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On the viscous flows of leak-out and spherical cap natation

Published online by Cambridge University Press:  12 December 2017

Rolf J. Ryham Open the ORCID record for Rolf J. Ryham [Opens in a new window]
Rolf J. Ryham*
Affiliation:
Department of Mathematics, Fordham University, Bronx, NY 10458, USA
*
†Email address for correspondence: [email protected]
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Abstract

This paper deals with the hydrodynamics of a viscous liquid passing through the hole in a deflating hollow sphere. I employ the method of complementary integrals and calculate in closed form the pressure and streamfunction for the axisymmetric, creeping motion coming from changes in radius. The resulting flow fields describe the motion of a deformable spherical cap in a viscous environment, where the deformations include changes in the size of the spherical cap, the size of the hole and translation of the body along the axis of symmetry. The calculations yield explicit expressions for the jumps in pressure and resistance coefficients for the combined deformations. The equation for the translation force shows that a freely suspended spherical cap is able to propel as an active swimmer. The expression for pressure contains the classic Sampson flow rate equation as a limiting case, but simulations show that the pressure must also account for the velocity of hole widening to correctly predict outflow rates in physiology. Movies based on the closed-form solutions visualize the flow fields and pressures as part of physical processes.

JFM classification

Mathematical Foundations: General fluid mechanics Biological Fluid Dynamics: Propulsion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 502 - 531
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Copyright
© 2017 Cambridge University Press 

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Ryham supplementary movie 1

A contracting spherical cap (thick black arc) expels fluid by decreasing in size. The spherical radius R is initially 20 μm and decreases monotonically to the final value 12 μm. Because the cap is freely suspended, the distance traveled by the centre of the sphere is identical to the initial minus the final radius (see (4.21)). The top of the spherical cap is a stagnation point and does not move. In this simulation, the angle α = 150° is constant. Tracer particles (small black dots) show the velocity field in the aqueous phase. The simulation uses the creeping motion (2.3, 2.4) with U1 = d R/dt, U2= 0 and U1 + U3= 0 to define the particle trajectories. The particles pass through the hole opening and do not cross the spherical cap surface.

Download Ryham supplementary movie 1(Video)
Video 7.3 MB

Ryham supplementary movie 2

Complementary to the flow pattern in Movie 1, hole widening draws the spherical cap (thick black arc) and its contents backwards. In this simulation, spherical radius R = 15 μm is constant and the hole radius grows from 3 μm to 15 μm (the angle α decreases monotonically from 168° to the final angle 90° over time). The creeping motion (2.3, 2.4) driven by changes in the hole angle uses U1 = 0 and U2 = Rdα/dt. The translation velocity U3 derives from momentum conservation (3.16) with F3 = 0.

Download Ryham supplementary movie 2(Video)
Video 6.9 MB

Ryham supplementary movie 3

The stretched liposome is under tension. At first, leak-out cannot relieve the tension because the initial pore (radius 0.05 μm) is too small to allow passage of the viscous fluid. Instead, the pore widens, decreasing surface area and releasing energy stored in stretching. Leak-out initiates once the pore is sufficiently wide, and continues to expel fluid until the pore has collapsed. The liposome and its contents shift backwards in hole widening and shift forwards during hole closure. In this simulation, the parameters R, α and z0 are the time course for kinetic equations (7.2-7.4). The creeping motion (2.3, 2.4) defines the tracer particle trajectories in the aqueous phase. In this slow motion movie, one second of realtime corresponds to 3.3 milliseconds. But to make the movements more apparent, the slow, linear closure stage plays ten times faster. Once the pore has collapsed, the final liposome volume is about 80 % that of the initial volume.

Download Ryham supplementary movie 3(Video)
Video 10.6 MB

Ryham supplementary movie 4

The stretched liposome exerts a pressure on the fluid inside the sphere. Initially, intracellular pressure (red, about 5 Pa) behaves as the Young-Laplace pressure for an intact sphere, and dissipates to about 0.1 Pa with the decreasing surface area and decreasing intracellular volume. The ambient reference pressure is 0 Pa (white). The blue colouration shows the inverse square root singularity just outside the edge of the hole. There, the pressure is negative because the edge is retracting. Around 15 s realtime, the edge singularities are positive (red spots) as the surface extends into the fluid. Hole closure increases surface area at a rate greater than that of leak-out, restoring some stretching. The pressure is about 0.5 Pa (pink) at the time of hole collapse. This movie uses the same velocities as in movie 3, and the analytical expression (5.9) to plot the pressure fields.

Download Ryham supplementary movie 4(Video)
Video 2.8 MB