Wet-tip versus dry-tip regimes of osmotically driven fluid flow

Oleksandr Ostrenko,Jochen Hampe,Lutz Brusch

doi:10.1038/s41598-019-40853-7

Oleksandr Ostrenko, Jochen Hampe + Show 1 more

Open Access

https://doi.org/10.1038/s41598-019-40853-7

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Abstract

The secretion of osmolytes into a lumen and thereby caused osmotic water inflow can drive fluid flows in organs without a mechanical pump. Such fluids include saliva, sweat, pancreatic juice and bile. The effects of elevated fluid pressure and the associated mechanical limitations of organ function remain largely unknown since fluid pressure is difficult to measure inside tiny secretory channels in vivo. We consider the pressure profile of the coupled osmolyte-flow problem in a secretory channel with a closed tip and an open outlet. Importantly, the entire lateral boundary acts as a dynamic fluid source, the strength of which self-organizes through feedback from the emergent pressure solution itself. We derive analytical solutions and compare them to numerical simulations of the problem in three-dimensional space. The theoretical results reveal a phase boundary in a four-dimensional parameter space separating the commonly considered regime with steady flow all along the channel, here termed “wet-tip” regime, from a “dry-tip” regime suffering ceased flow downstream from the closed tip. We propose a relation between the predicted phase boundary and the onset of cholestasis, a pathological liver condition with reduced bile outflow. The phase boundary also sets an intrinsic length scale for the channel which could act as a length sensor during organ growth.

Highlights

Taking the definite integral with respect to the axial coordinate from 0 to z on both sides of Eq 4, we can solve for solute concentration c(z) at steady state
We introduce the term “wet-tip regime” for an unobstructed steady state outflow from the very tip of the channel
Our model suggests that the dry-tip regime may arise either due to reduction in the width of the channel thereby reducing Lmax ∼ a3/2 or by increasing the viscosity of bile thereby reducing Lmax ∼ μ−1/2

Summary

Methods

Fluid flow in long and narrow cylindrical channels allows for approximations to the Navier-Stokes equations. U(x) = (u, v, w) is the fluid velocity in cylinder coordinates, p(z) is the fluid pressure and μ is the dynamic viscosity. We consider the steady laminar flow that preserves the axial symmetry of the channel geometry since the Reynolds numbers for the considered biological systems are low, e.g. for bile flow it was estimated as Re~10−6 which lies 9 orders of magnitude below the threshold to turbulent flow[25]. For a L we can neglect the radial components of the fluid velocity, thereby reducing the problem to one spatial dimension. As it suffices to analyze the cross-section average w of the axial velocity component, we set it proportional to the negative local pressure gradient, following Darcy’s law w(z) = − a2 ∂p(z)

Results

Discussion

Conclusion