Helical Movements of Flagellated-Propelling Microorganisms

Citation

Chwang, Allen Tse-Yung (1971) Helical Movements of Flagellated-Propelling Microorganisms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JWFS-FA36. https://resolver.caltech.edu/CaltechETD:etd-03262007-141206

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The helical motion of an infinitely long flagellum with a cross-sectional radius b, along which a helical wave of amplitude h, wave-length [lambda] and phase velocity c is propagated, has been analyzed by using Stokes' equations in a helical coordinate system (r,[xi],x). In order to satisfy all the boundary conditions, namely the no-slip condition on the flagellum surface and zero perturbation velocity at infinity, the flagellum must propel itself with a propulsion velocity U in the opposite direction to the phase velocity c. For small values of kb (where k = 2[pi]/[lambda] is the wave number), by a single-harmonic approximation for the outer region (r > h), the ratio of the propulsion velocity U to the phase velocity c is found to be[...], where K[n](kh) is the modified Bessel function of the second kind. A modified and improved version of the Gray and Hancock method has been developed and applied to evaluate helical movements of a freely swimming microorganism with a spherical head of radius a and a tail of finite length and cross-sectional radius b. The propulsion velocity U and the induced angular velocity [omega] of the organism are derived. In order that this type of motion can be realized, it is necessary for the head of the organism to exceed a certain critical size, and some amount of body rotation is inevitable. For fixed kb and kh, an optimum head-tail ratio a/b, at which the propulsion velocity U reaches a maximum, has been discovered. The power required for propulsion by means of helical waves is determined, based on which a hydromechanical efficiency [eta] is defined. This [eta] reaches a maximum at kh [...] 0.9 for microorganisms with optimum head-tail ratios. In the neighborhood of kh = 0.9, the optimum head-tail ratio varies in the range 15 < a/b < 40, the propulsion velocity in 0.08 < U/c < 0.2, and the efficiency in 0.14 < [eta] < 0.24, as kb varies over 0.03 < kb < 0.2. The modified version of the Gray and Hancock method has also been utilized to describe the locomotion of spirochetes. It is found that although a spirochete has no head to resist the induced viscous torque, it can still propel by means of helical waves provided that the spirochete spins with an induced angular velocity [omega]. Thus the 'Spirochete paradox' is resolved. In order to achieve a maximum propulsion velocity, it is discovered that a spirochete should keep its amplitude-wavelength ratio h/[lambda] around 1:6 (or kh [...] 1). At kh = 1, the propulsion velocity varies in the range 0 < U/c < 0.2, and the induced angular velocity in 0.4 < [omega]/[low-case omega] < 1 (where [low-case omega] = kc is the circular frequency of the helical wave), as the radius-amplitude ratio varies over 0 < b/h < 1. A series of experiments have been carried out to determine by simulation the relative importance of the so-called 'neighboring' effect and 'end' effect, and results for the case of uniform helical waves are presented

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