On a steady, viscous flow in two-dimensional collapsible channels
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Two characteristic cases of steady, viscous flow in 2-D collapsible channels are considered in the paper: low Reynolds number, lubrication type flow, and moderately high Reynolds number, boundary-layer type flow. In the latter case the boundary layer occupies the whole cross-section of the channel, and, in order to treat the flow analytically, we apply an approximate method based strongly on the Karman-Pohlhausen method, known from early developments of the classical boundary-layer theory. In modeling the elastic behavior of channel walls, we show how the careful scaling employed in studying the fluid flow imposes the application of geometrically nonlinear Karman shell theory. For the lubrication type flow we derive a single integro-differential equation describing the wall configuration, while in the boundary-layer type flow a similar integro-differential equation is supplemented by two first order, nonlinear ordinary differential equations. In both cases the governing equations are solved numerically by finite differences, for a wide range of control parameters. At that, steady flow was found to exist in all cases considered (we did not perform the stability analysis of the obtained solutions). Also, in the boundary-layer type flow the boundary layer was shown to be very resistant to separation, so that the presented theory was valid over the whole length of the channel. The effect of control parameters, in particular of the volume flow rate and the entrance transmural pressure, upon the wall configuration was discussed.
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