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Extensive numerical simulations of the 2-D laminar flow of power-law fluids over elliptical cylinders with different aspect ratios have been carried out to establish the conditions for the onset of wake formation and the onset of vortex shedding. The continuity and momentum equations were solved numerically using FLUENT (version 6.3.26). The influence of the power-law index (0.3 ≤ n ≤ 1.8) and the aspect ratio (E = b/a; 0.2 ≤ E ≤ 5) of the cylinder on the critical values of the Reynolds number denoting the onsets of flow separation and vortex shedding are presented. For shear-thinning (n < 1) fluid behavior, the onsets of wake formation and vortex shedding are both seen to be postponed to higher Reynolds numbers as compared to those in shear-thickening fluids (n > 1). Also, the values of the Strouhal number (St c) and the time-average drag coefficient (C̅ D) corresponding to the cessation of the steady-flow regime are presented. Velocity vector plots denoting the flow separation and vorticity profiles showing the vortex shedding are also included. The delineation of different flow regimes also helps identify the range of validity of some of the results on flow and heat transfer available in the literature.

The linear stability of a liquid layer flowing down an inclined plane lined with a deformable, viscoelastic solid layer is analyzed in order to determine the effect of the elastohydrodynamic coupling between the liquid flow and solid deformation on the free-surface instability in the liquid layer. The stability of this two-layer system is characterized by two qualitatively different interfacial instability modes: In the absence of the deformable solid layer, the free surface of the liquid film undergoes a long-wave instability due to fluid inertia. With the presence of the deformable solid layer, the interface between the fluid and the solid undergoes a finite-wavelength instability when the deformable solid becomes sufficiently soft. The effect of the solid layer deformability on the free-surface instability of the liquid film flow is analyzed using a long-wave asymptotic analysis. The asymptotic results show that for a fixed Reynolds number and inclination angle, the free-surface instability is completely suppressed in the long-wave limit when the nondimensional (inverse) solid elasticity parameter Gamma=Va eta/(GR)increases beyond a critical value. Here, Va is the average velocity of the liquid film flow, eta is the viscosity of the liquid, G is the shear modulus of the solid layer, and is R the thickness of the liquid layer. The predictions of the asymptotic analysis are verified and extended to finite wavelengths using a numerical solution, and this indicates that the suppression of the free-surface instability indeed continues to finite wavelength disturbances. Further increase of Gamma is found to have two consequences: first, the interface between the liquid and the deformable solid layer could become unstable at finite wavelengths; second, the free-surface interfacial mode could also become unstable at finite wavelengths due to an increase in solid layer deformability. However, our numerical results demonstrate that, for a given average velocity, there exists a sufficient window in the value of shear modulus G where both the unstable modes are absent at all wavelengths. Our study therefore suggests that soft solid layer coatings could potentially provide a passive method of suppressing free-surface instabilities in liquid film flows.

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