Shear flow past a sphere is one of the fundamental problems in fluid dynamics, and the singularity method has been used [2] to construct exact solutions of the velocity field to several Stokes flow problems. Building on the exact velocity field, we explore the analytic and numerical results of the fluid particle trajectories for linear shear flow U = Ωzex + Uex past a stationary isolated object in Stokes regime.
Analytical and numerical results
Obtaining the explicit Lagrangian trajectory of fluid particles is not a trivial task and sometimes is impossible for a general 3D flow. Based on the geometry of the object, we rewrite the velocity field to an ODE system in spherical coordinates (r,θ,ϕ), and choose radius r as the new independent variable. We decouple the obtained ODEs and solve the system analytically when the sphere is centered on the zero velocity plane of the background shear flow (U = 0). The explicit closed formula of the Lagrangian trajectories is
in which r is the new independent variable and x2 = r2 - y2 - z2. Since the system is autonomous, equations of fluid particle trajectory are equivalent to stream functions.
Using the particle trajectory formula, we attain streamline separation as in Figure 1. Between the separatrix surfaces, fluid particles cannot go through the sphere. This blocking behavior was discussed by Chwang &Wu [2] and Jeffrey & Sherwood [3] for the two-dimensional cylindrical case, and we show rigorously that a similar blockage exists in three-dimensional flows. The two separation surfaces intersect at the y-axis, and any point on the y-axis out of the sphere is a hyperbolic critical point in the flow. In the y = 0 symmetry plane, the distance between the separation surfaces converges to a finite value as |x|→∞. This is significantly different from previous two-dimensional results. For a two-dimensional flow, the distance of separation streamlines goes to infinity far from the sphere. This is known as the Stokes Paradox in two-dimensional space. For the three-dimensional case, there are six stagnation points on the sphere, which are independent of the ratio of the shear flow. These stagnation points are either nodal points or higher order hyperbolic singularities. By rescaling the velocity, we find the stagnation lines connecting these stagnation points on the sphere. These trajectories provide information on how stagnation lines split or merge on the sphere and show the trend of the flow near the no-slip surface of the sphere. Using a vertical plane put at x = ∞perpendicular to the shear direction as Figure 1, we measure the area of intersection of the blocked flow and the vertical plane using the explicit trajectory formula, and find it is infinite. This illustrates an unreported property of the velocity field of 3D Stokes flow, which is not physically observed.
Figure 1: Separation surface generated by stagnation lines(streamlines) in flows.
When the center of the sphere is out of the zero velocity plane of the non-dimensional background shear flow U = zex + Uex, the problem is complicated by the fact that U is not zero. Without loss of generality, we assume U > 0. The number of stagnation points on the sphere is reduced from six to two when U ≥ 8∕3. The critical points in the interior of the flow are still in the y-z plane but are more complex than points on the y-axis. They are found explicitly as a parameter function of r and U. These critical points are classified as elliptical, hyperbolic, cusp, or higher order hyperbolic critical points based on r and U. Table 1 shows the lateral view of streamlines in the symmetry plane with different U. Intricate local and global bifurcations are found in the particle trajectories below the sphere as the critical constant U varies.
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Table 1: Streamlines in y = 0 plane as U increases (from left to right then top to bottom). The dash lines indicate where the center of the shear flow is relative to the sphere. The black dots are the stagnation points on the sphere in this plane.
[1] Camassa, McLaughlin, R.M. & Zhao, L 2011 Lagrangian blocking in highly viscous shear flows past a sphere. J. Fluid Mech., 669, 120-169.
[2] Chwang, A.T. and Wu, T.Y. 1975 Hydrodynamics of low Reynolds-number flow. Part 2. Singularity method for Stokes flow, J. Fluid Mech., 67, 787-815.
[3] Jeffrey, D.J. and Sherwood, J.D., 1980 Streamline patterns and eddies in low Reynolds number flow. J. Fluid Mech. 96, 315334.
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