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3. Asymptotic Behavior Before calculating thermocapillary flows in a fully numerical way, it is useful to consider possible singularities and the scaling behavior of the flow. A fundamental problem is the fluid flow near the contact line, i.e. the flow in a wedge made by an isothermal wall and a plane thermocapillary surface. Since the velocity field must vanish on the rigid wall (u = O) one is tempted to assume that the velocity in a small vicinity of the corner remains small. If the derivatives of the velocity field also remain small the flow should be well approximated by the Stokes equation. This can be solved locally, close to the corner. One major difficulty is posed, however, by the discontinuity of the boundary conditions at the contact line. Another aspect is the appearance of different boundary layers when the Reynolds and Marangoni numbers are large. When convective heat transport becomes important, for instance, the free-surface isotherms are convectively compressed towards the cold contact line. With the help of scaling analyses it is possible to calculate the orders of magnitude of the boundary layer thicknesses and velocity fields. 3 . 1 F l o w N e a r t h e C o n t a c t L i n e Consider, for the moment, the local flow in the vicinity of a cold corner with contact angle c~.1 In the limit r --~ R, z --~ - d / 2 , a Cartesian coordinate system can be used. Corrections are of vanishing order O[(R - r)/R]. The fixed contact point (r, z) = (R, - d / 2 ) in the (r, z) plane is taken as the origin of a