Theory of hydrodynamic interaction of two spheres in wall-bounded shear flow

The seminal Batchelor-Green's (BG) theory on the hydrodynamic interaction of two spherical particles of radii $a$ suspended in a viscous shear flow assumes unbounded fluid. In the present paper we study how a rigid plane wall modifies this interaction. Using an integral equation for the surface traction we derive the expression for the particles' relative velocity as a sum of the BG's velocity and the term due to the presence of a wall at finite distance, $z_0$. Our calculation is not the perturbation theory of the BG solution, so the contribution due to the wall is not necessarily small. We indeed demonstrate that the presence of the wall is a singular perturbation, i.e., its effect cannot be neglected even at large distances. The distance at which the wall significantly alters the particles interaction scales as $z_0^{3/5}$. The phase portrait of the particles' relative motion is different from the BG theory, where there are two singly connected regions of open and closed trajectories both of infinite volume. For finite $z_0$, besides the BG's domains of open and closed trajectories, there is a domain of closed (dancing) and open (swapping) trajectories that do not materialize in an unbounded shear flow. The width of this region grows as $1/z_0$ at smaller separations from the wall. Along the swapping trajectories, which have been previously observed numerically, the incoming particle is turning back after the encounter with the reference particle, rather than passing it by, as the BG theory anticipates. The region of dancing trajectories has infinite volume and is separated from a BG-type domain of closed trajectories that becomes compact due to presence of the wall. We found a one-parameter family of equilibrium states that were previously overlooked, whereas the pair of spheres flows as a whole without changing its configuration. These states are marginally stable and their perturbation yields a two-parameter family of the dancing trajectories, whereas the test particle is orbiting around a fixed point in a frame comoving with the reference particle. We suggest that the phase portrait obtained at $z_0\gg a$ is topologically stable and can be extended down to rather small $z_0$ of several particle diameters. We confirm this hypothesis by direct numerical simulations of the Navier-Stokes equations with $z_0=5a$. Qualitatively the distant wall is the third body that changes the global topology of the phase portrait of two-particle interaction.

Figure 1

Setup of a particle pair in the Poiseuille flow (depicted in a comoving reference frame). In this work we study the case $z_0\gg a$, where $a$ is the particle radius, however, does not necessarily require $z_0\gg r$. The upper wall is assumed to be much further away from the particle pair than $z_0$.

Figure 2

Phase portrait of the trajectories in the symmetry $xz$ plane, within the BG approximation of $z_0=\infty$. All lengths are scaled with the particle radius $a$. Throughout the paper the reference sphere is at the origin and the trajectories of the second sphere are shown. Due to the fore-and-aft and top-down symmetries only one quadrant is depicted. The two types of the trajectories—closed (blue) and open (red), are separated by the separatrix, the open trajectory that asymptotically approaches the $x$ axis [4, 6]. For two spheres at the same vertical line, the maximal separation for closed trajectories is of order of ${10}^{-5}$; see Ref. [5]. As a result, at this resolution, the trajectories are indistinguishable when approaching the $z$ axis. This includes the shown open trajectory that crosses the $z$ axis slightly above the closed trajectories. The time-period of revolution along the shown closed trajectory is more than 700 (here and thereafter the time units of inverse shear rate ${\dot{\gamma }}^{-1}$ are used).

Figure 3

The phase portrait in the symmetry $xz$ plane at finite $z_0=5$. The fore-and-aft and top-down symmetries survive the wall perturbation in the leading order. The phase portrait exhibits two critical (equilibrium) points: the saddle (hyperbolic) point $r_s$ and the neutral equilibrium (elliptic) point $r_c$, representing a completely different topology from the BG theory in Fig. 2. As $z_0\to \infty$, the topology of the phase portrait is preserved, while the critical points are being shifted to infinity.

Figure 4

(a) There is a unique value of the streamwise separation distance between two particles, $x=r_c$, flowing along the same streamline of the Poiseuille flow, for which they flow steadily without changing their configuration. The stability of such motion is marginal, cf. panel (b). The value $r_c=4z_0$ is confirmed in the numerical simulations of the Navier-Stokes equations for $z_0=5$. (b) Trajectories that pass through points around the stationary point $x=r_c$, $z=0$, crossing $x$ axis at distance larger than $2\sqrt{2}{z}_0$, exhibit a peculiar dancing dynamics. In the coordinate frame comoving with the trailing (left) particle, the leading (right) sphere follows an elongated closed orbit around $(r_c,0)$. Similar trajectories hold outside the symmetry plane.

Figure 5

The blue line depicts the closed trajectory that forms a loop around the critical circle of radius $r_c$ (whose segment is shown by the green line) for $z_0=5$. The trajectory can be shrunk to a (necessarily critical) point by continuously changing the initial conditions. The red line shows the BG trajectory that starts from the same initial condition as the blue line. The period of revolution along the closed trajectory is 1165.

Figure 6

Three-dimensional BG trajectory (red curve) of the sphere in the frame comoving with the reference sphere. The closed trajectory has a geometrical center on the $y$ axis. In contrast to the trajectory in Fig. 5, this curve cannot be shrunk to a point by a continuous change of the initial conditions. That point would have to be critical and the BG phase space does not admit those. For this trajectory, a distant wall is only a regular small perturbation. The period of revolution is 120.

Figure 7

The ratio $V_z/V_z^0$ for $z_0=20$ (a) in the $xz$ plane and (b) along the $x$ axis. The presence of the wall increases the velocity at $x\gg z_0$ by a constant large factor of order $z_0^2$. The two critical points are the neutral equilibrium point $r_c=4z_0=80$ and the saddle point $r_s={(32z_0^3/15)}^{1/5}$ [see the inset in (b)].

Figure 8

The phase portrait in the $xz$ plane for $z_0=5$ (a) and $z_0=10$ (b). The BG separatrix obeying $z^2=16/\left(9x^3\right)$ at large $x$ is depicted by the dashed (grey) line. For any finite $z_0$, the phase portrait contains two disconnected regions of closed trajectories, in contrast to one region at $z_0=\infty$. Region I, where all trajectories are closed and the spheres are close to each other, is similar to that at $z_0=\infty$. Region II is also similar to the $z_0=\infty$ case: all the trajectories are open and the vertical separation after the interaction returns to its original value. Region III has no counterpart at $z_0=\infty$. This region contains both closed and open trajectories (see Fig. 12 for a more detailed description). The trajectories passing not far from the stationary point $r_c$ are closed, orbiting around this point. The swapping open trajectories instead are characterized by a sign reversal of the vertical component of the separation vector after the encounter. The region of swapping trajectories is bounded from one side by the closed trajectories around $r_c$ and from the other side by open nonswapping trajectories.

Figure 9

The separatrices for $z_0=20$ corresponding to the initial point $(r_{s}cos\varphi ,r_{s}sin\varphi ,0)$ with $\varphi =7\pi /90$.

Figure 10

Surface of rotation formed by the separatrices that pass through $(r_{s}cos\varphi ,sin\varphi ,0)$ with $0\le \varphi \le 2\pi$ at $z_0=20$. The green curves represent separatrices corresponding to $\varphi =0,\pi /36,\pi /18,5\pi /36,\pi /4$. The trajectories inside the region formed by the blue surface are closed BG-type orbits, whereas the trajectories inside the orange surface are either open swapping or closed dancing trajectories, as in region III in Fig. 12, cf. Fig. 9.

Figure 11

Comparison of representative open trajectories in the BG case (red) and in the case of a wall at a finite distance $z_0$ (blue): (a) typical open trajectories are qualitatively similar in both cases ($z_0=20$); (b) for some initial conditions the presence of the wall results in the appearance of a swapping trajectory ($z_0=5$). The black arrows indicate the direction of motion.

Figure 12

The dancing-swapping region III of Figs. 8 for $z_0=20$. The red line is the region's boundary that crosses the $x$ axis at ${\left(32z_0^3/15\right)}^{1/5}$. The dashed blue line separates open swapping and closed dancing trajectories and crosses the $x$ axis at $x_s=2\sqrt{2}{z}_0$. The black dot is the equilibrium point $(r_c=4z_0,0)$.

Figure 13

Asymptotic vertical velocity of particle 2 relative to particle 1 as a function of the horizontal separation, at $z_0=5a$ (cf. Fig. 1). We focus on the range where the theory predicts change of sign of the velocity and the associated critical point. The theory is seen to hold accurate predictions even in geometrically confined Poiseuille flows.