Solutions of the Oberbeck-Boussinesq equations with a linear dependence of temperature on one of the space coordinates firstly were studied by G.A. Ostroumov (1952). Their exact solution, which describes plane stationary flow in a strip under action of longitudinal temperature gradient and transversal gravity field, was obtained by R.V. Birikh (1966). The upper strip boundary can be a non-deformable free surface. Asymptotical character of this solution was confirmed by both experimental and numerical methods (A.G. Kirdyashkin, V.I. Polezhaev and A.I. Fedyushkin, 1983). Generalization of mentioned solutions for flows in a horizontal cylindrical tube with an arbitrary cross section is given by V.V. Pukhnachev (2000). Here the velocity vector has three components but they do not depend on the axial coordinate while the temperature and pressure depend on it linearly. One can hope that these solutions give a good flow description in the main part of a long tube, which ends are solid impermeable isothermal walls. Another example of three-dimensional convective flow obtained on the base of solving two-dimensional equations gives the problem of thermal convection in a circular rotating tube with an axial temperature gradient (R.V. Birikh and V.V. Pukhnachev, 2011). Angular velocity of the tube rotation and gravity acceleration in this problem can depend on time arbitrarily. If the gravity is absent the problem becomes a linear one and admits exact solutions. From the point of interpretation, the solution, in which the liquid flux through the tube cross section is equal to zero, has a special interest. The characteristic peculiarity of the considered class of solutions is the possibility of passive admixture transport on large distance along the tube under joint action of longitudinal temperature gradient and transversal centrifugal or gravity forces (moreover the values of the latter can be very small). Exact solutions of plane and axially symmetric problems for uniform and two-layered liquids, in which longitudinal temperature gradient is a function of time, are considered also. In the last case, thermocapillary effect on the interface is taken into account.
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