The Penetration of a Finger into a Viscous Fluid in a Channel and Tube

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 equations for low Reynolds number flow. To solve the equations, an assumption for the shape of the finger is made and the normal-stress boundary condition is dropped. The remaining equations are solved numerically by covering the domain with a composite mesh composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. The shape of the finger is then altered to satisfy the normal-stress boundary condition by using a nonlinear least squares iteration method. The results are compared with the singular perturbation solution of Bretherton (J. Fluid Mech., 10 (1961), pp. 166–188). When the axisymmetric finger moves through a 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 (J. Fluid Mech., 10 (1961), pp. 161–165) as a function of the dimensionless parameter µU/T. The numerical results are compared with the experimental results of Taylor.