Inertial effects in closed cavity flows and their influence on drop breakup

An exact similarity solution to the steady Navier-Stokes equations for the flow in an infinitely long channel or tube with an accelerating surface velocity is presented. Axisymmetric nonswirling flow in a tube is investigated. The structure of the flow in a very long tube when the Reynold number R exceeds 10.25 is studied. It is shown that the problem of determining the motion in a long finite tube is equivalent to that of selecting the initial condition for the boundary layer equations that properly takes into account the presence of the reverse flow. Numerical solutions are obtained for the flow in a finite tube for Reynolds numbers up to 70. The same change in structure brought about by the returning fluid occurs in a finite two dimensional channel at R = 57. The influence of the internal inertia on the deformation and breakup of a slender drop suspended in an axisymmetric extensional flow is studied. The deformation and critical conditions at breakup are obtained. It is proven that the shape of a slender drop in an extensional flow must be an analytic function. A family of solutions to the Stokes equations for cylindrical geometry is developed.