The phase separation of a binary fluid can lead to the creation of two horizontal fluid layers with different concentrations resting on a solid substrate and divided by a diffuse interface. In the framework of the Cahn-Hilliard equation, it is shown analytically and numerically that such a layered system is subject to a transverse instability that generates a slowly coarsening multidomain structure. The influence of gravity, solutocapillary effect at the free boundary, and Korteweg stresses inside the diffuse interface on the stability of the layers is studied using the coupled system of the hydrodynamic equations and the nonlinear equation for the concentration ͑H model͒. The parameter regions of long-wave instabilities are found.
Abstract:The qualitative properties of solutions of a hereditary model of motion of aqueous solutions of polymers, its modification in the limiting case of short relaxation times, and a similar second grade fluid model are studied. Unsteady shear flows are considered. In the first case, their properties are similar to those of motion of a usual viscous fluid. Other models can include weak discontinuities, which are retained in the course of fluid motion. Exact solutions are found by using the group analysis of the examined systems of equations. These solutions describe the fluid motion in a gap between coaxial rotating cylinders, the stagnation point flow, and the motion in a half-space induced by plane rotation (analog of the Karman vortex). The problem of motion of an aqueous solution of a polymer in a cylindrical tube under the action of a streamwise pressure gradient is considered. In this case, a flow with straight-line trajectories is possible (analog of the Hagen-Poiseuille flow). In contrast to the latter, however, the pressure in the flow considered here depends on all three spatial variables.
Axisymmetric motion of a leading fluid drop and a trailing gas bubble (or thermally nonconducting drop) in a viscous fluid under the combined effect of gravity and thermocapillarity is considered under the assumption of negligible inertia effects and of nondeformable interfaces. The ambient fluid far from the inclusions is isothermal and the temperature of the leading particle differs from that of the continuous medium. At large Peclet number, thermal boundary layers are present along the fluid-liquid and the gas-liquid interfaces, and thermal wakes are formed downstream from the particles. The interaction of the thermal wake, shed from the leading inclusion, with the thermal boundary layer on the surface of the trailing one causes a nonuniform temperature distribution on the surface of the latter. The induced Marangoni flow results in the change of the flow pattern, the velocity of both particles, and the equilibrium separation distance. In the present paper, the influence of the Marangoni effect on the drag force and the rate of heat transfer from the drops translating at given velocity is studied as well as on the equilibrium velocities and separation distance of the drops freely migrating under the action of gravity. The analysis considers particles at arbitrary separation distance and takes full account of thermal and hydrodynamic interactions.
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