A model of Stokesian peristalsis and vesicle transport in a three-dimensional closed cavity

Abstract

The complexity of the mechanics involved in the mammalian reproductive process is evident. Neither an ovum nor an embryo is self-propelled, but move through the oviduct or uterus due to the peristaltic action of the tube walls, imposed pressure gradients, and perhaps ciliary motion. Here we use the method of regularized Stokeslets to model the transport of an ovum or an embryo within a peristaltic tube. We represent the ovum or the embryo as a spherical vesicle of finite volume - not a massless point particle. The outer membrane of the neutrally buoyant vesicle is discretized by nodes that are joined by a network of springs. The elastic moduli of these springs are chosen large enough so that a spherical shape is maintained. For simplicity, here we choose an axisymmetric tube where the geometry of the two-dimensional cross-section along the tube axis reflects that of the sagittal cross-section of the uterine cavity. Although the tube motion is axisymmetric, the presence of the vesicle within the tube requires a fully three-dimensional model. As was found in Yaniv et al. (2009, 2012) for a 2D closed channel, we find that the flow dynamics in a 3D peristaltic tube are strongly influenced by the closed end and the manner in which the peristaltic wave damps out towards the closure. In addition, we demonstrate that the trajectory of a vesicle of finite volume can greatly differ from the trajectory of a massless fluid particle initially placed at the vesicle׳s centroid.

A posteriori error analysis of fluid–structure interactions: Time dependent error

“…The basis vector,ê (k) with k ∈ {1, 2, 3} is defined such thatê (1) is understood to be the basis vector that points along the x-axis,ê (2) along the y-axis, andê (3) along the z-axis. The symbol, δ(·) represents the Dirac delta distribution.…”

“…Of interest here and in many applications [21,20,2], is the quasi-steady state situation where the fluid velocity is found from Stokes equation, but the boundary or particle positions and forces may vary over time. This quasi-steady state assumption is applicable for slowly moving immersed structures and small length scales.…”

“…The singular fundamental solution is regularized by convolution with a smooth, radially symmetric function that satisfies certain moment conditions (e.g. [7,8]), and the integral is approximated as a summation over a finite number of points 2…”

“…In this paper we develop and apply a posteriori error estimation techniques described in [5] to the ordinary differential equations that arise from the spatial discretization of the singular integral operators associated with the MRS. Such systems frequently arise in the study of quasi-steady state phenomena in biofluids [2,6,7,8,21,20]. However, little work has been applied towards understanding the error in specific simulations that use the MRS (although see [8] for some examples of numerical convergence).…”

Morphogenetic implications of peristaltic fluid–tissue dynamics in the embryonic lung

“…There are a few studies of blind-ended peristalsis in a channel of fixed length (Yaniv et al, 2012; Yaniv et al, 2009). However, to our knowledge, the mechanics of blind-ended peristalsis in a tube has until very recently (Aranda et al, 2015) not been studied.…”