Optimal disturbance growth in shear‐imposed falling film

Abstract

A linear stability analysis of a three‐dimensional shear‐imposed fluid flowing down an inclined plane is studied based on the evolution equations for normal velocity and normal vorticity components. Both modal and nonmodal stability analyses are carried out. The modal stability analysis demonstrates that shear‐imposed flow is more unstable to solo streamwise perturbation. On the contrary, the nonmodal stability analysis demonstrates that shear‐imposed flow is more unstable to solo spanwise perturbation. There is evidence of existing transient energy growth in the wavenumber space that intensifies in the presence of imposed shear stress. Further, the boundary for the zone of transient growth appears far ahead of the boundary for the zone of exponential growth. This fact indicates that the onset of instability for the shear mode may occur before than that predicted from the eigenvalue analysis. A new critical surface parameter involving imposed shear stress is determined, which in fact reveals the existence criterion of instability for the surface mode. Moreover, we have found three different regimes of surface mode instability, shear mode instability, and transient growth phenomenon in the wavenumber space. The unstable region for the surface mode reduces while the unstable region for the shear mode enhances in the wavenumber space with the increasing value of surface parameter, or equivalently, with the increasing value of imposed shear stress. Finally, the unstable region for the surface mode disappears from the wavenumber space as soon as the surface parameter exceeds its critical value; instead, the wavenumber space is occupied by the regime of transient energy growth. In addition, the incident of pseudoresonance takes place on the response curve to external harmonic forcing.

Instability of a shear-imposed flow down a vibrating inclined plane

“…It should be fruitful to mention here that spurious eigenvalues may appear in the numerical solution because of the homogeneous boundary conditions (3.11) used in the rows of matrix A. However, these spurious eigenvalues are mapped to the arbitrary irrelevant stable modes by carefully selecting the complex multiple for the corresponding rows of matrix B (Schmid & Henningson 2001). In this way, one can avoid spurious eigenvalues from the matrix eigenvalue problem (4.4).…”

“…The nonlinear travelling wave solution was found under the reference frame moving with the speed of the travelling wave. Recently, the non-modal analysis for a shear-imposed liquid flowing down an inclined plane was performed by Samanta (2020 c ). It is reported that the transient growth exists and intensifies in the presence of external imposed shear stress.…”

Modal analysis of a viscous fluid falling over a compliantwall

“…To close the flow system, the following boundary conditions are employed. At the fluid surface, the hydrodynamic stress of the mainstream fluid is balanced by the surface stress, which yields the tangential and normal stresses dynamic boundary conditions [25,26]…”

Effect of porous layer on the Faraday instability in viscous liquid

“…(ii) The evolution of liquid surface, y = h ( x , z , t ), is described by the kinematic boundary condition [25] ∂th+bold-italicu⋅bold∇h=bold-italicu⋅bold-italicjboldfalse^. (iii) Following Beavers & Joseph [26], an empirical boundary condition for shear stress is imposed at the liquid–porous interface, y = − d , ∂yuH=(αBJκ)false(uH−upHfalse),1emv=vp, p=pp, where α BJ is an empirical dimensionless coefficient, which, in fact, depends on the local structure of the porous material close to the liquid–porous interface [26,27]. In fact, this empirical boundary condition (2.8) is considered to take into account the jump of velocity in liquid and porous layers due to the boundary layer developed by the viscous friction effect in the vicinity of the liquid–porous interface.…”