Mathematics & Physics 2015, 8(3), 303–311 УДК 517.532 Axisymmetric Thermocapillary Motion in a Cylinder at Small Marangoni Number Evgeniy P.Magdenko∗ Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Russia Received 21.04.2015, received in revised form 05.05.2015, accepted 20.06.2015 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. <...> All rights reserved c – 303 – (1.5) Evgeniy P.Magdenko Axisymmetric Thermocapillary Motion in a Cylinder at Small Marangoni Number In this case, it follows from the equation of conservation of mass (1.3) that v1 is a linear function of z: v1 = w(r, t)z +w1(r, t). <...> We use solutions (1.5)–(1.9) to describe the motion in a cylindrical tube of radius b with the fluid-fluid interface at a < b. <...> If to write down the problem in dimensionless form, then the nonlinear term will stand Marangoni number M = жθa2/ρνχ. <...> They allow us define the functions h1(t) and h2(t) if function a1(r, t) is known. <...> The stationary solution Let us assume that all functions do not depend on time. <...> Solution of the problem by the method of the Laplace transform To solve linear adjoint problems one can use the Laplace transform [3]. <...> This means that function aj(r, t) tends to constant value as time increases [3]. <...> The motion arises only under the action of thermocapillary forces, that is, initial conditions (1.12) are zero: wj0(r) = 0, j = 1, 2. <...> This means that the solution tends to the stationary solution as time increases. <...> Figs. 1, 2 show the dimensionless function wj = a2wj/ν1 for silicon-water system at temperature of 20oC. <...> The author is grateful <...>