Figure 1
From: Wet-tip versus dry-tip regimes of osmotically driven fluid flow
Model and solutions of osmotically driven fluid flow. (A) Model of a secretory system (cylinder with radius a, length L in cylinder coordinates with z along the centerline of the channel) with osmolyte (green) and water (blue) transport coupled through mechanical feedback loops (red arrows). Water influx j(x) through the lateral surface is driven by local osmotic pressure posm(x) from osmolyte concentration c(x) and opposed by hydrostatic fluid pressure p(x), therewith forming one negative feedback loop (upper red arrows). Washout of osmolytes (c(x)) by flow (u(x)) constitutes a second negative feedback loop (lower red arrow). Boundary conditions for flow are mixed: zero velocity at the closed tip (\(z=0\)) and ambient pressure (set to zero) at the open outlet (\(z=L\)), hence a pressure gradient between the boundaries is not imposed but self-organizes. (B,B’) Typical results of 3D flow simulations depicted as streamlines on central plane for parameters \(c=const\) and \(M={M}_{0}\) (B,C), \(M=100\,{M}_{0}\) (B’,C’) as given in Tables 1 and 2, color denotes velocity. Two qualitatively different flow regimes, a wet-tip regime with flow all along the channel and a dry-tip regime with ceased flow near the closed tip of the channel, are encountered. Note the beginning of streamlines at the channel surface, see Methods for details. (C,C’) Analytical solutions for pressure \(p(\tilde{z})\) (Eq. 12), average axial velocity \(\overline{w}(\tilde{z})\) (Eq. 27), osmotic water influx \(j(\tilde{z})\) (Eq. 31) and osmolyte secretion \(g(\tilde{z})\) (Eq. 29) profiles compared to results of numerical simulations as in (B,B’).
