Linear Stability Analysis of Buoyancy-Thermocapillary Convection driven by Bidirectional Temperature Gradient

Jia Liu, College of Energy and Power Engineering, Chongqing University

Hopeman 224

Study of temperature-driven buoyancy-thermocapillary convection is a rapidly developing filed
with broad application in engineering (crystal growth process, chemical engineering, heat pipes
etc.) The investigation of the convective patterns and transitions is of great importance in many
natural and technological processes.
There are a lot of works devoted to the study of the buoyancy-thermocapillary flow in a horizontal
liquid layer, but most of them confine to either horizontal or vertical temperature gradient driven
convection, only a few considered convection with joint action of vertical and horizontal
temperature gradient. In reality, it is hard to guarantee that the temperature is directed strictly
unilaterally to the surface, so the problem of bidirectional temperature gradient driven buoyancythermocapillary
flow is deserved to be explored.
Here we present the linear stability analysis of the buoyancy-thermocapillary convection in a
horizontal infinite liquid film subject to bidirectional temperature gradient. An eigenvalue problem
is obtained through this analysis and is solved numerically using the Chebyshev collocation
spectral method. The stability threshold of the flow and their variation with the control parameters
are determined and studied. The control parameters include temperature gradient inclination (ratio
of vertical temperature gradient to horizontal temperature gradient b), buoyancy (dynamic Bond
number BoD), heat exchange at the free surface (Biot number Bi) and fluid property parameter
(Prandtl number Pr).
It is found that with the increase of the control parameters, the liquid layer will change from being
dominated by the horizontal temperature gradient to be dominated by the vertical temperature
gradient. For the onset of transverse rolls, when the horizontal temperature gradient dominates the
system, the buoyancy enhances the stability of the flow as most literature suggested, while for
liquid layers dominated by vertical temperature gradient, buoyancy’s influence on flow stability is
less evident. For the onset of longitudinal rolls, buoyancy plays little effect on the flow stability.
In terms of the effect of Bi (heat exchange at the free surface), the increase of Bi stabilizes the flow
when horizontal temperature gradient is dominant in the system, and destabilize the flow when
vertical temperature dominates. As for Prandtl number, for the development of transverse rolls, its
influence on flow stability is similar to that of buoyancy, and for the onset of longitudinal rolls,
the increase of Pr first stabilizes then destabilizes the flow when the dominant temperature gradient
in the liquid layer changes from horizontal to vertical. It is also observed that when the temperature
gradient inclination exceeds certain value, the critical Marangoni number (value of
thermocapillary stress) will experience a “sudden drop” phenomenon, where the flow instability
takes place at a very small horizontal temperature gradient, and the corresponding critical
frequency is zero for longitudinal rolls.