Effect of thermal convection

The discrepancies between the experimental trajectory and the numerical prediction may be explained by taking account of the approximation of the slender body theory, the dynamics of the experiments, the buoyancy effect, and thermal effects in highly viscous fluids. However, the thermal effect is not controlled in our experiment. In this section, we briefly study the thermal effects with a simplified situation.

When the Rayleigh number exceeds some critical value, the background flow is no longer stationary. The convection is a very slow motion. For the experiments, thermal effect convection was indeed detected in our highly viscous fluids. Unfortunately, it is hard to measure globally. Fortunately, our model has the ability to handle any given background flow, even if time dependent. To better understand the thermal effect, we check the flow with a straight rod sweeping out a double cone with a uniform background flow in free space.

−2 −1 0 1 2 3 4 −0.5 0 0.5 1 1.5 2 0.150.2 0.250.3 0.35 x N=50, z 0=0.25, U0=(0.05,0,0) B:r 0=1; M:r0=1.5,G:r0=2 y z

Figure 7.23: Fluid particle trajectories with a straight rod sweeping out a double cone in free space. The number of revolution of the rod isN = 40, and the uniform background flow is (0.05,0,0). The particles are initialized at red stars with the initial the initial height z0 = 0.25. For blue trajectories, r0 = 1 in the cylindrical coordinates for the

initial positions, andr0 = 1.5,r0 = 2 for the magenta and green, respectively.

Figure 7.23 show the fluid particle trajectories when a straight rod sweeps out a double cone in free space. This verifies that if the flow induced by the rod is dominant then there are closed orbits near the cone. While the particle moves away from the rod, the background flow is comparable to the flow induced by the rod, then the particles move along open trajectories. Figure 7.24 shows the effect of the magnitude of the uniform background flow to the fluid particle trajectories.

Go back to Chapter 6, where we compare the experimental trajectory with the model for the straight case. The trajectories in Figure 6.8 are effected by the convection based on the asymmetry of the trajectory in intermediate time.

Figure 7.24: Fluid particle trajectories with a straight rod sweeping out a double cone in free space. The fluid is at rest for the black trajectories. The uniform background flow is (0.0005,0,0), (0.005,0,0), and (0.05,0,0) for the blue, yellow, and red trajectory, respectively.

Chapter 8

A swimming related application of

the slender body theory in Stokes

flows

The slender body theory has been developed for a long time [75, 21, 74, 3, 39]. It provides a good asymptotic solution for flows generated by a slender body, if ignoring the end effect. The end effects can be attained with higher order singularities [38]. In this chapter, we report a swimming-related application of the slender body theory. We study the velocity field, the fluid particle trajectory and the flux introduced by the periodic motion of a slender body. This study may shade light on the efficiency of swimming or propelling in the low-Reynolds-number regime.

8.1

The problem

The cylindrical slender body (red) is attached to a fuselage (gray) by a fine wire and moves along the track (green) on the fuselage as shown in Figure 8.1. In the fuselage’s body frame, the cylindrical slender body moves periodically in a plane. Assume in the fuselage’s body frame the surrounding fluid is stationary, the configuration of the

(a) (b) (c) (d) Figure 8.1: Swimmer.

periodic motion of the rigid slender body is show in Figure 8.2. We examine on the flow induced by the slender body in this chapter. Study of the effect of the fuselage and the whole system will be pursued in the future work.

The body moves periodically from step (a) to step (e), divided into 4 phases con- sidering its motion and direction. After one revolution, the body returns to its original position in the fuselage frame. If the fuselage is fixed and the surrounding fluid is at rest, the background flow in the fuselage frame is stationary. If the fuselage is free in a stationary fluid, the background flow in the fuselage frame depends on the motion of the system. We focus on the case when the background flow in the fuselage frame is at rest, and refer the fuselage frame as the lab frame in the rest of this chapter. Since the flow is in the Stokes regime, inertia effect is negligible and transient is neglect. We focus on the steady Stokes flow for each phase. Generally, two types of motion are considered, uniform translation and pure rotation.

The half length of the cylindrical slender body ` is much large compared with its cross-sectional radiusr (r`). We study the flow induced by the slender body when it moves periodically in the laboratory frame, which is selected that the slender body is always in the x-y plane. Beside the laboratory frame (the fuselage frame), another important reference frame used here is the body frame, which is a moving frame. The

(a) (b) (c) (d)

(e=a) (a) (b) (c) (d)

(e)

Figure 8.2: Configuration of the periodic motion of the slender body in the x-y plane.

velocity field is presented with explicit formulae using the slender body theory.