20191201_00414281.pdf - 158 6.2.4 Motion of a suspended...

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158 6.2.4 Motion of a suspended rigid fiber The motion of ellipsoids in uniform, viscous shear flow in a Newtonian fluid was analyzed by Jeffery [152-153] in 1922. For a prolate spheroid of aspect ratio r a (defined as the ratio between the mayor axis and the minor axis) in simple shear flow, ( ) 0 , 0 , 3 γ & x = u , the angular motion of the spheroid is described by [152-153], φ φ θ 2 2 2 sin cos tan + = r r J a a K (6.18) and = J r T t a π φ 2 tan tan (6.19) where θ is the angle between the fiber’s major axis and the vorticity axis, i.e. 2 x axis, φ is the angle between the 3 x axis and the 3 1 x x projection of the fiber axis (see Figure 6.14), J T is the orbit period, + = r r J a a T 1 2 γ π & (6.20) and J K is the orbit constant, determined by the initial orientation by, r J a K 0 2 0 2 0 sin cos tan φ φ θ + = (6.21) These equations predict that the spheroid will repeatedly rotate through the same orbit, the particle will not migrate across the streamline, and that the orbit period is independent of the initial orientation.
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161 as the application framework. Thus, network delay is negligible. Their arrival rate λ follows the exponential distribution. Each test has duration of 5 minutes. All the assumptions considered during this technical evaluation are listed in Section .
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