Steady axisymmetric motion of a small bubble in a tube with flowing liquid

The steady axisymmetric behaviour of a relatively small bubble moving with a. owing liquid in a straight round tube is studied by computationally solving the nonlinear Navier-Stokes equations, using a Galerkin finite-element method with boundary-fitted mesh, for wide ranges of capillary number Ca and Reynolds number Re. Here a bubble is considered relatively small when its volume-equivalent radius is less than that of the tube. At small values of Re, the velocity of a bubble increases with bubble size for large values of Ca but decreases with bubble size for small values of Ca. At large values of Re, however, a bubble of large size appears to move at a slower velocity for any given value of Ca. When Re is large (e. g. Re = 100) and Ca > 0.1, a bubble of radius greater than half of the tube radius moves at a velocity that seems to be independent of bubble size. The strong inertial effect at large Re makes a small bubble of radius greater than a quarter of the tube radius to deform into a noticeable oblate shape as Ca increases from very small value, and then to be elongated into a bullet shape with further increasing Ca after Ca reaches an intermediate value. Even very small bubbles (e. g. of radius equal to one-tenth of the tube radius) can still be significantly deformed provided that the value of Ca is adequately large. Despite significant shape deformations that may still occur, bubbles of radius less than a quarter of that of the tube almost always move at the same velocity as that of the local liquid flow at the tube centreline (i.e. twice that of the average liquid velocity), regardless the values of Re and Ca. This fact suggests that very small bubbles are basically carried by the local liquid flow.