STUDY OF OSCILLATORY THERMOCAPILLARY FLOW OF HIGH PRANDTL NUMBER FLUID

Y. Kamotani¹ and S. Yoda²

¹Case Western Reserve University, Cleveland, OH 44106, USA
²National Space Development Agency of Japan, Tsukuba, Ibaraki, Japan

Oscillatory thermocapillary flow is investigated for high Prandtl number (Pr) fluids. The investigation is done experimentally as well as numerically. Many experiments have been performed in the past with high Pr fluids because it is easier to perform thermocapillary flow experiments with those fluids due to their insensitivity to surface contamination. Those experiments have shown that thermocapillary flow in the half-zone configuration becomes oscillatory. However, despite the fact that much experimental information is available, the cause of oscillations is not yet fully understood. The main objective of the present work is to clarify the cause of oscillations in the half-zone and similar configurations for high Pr fluids.

The basic oscillation mechanism is analyzed first. One important feature of thermocapillary flow of high Pr fluid at high Marangoni number (Ma) is the existence of a hot corner where the flow driving force is concentrated. During oscillations, the whole flow field is known to change cyclically. Since the flow is driven mainly in the hot corner, the driving force there must also be altered periodically. The only way to change the hot corner temperature distribution, thus the driving force, is to change the thermal boundary layer thickness along the hot wall. Therefore, the period of oscillations should scale with the time scale associated with the thermal boundary layer. In order to find the boundary layer time scale numerically, we first obtain a steady solution for the conditions that are known to be close to the onset of oscillations. Then, we suddenly increase Ma by a small amount (10%) and monitor the subsequent variation of the total heat transfer rate (Nusselt number) at the hot wall. It is shown that the time constants computed for various conditions are indeed proportional to experimentally observed oscillation periods.

Next we discuss how the flow adjusts itself after it is disturbed. It is important to understand this adjustment process before we investigate the oscillation process. After a steady flow is obtained, the value of Ma is increased by 30% for a certain time. After that, Ma is decreased to the original value. Various dimensionless quantities, such as the maximum stream function, Nusselt number at the hot wall, and the total shear force at the hot wall, are monitored in the subsequent time. Those quantities are shown to change periodically with decreasing amplitudes. Figure 1 shows the phase relations among various quantities during the oscillatory decaying process. The maximum stream function variation and the total shear variation at the hot wall are nearly out-of-phase. This means that when the shear is large (small), the bulk flow is weak (strong). The total shear variation and the heat flux variation are nearly in-phase. This means that when the thermal boundary layer is thin (which increases the heat transfer), the hot corner is narrower, which increases the wall shear effect. Those phase relations show that when the bulk flow becomes strong, the thermal boundary layer becomes thinner after some time, which increases the wall shear and slows the bulk flow. The opposite happens when the bulk flow becomes weak. This mechanism simply tries to bring the whole situation back to the original state after it is disturbed. In order to oscillate, the adjustment process must overshoot and undershoot the original state (non-linear effect). In the process shown in Fig. 1, the non-linear effect comes from the inertia forces, but the inertia forces are not strong enough to sustain steady oscillations.

Two-dimensional liquid bridges with flat and curved free surfaces are analyzed numerically. It is shown that the flow becomes oscillatory when the free surface is concave. It is shown that the oscillations are caused by the interactions of flow cells across the liquid bridge center plane. When the surface is highly concave, the flow becomes oscillatory without the inertia forces. The numerical analysis explains the experimentally known shape effect on the onset of oscillations qualitatively.

Finally the effect of flow unsteadiness on the onset of oscillations is investigated experimentally. The flow unsteadiness is caused by the fact the hot wall temperature is increased too fast (in the process of finding the critical temperature difference) compared to the characteristic time scale of the system. It is shown that although it takes a long time for the oscillations to appear (or to become observable) very near the critical conditions, the oscillations grow very quickly above those conditions so that the waiting time (to observe the onset of oscillations) is drastically reduced. The result tells us that if a test is performed with due consideration to the time of convection, as opposed to the thermal diffusion time, the error in the critical temperature measurement is relatively small.

Figure 1 Variations of dimensionless maximum stream function, total shear and heat flux at hot wall after disturbance