Two-Fluid Viscous Flow in a Corner
We consider steady, two-dimensional viscous flow of two fluids near a corner. The two fluids meet at the wedge vertex and are locally in contact with each other along a straight line emanating from the corner. The double wedge, treated in polar coordinates, admits separable solutions with bounded velocities at the corner. We seek local solutions which satisfy all local boundary conditions, as well as partial local solutions which satisfy all but the normal-stress boundary condition. We find that local solutions exist for a wide range of total wedge angles and that a class of of individual wedge angles and stress exponents is selected. Partial local solutions exist for all combinations of individual wedge angles and the stress exponents are determined as functions of these angles and the viscosity ratio. In both cases, Moffatt vortices can be found. Our aim in this work is to describe local two-fluid flow by determining for which wedge angles solutions exist, identifying singularities in the stress at the corner, and identifying conditions under which Moffatt vortices can be present in the flow. Furthermore, for the single-wedge geometry, we identify for small capillary number non-uniformities present in solutions valid near the corner.
An article on this work has been published in J. Fluid Mech. 257 (1993) 1-31. PDF version of this manuscript
Local Fluid Flow and Heat Transfer Near Contact Lines
We consider steady two-dimensional fluid flow and heat transfer near contact lines in single-phase and two-phase systems. both single- and double-wedge geometries admit separable solutions in plane polar coordinates for both thermal and flow fields. We consider the class of functions which have bounded temperatures and velocites at the corner. When free surfaces are present, we seek local solutions, those that satisfy all local boundary conditions, and partial local solutions, those that satisfy all but the normal-stress boundary condition. Our aim in this work is to describe local fluid flow and heat flow in problems where these fields are coupled by determining for which wedge angles solutions exist, identifying singularities in the heat flux and stress which are present at contact lines, and determining the dependence of these singularities on the wedge angles. For thermal fields in two phases we identify two modes of heat transfer that are analogous to the two modes identified by Proudman and Asadullah (1988) for two-fluid flow. For non-isothermal flow, locally, convection does not play a role but coupling through thermocapillary effects on non-isothermal free surfaces can arise. We find that under non-isothermal conditions a planar free surface must leave a planar rigid boundary at an angle of 180 degrees, the same angle found by Michael (1958) for an isothermal rigid/free wedge, in order to satisfy all local boundary conditions. Finally, we find that situations arise where no coupled solutions of the form sought can be found; we discuss means by which alternative solutions can be obtained.
An article on this work has been published in J. Fluid Mech. 268 (1994) 231-265. PDF version of this manuscript See also an addendum to this article in J. Fluid Mech. 371 (1998) 377-378. PDF version of this addendum
Fluid Flow, Heat Transfer and Solidification Near Tri-junctions
Steady, two-dimensional fluid flow and heat transfer are considered near tri-junctions at which solidification is occurring. Meniscus-defined configurations as well as closed configurations such as directional solidification are examined. The local wedge geometry admits separable solutions in plane polar coordinates. Over the class of functions which have bounded temperatures and velocities at the corner, local solutions, those which satisfy all local boundary conditions, and partial local solutions, those which satisfy all but the normal-stress boundary condition, are considered. The aim in this work is to describe local fluid flow and heat transfer in problems where solidification is occurring by identifying singularities in the heat flux and stress which are present at the tri-junction, determining the dependence of these singularities on the wedge angles, and determining when specific wedge geometries are selected. It is found that the locally dominant flow is that due to the expansion or contraction of the material upon solidification.
An article on this work has been published in J. Cryst. Growth 142 (1994) 245-252. PDF version of this manuscript
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