FINITE-ELEMENT CALCULATIONS OF FLOW PAST A SPHERICAL BUBBLE RISING ONTHE AXIS OF A CYLINDRICAL TUBE

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated b y solving the Navier-Stokes equations and the continuity equation usin g the finite element technique with the penalty function method. Termi nal rise velocities of spherical air bubbles in water have been calcul ated as function of the bubble radius and some conclusions have been d rawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined a nd a new empirical equation is derived for the waif correction factor for a sphere rising with free slip at its surface at low Reynolds numb er.