The steady-state shape of a finger penetrating into a region filled with a viscous fluid is examined. The two-dimensional and axisymmetric problems are solved using Stokes's equations for low Reynolds number flow. Since the viscosity of the fluid inside the finger is assumed to be negliglible in comparison with the viscosity of the fluid exterior to the finger, boundary conditions for a liquid-gas interface are applied on the free surface of the finger.
The two-dimensional case is solved analytically, by using singular perturbation methods. Inner and outer expansions are developed in terms of the small parameter µU/T. An ordinary differential equation for the shape of the finger is solved numerically in order to determine the inner solution. The method of matched asymptotic expansions is used to match the inner and outer solutions.
To solve the fingering problem numerically, an initial guess for the shape of the finger is made by using the perturbation solution. Since the shape of the finger has been fixed, we are forced to drop one of the three boundary conditions on the curved interface. The normal-stress boundary condition is dropped. To solve the resulting problem, the domain is covered with a composite mesh. It is composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. These overlapping grids are stretched so that the number of grid points is greatest in regions where they are needed most. Interpolation equations connect the two grids. Finite difference methods are used to calculate the numerical solution.
The curved interface of the finger is expanded in terms of Tchebycheff polynomials and the known asymptotic behavior of the finger as x → -∞. Using the solution calculated on the fixed domain, the expansion of the interface, and the normal-stress boundary condition, the correct shape of the finger is determined using Newton's method.
When the axisymmetric finger moves through the tube, a fraction m of the viscous fluid is left behind on the walls of the tube. The fraction m was measured experimentally by Taylor [15] as a function of the parameter µU/T. The numerical results show excellent agreement with the experimental results of Taylor.
