Linear and Nonlinear Longwave Marangoni Stability of a Thin Liquid Film Above or Below a Thick Wall with Slip in the Presence of Microgravity

The linear and nonlinear thermocapillary instability of a liquid layer located above or below a thick horizontal wall with slip effect under gravity is investigated for the first time. A nonlinear evolution equation for the free surface deformation is obtained under the lubrication approximation. The curves of linear growth rate, maximum growth rate and critical Marangoni number are calculated under two physical conditions. First, gravity is directed from the liquid to the wall and has a stabilizing effect. Second, gravity is directed from the wall to the liquid (Rayleigh-Taylor instability). In the latter case, the liquid film is also subjected to stabilizing and destabilizing Marangoni numbers. The film is subjected to slip at the thick wall with finite thermal conductivity. A very important result is that slip is not always destabilizing. From the point of view of the growth rate, the slip parameter beta changes its destabilizing role into a stabilizing one at a particular wavenumber, independent of beta. Also, from the standpoint of the maximum growth rate, the slip parameter beta is not always destabilizing. Magnitudes of the Marangoni number were found where beta is stabilizing. Formulas to delimit where the role of beta changes are derived analytically. The free surface profiles obtained from the nonlinear evolution equation show the stabilizing and destabilizing effects of the parameters. It is revealed that the Rayleigh-Taylor instability can be delayed, before the film free surface touches the wall.

Automated summary

The linear and nonlinear thermocapillary instability of a liquid layer above or below a thick horizontal wall with slip effect under gravity is investigated. Important findings regarding the stabilizing and destabilizing effects of parameters are revealed.