Abstract
Let us consider the problem of the development of laminar threelayer flow of an incompressible viscous fluid in a fiat channel x ~ > > 0, ly'[ < h. Let the fluid parameters in the inner layer (ly ~ < y;) be u[, v], 9~, P~ /@ and those in the outer wall layers u[, v [ , p[ , p~ g~. Here u ~ v" are the velocity components, p~ is the viscosity coefficient, p~ is density, p~ is pressure. If the channel is sufficiently narrow the pressures in the Myers are the same. We shall consider the flow symmetric relative to the x ~ axis. Let us assume that the transverse velocity component is small in comparison with tile streamwise component, and that the line of separation of the inner and outer layers, determined by the equation y" = y0(x~ differs little from the straight line y~ = y~* over the length of the initial zone. Therefore, in the first approximation we shall consider the line of separation to be straight. Let us also assume that the thickness of the outer layer is small in comparison with the thickness of the inner layer (h y~ << y~), and that the flow in the outer layer may be described by the equations of hydrodynamic lubrication theory. With account for these assumptions we obtain the following dimensionless equations and boundary conditions [1] :
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