Hydromagnetic thin film flow: Linear stability

Abstract

This paper deals with the long wave instability of an electroconductor fluid film, flowing down an inclined plane at small to moderate Reynolds numbers, under the action of electromagnetic fields. A coherent second order long wave model and two simplified versions of it, referred to as first and second reduced models (FRM and SRM), are proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The full model is naturally unique, i.e., independent of the particular form of the trial functions. The linear stability of the problem is investigated, and the influence of electromagnetic field on the flow stability is analyzed. Two cases are considered: the applied magnetic field is either normal or parallel to the fluid flow direction, while the electric field is transversal. The numerical solution of the Orr-Sommerfeld (OS) eigenvalue problem and those of the depth averaging model are used to assess the accuracy of the reduced models. It is found that the current models have the advantage of the Benney-like model, which is known to asymptote the exact solution near criticality. Moreover, far from the instability threshold, the current reduced models continue to follow the OS solution up to moderate Reynolds numbers, while the averaging model diverges rapidly. The model SRM gives better results than FRM beyond sufficiently high Reynolds numbers.

Stabilising falling liquid film flows using feedback control

“…In order to compute the stability of travelling waves, we transform to the frame moving at speed U , and then identify x with ζ. For the Benney equation, the generalised eigenvalue problem (24) becomes…”

“…Thermal effects and topography are often combined with each other 11,12 or with electric fields [13][14][15][16][17] . Other physical mechanisms that have been investigated within the context of thin-film flow down inclined planes include chemical coatings or microstructure to induce effective slip 18 , surfactants 19 , porous 20,21 or deformable 22 walls, explicit injection/suction through the planar wall 23 and magnetic fields 24 . For flat homogeneous walls, a steady uniform flow solution exists known as the Nusselt solution, which can of course be unstable.…”

Optimal Control of Thin Liquid Films and Transverse Mode Effects

“…The film flow problem has also been considered with the addition of heating at the wall surface which is not necessarily uniform [66,67,68,69], and even in conjunction with substrate topography [70]. Other possibilities for influencing the dynamics of thin films include the introduction of magnetic fields [71], surfactants [72], and substrate microstructure or coatings to induce effective slip [73]. In the current work, we focus on the control of the film interface using same-fluid blowing and suction controls at the substrate surface.…”