Numerical investigation of the non-isothermal contact

Supported by: BMBF-DLR: ,,Treibstoffverhalten in Tanks von Raumtransportsystemen -- Comportement des Ergols dans les Réservoirs''

It is known that thermal effects have a considerable impact on the shape of a gas-liquid phase boundary and on the angle between this surface and a solid wall at the contact line. Gerstmann et al. investigated in [1] the reorientation of such a phase boundary upon step reduction of gravity in the non-isothermal case experimentally. Our aim is similar, but we want to focus on the effect of Marangoni convection on the shape of the phase boundary. Therefore, we limit ourselves to the case where no external body forces act on the fluid.

Consider a circular cylinder, partly filled with liquid. We assume isothermal initial conditions. In the absence of gravity, the liquid surface is spherical due to surface tension. Now, the upper part of the cylinder walls, starting from a point below the liquid surface, is going to be heated.

The temperature gradient at the gas-liquid phase boundary induces a Marangoni stress, exciting a flow in the liquid and a deformation of the free surface (Figure 1). As indicator for the deformation of the phase boundary, we study the vertical coordinate of the center of the free surface z_c and of the contact point of the surface with the cylinder wall z_w. The initial deformation caused by the raise of temperature in the heated part of the wall is characterized by a raise of z_c and a recede of z_w. The angle of contact between the free surface and the wall appears to be larger. After this deformation, the shape of the free surface changes only slowly in time, so that one may speak of a quasi-stationary state.

**Fig. 1:** Shape of the free surface, isolines of temperature and velocity field in the isothermal initial configuration (left) and after the onset of thermocapillary convection (right)
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The shape of the non-isothermal phase boundary has been evaluated. It is significantly flattened near the center compared to the isothermal initial configuration, resulting in a higher (flatter) angle between the tangent to the surface and the vertical cylinder wall (Figure 2, center). While the curvature of the free surface is constant in the isothermal configuration, it becomes much larger close to the cylinder wall in the non-isothermal case (Figure 2, right). One might assume that this strong variation in curvature within a small layer close to the wall is hardly visible to the eye. This could explain why the contact angle in the non-isothermal configuration appears to be higher than in the isothermal case, although the same contact angle was prescribed in both cases.

**Fig. 2:** Free surface (left). Angle of the tangent to the free surface and the vertical cylinder wall (center). Curvature of the free surface (right).
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**Fig. 3:** Definition of the apparent contact angle: Approximation of the free surface by the circle connecting the center and the contact point (left). Deviation of the apparent contact angle from the prescribed static contact angle vs. Re_M for different static contact angles (right).
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In order to get a notion of a contact angle that corresponds to what may be observed in experiments, we define an ``apparent contact angle'' as follows: We approximate the free surface by the circle connecting the center and the contact point (Figure 3, left). Now the apparent contact angle $\gamma_{{app}}^{}$ is defined to be the angle between the tangent of this circle and the vertical cylinder wall at the contact point. Using this definition, the apparent contact angle is given by

Figure 3 (left) shows the deviation of $\gamma_{{app}}^{}$ from the prescribed static contact angle $\gamma_{s}^{}$ for different thermocapillary Reynolds numbers and for different values of $\gamma_{s}^{}$ . The trend of a larger apparent contact angle with increasing Re_M, as observed in the experiments, is confirmed by the numerical simulations. But also the static contact angle $\gamma_{s}^{}$ turns out to play an important role: In case of a small $\gamma_{s}^{}$ , the convection role is more confined due to the narrower layer at the meniscus. In turn, the direction of the flow induced by the Marangoni stress that is pointing away from the wall forms a larger angle with the tangent of the surface, thus having a stronger effect on its shape. For larger contact angles, the situation is opposite. The flow direction is more parallel to the surface and thus has a weaker influence on its shape.

From our numerical investigations we can conclude that the enlargement of the apparent contact angle as observed in experiments in the scenario of a cold liquid meniscus on a hot solid wall is confirmed. We would like to emphasize that this effect does not depend on the specific model of a dynamic contact angle, since in our simulations a fixed static contact angle as boundary condition for the shape of the free surface has been used.

Numerical investigation of the non-isothermal contact angle