Abstract
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.
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Abbreviations
- b :
-
slip length
- Bi :
-
Biot number
- \(c_{p}\) :
-
heat capacity at constant pressure
- \(c_{pW}\) :
-
wall heat capacity at constant pressure
- d :
-
wall relative thickness
- \(d_f\) :
-
fluid thickness
- \(d_w\) :
-
wall thickness
- \(\delta\) :
-
denominator, \(\delta (x,y,t)\)
- \(\delta _L\) :
-
linearized denominator, constant
- g :
-
acceleration of gravity
- \(h= 1 + H\) :
-
free surface height
- H :
-
free surface deformation
- k :
-
wavenumber magnitude
- \(k_c\) :
-
critical wavenumber
- \(K_f\) :
-
fluid heat conductivity
- \(k_{max}\) :
-
maximum growth rate wavenumber
- \(K_W\) :
-
wall heat conductivity
- \(k_x\) :
-
wavenumber x-component
- \(k_y\) :
-
wavenumber y-component
- \(k_+\) :
-
wavenumber where the role of \(\beta\) changes
- Ma = \(Ma^*/Pr\) :
-
here called Marangoni number
- \(Ma^*\) :
-
usual Marangoni number
- \(Ma_c\) :
-
critical Marangoni number
- \(Ma_{++}\) :
-
Marangoni number where the role of \(\beta\) changes
- \(Ma_{--}\) :
-
Marangoni number where the role of \(\beta\) changes
- p :
-
fluid pressure
- \(P_A\) :
-
atmospheric pressure
- \(p_f\) :
-
parabola focal distance of \(\Gamma _{max}\)
- Pr :
-
Prandtl number
- \(Q_{h}\) :
-
free surface heat transfer coefficient
- S :
-
surface tension number
- t :
-
time
- T :
-
fluid temperature
- \(T_{0}\) :
-
basic temperature of the fluid
- \(T_{A0}\) :
-
lower atmosphere temperature
- \(T_{A1}\) :
-
upper atmosphere temperature
- \(T_{W0}\) :
-
wall basic temperature
- u :
-
x-velocity component
- v :
-
y-velocity component
- w :
-
z-velocity component
- x :
-
x-coordinate
- y :
-
y-coordinate
- z :
-
z-coordinate
- \(\beta\) :
-
dimensionless slip length
- \(\gamma\) :
-
surface tension
- \(\Gamma\) :
-
growth rate
- \(\Gamma _{max}\) :
-
maximum growth rate
- \(\Delta T\) :
-
\(T_{A0} - T_{A1}\)
- \(\varepsilon\) :
-
smallness parameter
- \(\lambda\) :
-
wavenumber
- \(\nu\) :
-
kinematic viscosity
- \(\rho\) :
-
fluid density
- \(\rho _W\) :
-
wall density
- \(\chi\) :
-
wall over fluid heat conductivities ratio
- \(\omega\) :
-
frequency of oscillation
- \(\Omega\) :
-
\(\Gamma +i \omega\)
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Acknowledgements
The authors would like to thank Alejandro Pompa, Cain González, Raúl Reyes, Ma. Teresa Vázquez and Oralia Jiménez for technical support. I. M. S. B. would like to thank a scholarship by CONACyT.
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Luis Antonio Dávalos Orozco: Problem idea and solution of nonlinear problem. Isabel Monserrat Sánchez Barrera: Solution of the linear problem.
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Dávalos-Orozco, L.A., Barrera, I.M.S. Linear and Nonlinear Longwave Marangoni Stability of a Thin Liquid Film Above or Below a Thick Wall with Slip in the Presence of Microgravity. Microgravity Sci. Technol. 34, 107 (2022). https://doi.org/10.1007/s12217-022-10022-z
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DOI: https://doi.org/10.1007/s12217-022-10022-z