Gravity-driven flow over heated, porous, wavy surfaces

Abstract

The method of weighted residuals for thin film flow down an inclined plane is extended to include the effects of bottom waviness, heating, and permeability in this study. A bottom slip condition is used to account for permeability and a constant temperature bottom boundary condition is applied. A weighted residual model (WRM) is derived and used to predict the combined effects of bottom waviness, heating, and permeability on the stability of the flow. In the absence of bottom topography, the results are compared to theoretical predictions from the corresponding Benney equation and also to existing Orr-Sommerfeld predictions. The excellent agreement found indicates that the model does faithfully predict the theoretical critical Reynolds number, which accounts for heating and permeability, and these effects are found to destabilize the flow. Floquet theory is used to investigate how bottom waviness influences the stability of the flow. Finally, numerical simulations of the model equations are also conducted and compared with numerical solutions of the full Navier-Stokes equations for the case with bottom permeability. These results are also found to agree well, which suggests that the WRM remains valid even when permeability is included. V

Falling liquid films on a slippery substrate with Marangoni effects

“…Samanta et al discussed the instability of an isothermal film falling down a slippery plane by deriving a weighted-residual model [23]. We observed that, the equations derived by Ogden et al [22] do not agree with the equations by Samanta et al [23] even if the substrate is flat and the thermal field is removed in the work of Ogden et al [22]. The inconsistency between the two models is due to the different consideration of the order of the slippery length.…”

“…A careful look at the literature indicates that studies on instability of liquid film flowing down a slippery heated substrate at a moderate flow rate are very limited. In addition, the work by Ogden et al [22] focused on a wavy porous substrate whereas the nonlinear dynamical behavior of the film on a flat slippery plane was not discussed. This paper investigates a falling liquid film on an inclined flat slippery heated/cooled substrate.…”

“…The slippery length in such a system [21] is very large. Recently, Ogden et al derived a weighted-residual model and investigated a falling film on a wavy porous heated substrate where the influence of the porous medium was reduced to the Navier-slip condition [22]. Samanta et al discussed the instability of an isothermal film falling down a slippery plane by deriving a weighted-residual model [23].…”

“…The inconsistency between the two models is due to the different consideration of the order of the slippery length. In the work of Ogden et al, terms up to second order of the slippery length were retained while higher order terms were neglected [22]. In Samanta et al's work [23], all the terms were retained.…”

“…In the work by Ogden et al, the term due to wall slippery is15 2 d 1 (d 1 is the dimensionless slippery length defined by Ogden et al) provided that the second order terms are neglected in their equation and the substrate is flat[22]. Clearly, our Eq.…”