You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Linear and non-linear theory of long wavelength Marangoni<br />
instability with Soret effect at finite Biot numbers.<br />
A. Podolny ∗ ,A.Oron † , A. A. Nepomnyashchy ‡<br />
We consider a system which consists of a layer of an incompressible binary liquid<br />
with a deformable free surface, and a solid substrate layer heated or cooled from<br />
below. Surface tension σ is assumed to be linearly depend upon both temperarute<br />
and solute concentration. The Soret effect is taken into account. It is assumed that<br />
the layer is sufficiently thin, so the effect of buoyancy can be neglected as compared<br />
to the Marangoni effect. The Dufour effect is neglected.<br />
We investigate the long wavelength Marangoni instability in the case of asymptotically<br />
small Lewis and Galileo numbers for finite surface tension and Biot numbers. We<br />
find both long wavelength monotonic and oscillatory modes of instability in various<br />
parameter domains of Biot and Soret numbers.<br />
The weakly nonlinear analysis is carried out in the limit of the small conductivity<br />
of the solid substrate. In the leading order of the problem we consider a particular<br />
solution corresponding to a pair of travelling waves with complex amplitudes H±, and<br />
angle θ between wave vectors. We obtain a set of two Landau equations that govern<br />
evolution of wave amplitudes:<br />
dH+<br />
dτ = λH+ + α(X, Z)|H+| 2 H+ + β(X, Z, θ)|H−| 2 H+, (1)<br />
dH−<br />
dτ = ¯ λH− +¯α(X, Z)|H−| 2 H− + ¯ β(X, Z, θ)|H+| 2 H−, (2)<br />
where α = αr +iαi, β = βr +iβi, X is rescaled Galileo number, Z is rescaled squared<br />
wave number. In the case of small gravity (X = 0) the weakly nonlinear theory<br />
predicts the appearance of stable supercritical solutions if (i) max βr(θ)
Change language