyer-Quil and Manneville [“Improved modeling of flows down inclined planes,” Eur. Phys. J: B 15, 357–369 (2000)] and energy integral method with parabolic profile [EIM(P)] by Usha and Uma [“Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method,” Phys. Fluids 16, 2679–2696 (2004)]. Furthermore, for any fixed inclination θ of the substrate, 0 < θ < π/2, there is no significant difference between the EIM(E) and EIM(P) results for weaker inertial effects, but when the inertial effects become stronger, the EIM(E) results for energy contribution from inertial terms to the perturbation at any streamwise location is enhanced. More detailed investigation on the model’s performance due to this enhancement in energy contribution and the assessment of the model as compared to the other existing theoretical models, experimental observations, and numerical simulations, in the inertia dominated regimes, will be reported in a future study.

Linear stability analysis of a surfactant-laden shear-imposed falling film

Abstract: A study of the linear stability analysis of a shear-imposed fluid flowing down an inclined plane is performed when the free surface of the fluid is covered by an insoluble surfactant. The purpose is to extend the earlier work [H. H. Wei, “Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane,” Phys. Fluids 17, 012103 (2005)] for disturbances of arbitrary wavenumbers. The Orr-Sommerfeld boundary value problem is formulated and solved numerically based on the Chebyshev spectral collocation method. Two temporal modes, the so-called surface mode and surfactant mode, are detected in the long-wave regime. The surfactant mode becomes unstable when the Peclet number exceeds its critical value. In fact, the instability of the surfactant mode occurs on account for the imposed shear stress. Energy budget analysis predicts that the kinetic energy of the infinitesimal disturbance grows with the imposed shear stress. On the other hand, the numerical results reveal that both surface and surfactant modes can be destabilized by increasing the value of the imposed shear stress. Similarly, it is demonstrated that the shear mode becomes more unstable in the presence of the imposed shear stress. However, it can be stabilized by incorporating the insoluble surfactant at the free surface. Apparently, it seems that inertia does not play any role in the surfactant mode in the moderate Reynolds number regime. Furthermore, the competition between surface and shear modes is discussed.

Linear stability of plane Poiseuille flow of two superposed fluids

Abstract: Stability of two superposed fluids of different viscosity in plane Poiseuille flow is studied numerically. Conditions for the growth of an interfacial wave are identified. The analysis extends Yih’s results [J. Fluid Mech. 27, 337 (1967)] for small wavenumbers to large wavenumbers and accounts for differences in density and thickness ratios, as well as the effects of interfacial tension and gravity. Neutral stability diagrams for the interfacial mode are reported for a wide range of the physical parameters describing the flow. The analysis shows also that the flow is linearly unstable to a shear mode instability. The dependence of the critical Reynolds number for the shear mode on the viscosity ratio is reported. Theoretical predictions of critical Reynolds numbers for both modes of instability are compared with available experimental data.

The phase lead of shear stress in shallow-water flow over a perturbed bottom

Abstract: The analysis of flow over a slowly perturbed bottom (when perturbations have a typical length scale much larger than channel height) is often based on the shallow-water (or Saint-Venant) equations with the addition of a wall-friction term which is a local function of the mean velocity. By this choice, small sinusoidal disturbances of wall stress and mean velocity are bound to be in phase with each other. In contrast, studies of shorter-scale disturbances have long established that a phase lead develops between wall stress and mean velocity, with a crucial destabilizing effect on sediment transport along an erodible bed. The purpose of this paper is to calculate the wall-shear stress under large length-scale conditions and provide corrections to the Saint-Venant model.

Stability of viscosity stratified flows down an incline: Role of miscibility and wall slip

Abstract: The effects of wall velocity slip on the linear stability of a gravity-driven miscible two-fluid flow down an incline are examined. The fluids have the matched density but different viscosity. A smooth viscosity stratification is achieved due to the presence of a thin mixed layer between the fluids. The results show that the presence of slip exhibits a promise for stabilizing the miscible flow system by raising the critical Reynolds number at the onset and decreasing the bandwidth of unstable wave numbers beyond the threshold of the dominant instability. This is different from its role in the case of a single fluid down a slippery substrate where slip destabilizes the flow system at the onset. Though the stability properties are analogous to the same flow system down a rigid substrate, slip is shown to delay the surface mode instability for any viscosity contrast. It has a damping/promoting effect on the overlap modes (which exist due to the overlap of critical layer of dominant disturbance with the mixed layer) when the mixed layer is away/close from/to the slippery inclined wall. The trend of slip effect is influenced by the location of the mixed layer, the location of more viscous fluid and the mass diffusivity of the two fluids. The stabilizing characteristics of slip can be favourably used to suppress the non-linear breakdown which may happen due to the coexistence of the unstable modes in a flow over a substrate with no slip. The results of the present study suggest that it is desirable to design a slippery surface with appropriate slip sensitivity in order to meet a particular need for a specific application.