Non-Newtonian Fluid Dynamics Analysis
Budget: ₹600 – ₹1,500 INR
We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography.
Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness
and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on
the steady free surface deformation. For weakly undulated walls we solve the governing equation analyt-
ically by a perturbation approach and find a resonant interaction of the free surface with the wavy bot-
tom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the
steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in impor-
tance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an ampli-
fication of the steady free surface. A linear stability analysis of the steady state shows that the bottom
undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance
this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear sta-
bility analysis. Finally, we show the important role of capillarity and discuss its influence on the steady
solution and on the stability.
Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness
and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on
the steady free surface deformation. For weakly undulated walls we solve the governing equation analyt-
ically by a perturbation approach and find a resonant interaction of the free surface with the wavy bot-
tom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the
steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in impor-
tance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an ampli-
fication of the steady free surface. A linear stability analysis of the steady state shows that the bottom
undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance
this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear sta-
bility analysis. Finally, we show the important role of capillarity and discuss its influence on the steady
solution and on the stability.