Axisymmetric thermocapillary motion in a cylinder at small Marangoni number

The solution to the linear problem of axisymmetric thermocapillary motion of two non-miscible viscous fluids in a cylindrical tube is presented. Their common interface is fixed and undeformable. This problem is an inverse problem because pressure gradients are unknown functions. The solution of the non-stationary problem is presented in the form of analytical expressions. They are obtained with the use of the method of Laplace transformation. If the wall temperature is stabilized then the general solution tends to the stationary solution as time increases. Numerical calculations confirm the theoretical results.