Hydromagnetic thin film flow: Linear stability

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.

Figure 1

Schematic of the physical problem.

Figure 2

Mean flow velocity vs $\text{Ha}\sin \theta$ for some values of the electric parameter.

Figure 3

Marginal wave number vs Reynolds number: influence of electromagnetic fields on the marginal stability of the flow for (a) and (c) perpendicular magnetic field ($\theta =\pi /2$) and (b) and (d) longitudinal magnetic field ($\theta =0$). $\beta =\pi /2$.

Figure 4

Growth rate vs wave number: influence of electromagnetic fields on the instability growth rate for (a) and (c) perpendicular magnetic field ($\theta =\pi /2$) and (b) and (d) and longitudinal magnetic field ($\theta =0$). $\text{Ha}=1$, $\beta =\pi /2$, and $\text{Re}=15$.

Figure 5

Comparison of the models' predictions with the Orr-Sommerfeld solution: (a) the magnetic field is perpendicular; (b) the magnetic field is parallel. In (a) and (b) $\beta =\pi /2$ and $\delta =0$.

Figure 6

Comparison of the FRM and IM predictions for the critical Reynolds number. The magnetic field is perpendicular. Thin lines are from the IM model, while thick lines are from the FRM model. Solid line, $\delta =0$; dotted line, $\delta =-1$; dashed line, $\delta =-3$; dot-dashed line, $\delta =-5$.