To visualize the motion of the fluid, visible markers of the order of 100 microns are placed at the desired locations in the fluid with a syringe to enable optical tracking of their Lagrangian trajectories. Two synchronized PointGray Dragonfly digital cameras are set up at the sides of the tank to capture videos with the program Ladybugrecord [20] as shown in Figure 5.1.
Main image analyses involved in the experiments are raw video processes and the 3D calibration, which has been documented in detail in the 3D calibration section. The raw image processes and the 3D data reconstruction are reported in this section.
The captured videos are originally saved as raw data files which can previewed by “rawplayer” and read by “raw2ppm” associated with Ladybugrecord. Each raw file includes 450 frames. At the current 15fps (frame per second) speed for our cameras, each file contains data for a 30-second movie. Every 100 raw files will be written to one folder. The first 100 raw files in folder “dir0”, and the second 100 files in folder “dir1”, etc. But the frame number is sequentially saved. Run “raw2txt” script to extract the information of these raw files and check if there are fames dropped during the filming. The raw files are then compressed into an AVI container from Matlab and processed
Figure 5.12: Snapshot of tracking.
with program Video Spot Tracker [73]. Video Spot Tracker program can read AVI video stream or one raw file with current version v06.02. Figure 5.12 is a snapshot of tracking with Video Spot Tracker program. The yellow trajectory is the trace of the marker. The red dot numbered with “0” is the tracker which is overlapped with the maker.
After tracking the marker from both cameras, based on the calibration data, we construct the 3D trajectory with the function “stereo triangulation” in the calibration toolbox. The parameters of the rod are tracked with the tracking program in Matlab from David Holz [35] with the calibration information.
The units of the output data from the calibration are consistent with the units imputed during the calibration. For the 7×11 checkerboard used in our experiment, the grid size is 5mm×5mm. If we enter 5mm for the size of each square, the reconstructed 3D trajectories of our tracer are in the units of mm. The origin of the reconstructed data is in the camera-centered coordinates. Proper rotation and translation are applied
Chapter 6
A straight rod sweeping a tilted
cone above a no-slip plane
A prolate spheroid or a straight slender rod sweeping out a cone or double cone in Stokes flow draw scientists’ attention due to its biological applications to flows gener- ated by spinning cilia [58, 69, 16]. Such fluid motions are fundamental to many living organisms. One important example is the left-right symmetry breaking in the early development of mammals, where primary nodal cilia exhibit canonical rotatory move- ment [16, 12]. Effectively, it is appropriate to approximate nodal cilia as rigid slender rods. We build a model to study the flow generated by a slender rod and compare the theoretical prediction to the experimental data. Such direct comparisons have not been found in the literature yet.
In free space, the exact velocity field for a spheroid sweeping out a double cone has been reported in Camassa et al. [14]. (In the appendix, we provide the error analysis of the velocity field if the spheroid is approximated with a slender body.) When the slender body precesses an upright cone above a no-slip plane, the velocity field and the properties of the fluid particle trajectories have been studied with the slender body theory and the image method [48, 9]. The velocity field is constructed in the body frame, where the rod is fixed and there is rotating background flow. In this frame, a
fundamental singularity Stokeslet is distributed along the center-line of the slender rod. Since the slender rod is sweeping out an upright cone, the distance between a point on the rod and the no-slip plane is fixed. So, the center-line of the rod and its image with respect to the no-slip plane have no time dependence in the body frame. Ultimately, the velocity field in the body frame is independent of time and can be constructed with Blakelet [5]. By applying the transformation between the body frame and the laboratory frame, the velocity in the laboratory frame is obtained from the velocity field in the body frame.
When the cone is tilted, the distance from a point on the rod to the no-slip plane varies while the rod sweeps. In such situations, the flow is fully time dependent in both the body frame and the laboratory frame. To construct the velocity field, we have to carry time information either in the body frame (to identify the position of the no-slip plane over time) or in the lab frame. The simplification in the body frame is no longer available, compared to the upright cone case. The tilt of the cone makes the construction of the flow velocity field complicated. The tilted cone is especially interesting, since most cilia sweep out tilted cones or extensive tilted cones in reality and the cilia themselves are bent. Results for bent cilia sweeping out a cone will be reported in the next chapter, which is much more complicated.
This chapter will continue with a brief review of results about the straight rod sweeping out an upright cone, and then focus on the straight rod sweeping a tilted cone above a no-slip plane. For a rod sweeping a tilted cone, we construct the velocity directly in the laboratory frame and show the properties of the flow with fluid particle trajectories. Distinct phenomena for a rod sweeping a tilted cone are deformations of the Lagrangian fluid particle trajectories and directional fluid transport induced by the rod. Since our model is flexible about the configuration of the cone, we run the model with tracked cone angle from the experiment and compare the numerical
trajectories with the experimental trajectories. Our model shows good agreement with the experimental data.