申論題

1.(50%) Lymphangions(淋巴管節),functional units of a lymph vessel(淋巴管)that lie between two semilunar values (瓣膜), are muscular and capable of contracting themself one by one to propel the lymph(淋巴液) forward and increase the fluid pressure (figure 1). A lymph vessel now is modeled by a micro-tube with a deformable wall as shown in figure 2. The micro-tube has a fixed length L and a radius R₀ when the wall is undeformed. We assume that a lymph vessel is very thin, Ie., R₀ ＜＜ L. We model the operations of lymphangions as a wall deformation in a form of traveling wave, i.e, △R = △R(kz - ωt), where z is the streamwise coordinate, k and ωare wavenumber and angular velocity of the traveling wave, whose traveling velocity is c = ω/k. Thus, the deformable wall is described by an axisymmetric surface at r = R(z, t) = R₀ + △R(kz - ωt). Consider the flow of lymph in the micro-channel is incompressible, axisymmetric and without swirling. The lymph is assumed to have constant densityρ and dynamics viscosity μ. 

The following incompressible axisymmetric no-swirling Navier-Stokes equations in the cylindrical coordinate will be helpful:
   where p is the pressure and up and uz are the radial and streamwise components of velocity.

(1) (5%) Let the scales of the streamwise and radial velocities of the lymph flow be U and W, respectively. Let the streamwise and radial positions in the lymph flow be scaled by R₀ and L, respectively. From scale analysis of the continuity equation, what can you tell about the magnitudes of U and W?