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The capillary dynamics of a thin layer of viscous liquid which wets a smooth solid surface is considered. To formulate the problems of wetting dynamics the method of matched asymptotic expansions is used. A general method of dosing the asymptotic theory of the dynamic wetting angle which was previously known in the first approximation of a small capillary number is described [1]. For the arbitrary external problem of thin-layer hydrodynamics, which describes the flow outside a small neighbourhood of a moving wetting line, asymptotic boundary conditions are formulated on the wetting line in the third approximation. The transient problem of the spontaneous spreading of a drop of viscous liquid over a solid surface, for which the first-approximation theory is known [2], is solved. The dynamics of the drop is determined for long times. The shape of the drop is obtained in three approximations in the capillary number. The effect of the gravity force on the shape of the spreading drop is determined. The limits of applicability of the model of the quasi-uniform shape of a drop with a dynamic wetting angle which depends on the velocity of the edge of the drop are found.

A general method based on the use of Lagrangian equations for determining hydrodynamic interaction between bodies in a fluid is presented. Formulas for the kinetic energy and the Lagrangian function are reduced to a form which permits an effective application of the method of small parameter. Additive components of kinetic energy and of the Lagrangian function, which determine the hydrodynamic interaction between two bodies, one of which is small in comparison with the distance between the two, are calculated. The method is used for considering the case of several bodies. The results are expressed in terms of coefficients of apparent mass of individual bodies in a boundless fluid. General formulas are derived for forces and moments acting on a body in a nonuniform stream. The feasibility of expressing hydrodynamic reactions on a body in a nonuniform stream in terms of apparent mass coefficients has not been, so far, established. Solutions were sought for bodies of particular form, while in the three-dimensional case no solution was found even for a small sphere. The description of a solid body in a perfect incompressible fluid by Lagrangian equations was first given in [1]. The forces and moments acting on such body moving in a boundless fluid were determined in [2]. The most rigorous proof of equations of motion of a solid body in a boundless fluid appears in [3]. The problem of a body in an arbitrary stream of fluid was apparently first formulated by Zhukovskii [4]. The motion of a body in a uniform accelerated stream was investigated in [5]. Formulas for forces and moments acting on a stationary elliptical cylinder in an arbitrary plane potential flow were derived in [6], while in [7] the force acting on an expanding circular cylinder moving in an arbitrary stream with constant vorticity was calculated. As survey of publications on the hydrodynamic interaction between bodies in a fluid appears in [8].

The process of expansion of a large number of gas bubbles in incompressible viscous liquid is considered. The free boundary shape is determined, and the formation of liquid films at convergence of two gas bubbles is investigated. A model of regularly packed bubbles is used for determining the process of their deformation and of formation of films.

We consider a problem concerning the temperature existing around a radiating sphere situated in an infinite, homogeneous medium in which, in addition to absorption of radiation, we have molecular energy transfer. Solution is obtained under the condition that the radius II of the sphere is small compared with the path length 1 /a of the photons within the medium. At the distances from the sphere which are large compared with i / a the solution corresponds approximately to the radiant heat transfer. When molecular transfer is absent in the problems on radiation, then the temperature discontinuity occurs at the surface of the sphere, but this is smoothed out when a finite value is assigned to the molecular heat transfer coefficient. If this coefficient is small compared with the radiant heat transfer coefficient, but their ratio exceeds considerably the value of the parameter an, then the temperature field near the sphere is defined solely in the terms of the molecular transfer. In this case the energy fIux is dominated by the molecular heat transfer near the sphere and by the radiant transfer away from the sphere.