Analytical Solution of a Creeping Flow Impinging on a Spherical Cap-Shaped Bubble on a Flat Solid Surface

In this study, a general, analytical solution of a steady creeping or Stokes flow impinging on a stationary spherical cap-shaped bubble on a solid flat surface is provided. The phenomena usually take place in bubble∕pore formation in materials and manufacturing processing and MEMS, boiling heat transfer, and nucleation and growth of gas bubbles in tissues of animals and human, etc. In view of high capillary pressure and small liquid pressure, the shape of the bubble in a microscale can be considered as a spherical cap on a surface. In this model, shear stresses associated with the no-slip condition, interfacial mass transport such as condensation and evaporation are absent on the bubble surface. An analytical solution of the Stokes equations for zero Reynolds number flow in a toroidal coordinate system is found by decomposing the flow into a stagnation flow and a flow disturbed by the bubble and applying the separation-of-variables method. The stream function can be expressed in terms of a difference in Legendre functions of the first kind. The effects of impinging velocity and contact angle of the bubble on the flow pattern and pressure distribution are provided.