A century ago, Joukowski and Chaplygin [1] found an exact solution to the problem of two-dimensional creeping flow between two eccentrically arranged cylinders (dowel and bearing) with fixed centers, one of which (dowel) rotates. This solution was based on the fundamental work of Petrov (1883) concerning the flow between two concentric cylinders and on the approximated solutions of Reynolds (1886) and Sommerfeld (1904) describing the influence of the eccentricity of the dowel and its bearing on the phenomenon under consideration. The exact solution to the problem was originally constructed in the form of a linear combination of partial solutions of the biharmonic equation obeyed by the stream function. Later, Jeffery [2] developed a general method of solving the biharmonic equation in a cavity between cylinders, using representation of the stream function by the sum of an infinite series of functions of bipolar coordinates. This method was applied in [3] for describing the motion and rotation of a cylinder on a plane. In this problem, the series has a finite number of terms. The purpose of this study is to construct a solution to the problem of flow between circular cylinders, with a free internal cylinder. The dowel is under the action of gravity, the force from the fluid filling the space between the dowel and the bearing, and the action of the moment of forces with respect to the center of the dowel. In the region between the cylinders, the velocity of the fluid is described by the Stokes equation and the continuity equation, with the boundary no-slip conditions on the walls of cylinders. The external cylinder rotates with a preset angular velocity around the immobile axis. The velocity vector of the axis of the free internal cylinder and its angular velocity are determined from the solution to the problem; however, they can be considered as known when solving the hydrodynamic part of the problem. The solution to the problem is found as a function of complex variables and contains two arbitrary harmonic functions. The Joukowski‐Chaplygin classical solution is constructed using a conformal mapping of a rectangle in the bipolar-coordinate plane to an eccentric ring with a slit. Instead of the indicated transformation, the mapping of the concentric ring with its center at the origin of coordinates on the plane is considered. An arbitrary analytical function in the ring can be represented in the form of the Loran series. In the classical problem, the series with material coefficients proves to be finite, which is completely consistent with the Joukowski‐Chaplygin exact solution. The constructed solution to the problem with the free dowel also has a finite number of series terms, but its coefficients are complex quantities. We consider a circular cylinder (bearing) of unit radius with a fixed axis, with a circular cylinder (dowel) of a preset radius d 1 placed inside. The axes of the cylinders are parallel, and the space between them is filled with a homogeneous incompressible highly viscous fluid. We admit that the fluid flow is independent of the coordinate along the axis. In the cross section perpendicular to the bearing axis, we introduce a two-dimensional Cartesian system of coordinates Z = X + iY with the origin on this axis. The dowel axis is projected into the point Z 1 = R 1 ( t ) e i θ ( t ) with the coordinates dependent on the time t , but the dowel axis always remains parallel to the bearing axis. By rotating and shifting the initial system of coordinates, we can construct the system of coordinates z = z 0 + Ze – i ( π + θ ) , in which the dowel and bearing images acquire the form