A significant feature of gravity-driven film flows of Newtonian and rheologically complex fluids down an inclined/vertical substrate is the instability of the free surface which manifests as surface waves having wavelengths much larger than the film thickness. There are a number of applications which can be modeled as thin film flow systems on porous substrates. Pascal [1] investigated the stability of a falling power-law film on an inclined porous substrate. This model for the fluid predicts a viscosity that goes to infinity as the shear rate approaches zero. There is a need to employ a more appropriate model to examine the effects of non-Newtonian rheology on the dynamics and stability of thin film free surface flows down inclined or vertical rigid/porous substrates. The four-parameter Carreau model predicts a viscosity that remains finite as the shear rate approaches zero and is given by
ηηη0η=[1+(δγ)˙2]n12.(1)
Weinstein [2] and Rousset et al. [3] have considered the Carreau model and have examined the temporal stability of a film flow down an impermeable rigid inclined substrate. The authors show that a shear-thinning Carreau fluid film on a rigid impermeable substrate is more unstable than a Newtonian film. This calls for an analysis that includes both the effects of Carreau non-Newtonian rheology and bottom permeability and the present study reports such an investigation of a Carreau non-Newtonian film on a porous inclined substrate.
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