Dynamics of a fluid sheet on a curved rigid wall with allowance for phase transformation on the free surface

Abstract

The flow of a three-dimensional sheet on a curved wall is considered . Gravity and surface tension forces act on the sheet while a droplet stream falls on its free surface. The systems of equations of viscous incompressible fluid dynamics on a curved rigid surface and the boundary conditions with allowance for the falling droplet stream are formulated . The problems of steady axisymmetric motion of the sheet on cylindrical and conical surfaces are considered . The effect of the curvature of the rigid wall on the solution is examined .Thin sheet motion on rigid surfaces in the presence of free-surface interphase exchange is of interest in connection with technical problems of heat and mass transfer . The problems of two-dimensional sheet motion on plane surfaces with allowance for various free-surface interactions with the contact medium have been considered (see detailed bibliography in [1]) . In contrast to previous investigations, below we consider the three-dimensional flow in a sheet on a curved wall with allowance for the falling droplet stream of the same fluid . The need to consider the three-dimensional sheet flow on a curved wall stems from the physical formulation of the problems, in particular, in investigating the processes taking place in open-type cooling radiators [2] . Below, omitting small terms of order higher than the first in the quantity c related with the ratio of the characteristic transverse and longitudinal dimensions, we obtain the equations and boundary conditions in the thin-sheet approximation . This system generalizes the well-known equations of thin-sheet dynamics on plane walls [1] . The solution of steady-state axisymmetric problems of sheet dynamics on cylindrical and conical surfaces is obtained. THE THIN-SHEET APPROXIMATIONAs in the case of a boundary layer near a curved surface, we introduce a special curvilinear coordinate system [3] . We denote the curvilinear coordinates of points on the rigid wall E by s t and s 2 , which have the dimension of length . As the spatial coordinate system we chooses', s 2 , and n, where n is the segment length reckoned from the wall surface along the normal to E .Then, the covariant space basis vectors E a =E a (s t ,s 2,n) are related with the basis vectors e a on E by the expressions :an Ea =ea + n , E3 =n (1 .1) as a and the components of the metric tensor g 9 can be determined from the expressions g a a =aae -2nbaa + n 2 c o >where a ag , b aa, and c ae are the coefficients of the first, second, and third quadratic forms of the surface E, respectively, n is the unit normal to E, the Greek subscripts vary from 1 to 2 and the Latin ones -from 1 to 3 .