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[2] C.2 a) Define the Reynolds number Re and explain its physical meaning. A swimming bacterium can be modelled as a spherical body of radius a pushed by a rotating helical filament. b) Estimate the Reynolds number for such a bacterium with a 1um moving with the speed v 20μm/s; the viscosity of water is 10-3 Pa.s. [4] c) The role of the filament is to generate a propulsive force F, applied to the fluid a distance L along the filament, propelling the bacterium in the opposite direction. Discuss the forces acting on the fluid and their direction. Neglecting the hydrodynamic interactions between the filament and the bacterial body, estimate the magnitude of the propulsive force Fp, if L 10μm. [6] N.B. The Stokes drag force on a sphere of radius a moving through a fluid with viscosity u is given by F = -67μaU, where U is the velocity of the sphere with respect to the fluid. d) Let e be a unit vector along the bacterial filament. Consider a coordinate system with the origin at the centre of the bacterial body. Demonstrate that the velocity field, created by the bacterium, at a position r far away from the bacterium is given, to linear order in L/r, by pr v(r) - [1-3 (-)²]. p3 where r = [r], and give an explicit expression for p. [8] N.B. You can use the velocity field v(s) at r due to a point force F applied to the fluid at the origin 1 XiXj - [815 + 122] F3. F₁. 8πμη Show that the flow field v(r) above is incompressible. [5] 1 1 }