Steady Thermocapillary Flows and Their Stability
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MATERIALS PROCESSING iN fHE REDUCED GRAVITY ENVIRONMENT OF SPACE Guy E. Rindone
173
STEADY THERMOCAPILLARY FLOWS AND THEIR STABILITY
A. K. SEN, M. K. SMITH AND S. H. DAVIS, Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois, USA and G. M. HOMSY, Department of Chemical Engineering, Stanford University, Stanford, California, USA
ABSTRACT Thermocapillarity can drive steady flows in float-zone configurations. These flows can become unstable and exhibit time-periodic oscillations. In order to obtain insight into such phenomena, studies are made of a simplified system consisting of liquid in a two-dimensional slot. Steady flows and their instabilities are examined.
INTRODUCTION In order to begin analysis of the thermocapillary flows generated in containerless-processing configurations, we consider a two-dimensional slot that contains a viscous liquid as shown.
y GAS
THX
ci
=LIQUID
---
Fig.
1.
"T TCC d _ýý X _L
|
"-- ........
-
L
I
zero heat
flux
Geometry and notation for steady slot flow.
The liquid has viscosity g, density p, and thermal diffusivity K all taken as constants. The surface tension a on the gas liquid interface has the equation of state a = o0[ 1 - y(T - TC)] where T is the temperature. The slot is differentially heated, TH> TC, causing a temperature gradient along the free surface. The surface-tension gradients induced generated motion in the bulk liquid. The parameters that occur in the problem are the Marangoni number M, M = Y(TH- TC)d2 /;IKL
(1)
the Prandtl number P,
(2)
P = s/pK the aspect ratio A, A = d/L
$
(3)
174 the capillary number C, (4)
C = Y(TH - TC)d/Lc0
L measures the heat transfer from the liquid to and the surface Biot number L. the gas (L =0 corresponds to zero heat flux into the gas) and C measures the degree of interfacial deformation. We study these steady thermocapillary flows when interfacial contamination Furthermore, we study instabilities of such is either absent or present. steady shear flows since instability can lead to temporal oscillations in the flow and thermal fields. STEADY SLOT FLOWS WITH NO CONTAMINATION We calculate [l] analytically steady slot flows without contamination effects for the case A -4 0. When the interfacial thermal condition matches the liquid heat transfer to a linear (in x) temperature distribution in the The gas, we can find nearly parallel flow under a nearly planar free surface. 2 flow and thermal fields are calculated through O(A ) for cases where the The degree of curvature of the surface is conMarangoni number M is O(A). trolled by the capillary number C, a measure of the mean surface tension. 4 Curvature is negligible only in the extreme case of CA- 4 0 as A + 0. SoluA sketch (not to tions are obtained for various contact angle conditions. scale) of the flow field is shown.
HOT
Fig. 2.
COLD
Sketch (not to scale) of steady slot flow.
Note that at the cold end, the analogue of the crystal face, the interface bulges and so that dynamic contact angle is alw
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