Abstract
We study the three-dimensional hydrodynamic interaction of a pair of identical, initially spherical capsules freely suspended in a simple shear flow under Stokes flow conditions. The capsules are filled with a Newtonian liquid (same density and viscosity as the suspending fluid). Their membranes satisfy the neo-Hookean constitutive law. We consider the rarely studied case where the capsule centres are initially located in (or near) the plane defined by the flow direction and the vorticity vector, i.e. in two different shear planes. The motion and deformation of the capsules are modelled by means of a boundary integral technique to compute the flows, coupled to a finite element method to calculate the force exerted by the membranes on the fluids. We follow the motion and deformation of the capsules as they are convected towards each other after a sudden start of the flow. Our main finding is that, depending on their initial position and deformability, the two capsules may oscillate slowly about the flow gradient axis, get nearer to each other at each oscillation to finally interact strongly and separate. This minuet motion had not been identified previously. We identify the regions of space where either simple crossing or minuet occurs. This phenomenon has a marked influence on the irreversible trajectory drift of two capsules after crossing: the minuet process leads to a significant trajectory displacement along the flow gradient when none was expected, based on the previous studies where the two capsules had a significant relative velocity.
Highlights
The hydrodynamics of pairwise interaction of deformable particles is a crucial topic for semi-dilute suspension rheology (Batchelor & Green 1972a; Guazzelli & Morris 2012)
The fluid–structure interaction problem is solved by means of the numerical scheme that couples a boundary integral method (BI) to solve the fluid flow and the finite element method (FE) to solve the membrane mechanics (Walter et al 2010; Hu et al 2012)
When two capsules with a NH membrane are in the same shear plane (X3(0)/a = 0), it is a well-established fact that, after crossing, the two capsules are irreversibly displaced from their initial trajectory: for a given value of Ca, the trajectory shift δ2 = |X2( f)| − |X2(0)| is maximum for X2(0)/a = 0, decreases when |X2(0)| increases and becomes almost zero for |X2(0)|/a > 2 (Lac & Barthès-Biesel 2008; Pranay et al 2010; Omori et al 2013; Gires et al 2014)
Summary
The hydrodynamics of pairwise interaction of deformable particles is a crucial topic for semi-dilute suspension rheology (Batchelor & Green 1972a; Guazzelli & Morris 2012). The crossing leads to an irreversible trajectory shift along the shear gradient (x2-direction) This effect, which decreases with an increase of the capsule deformability and/or the initial distance. Doddi & Bagchi (2008) used this technique to model the pair interaction of two initially spherical capsules, when the inertia of the flow was not negligible They found that, when the flow Reynolds number increased, the capsules did not cross, but reversed their motion. Lac & Barthès-Biesel (2008) modelled the three-dimensional motion of two capsules positioned in two different shear planes ( X3(0) = 0) In this case, a sideways leapfrog motion occurs with a maximum trajectory displacement along both the x2and x3-directions, which decreases as X3(0) and/or capsule deformability increase.
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