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a) Define the Reynolds number Re and explain its physical meaning. C.2 [2] b) A sphere of radius a moves through a viscous fluid of viscosity with a constant speed U. What is the Reynolds number associated with this motion? Estimate the Reynolds number of an insect flying through the air and the Reynolds number of a fish swimming in water. Assume that the density and viscosity of air are 1.2kg/m³ and 2-10-5 Pa-s, correspondingly; the viscosity of water is 10-³ Pa.s. [4] In the following, assume Re = 0. The radii of all spheres are a and the viscosity of the fluid is . c) The drag force on a sphere moving through a fluid is given by F = -6μаU, where U is the velocity of the sphere with respect to the fluid. Explain why this relationship is linear. [2] The velocity field created at a position r by a sphere located at ro and moving with a constant velocity U is well approximated by За [5] R₂R₁ dij + Uj, 4R R² if Ra, where R = r- ro, and R= |R|. d) Consider two spheres at (0,0,0) and (H,0,0) in a Cartesian coordinate system; H» a. Each sphere is subject to an external force (Fo, 0, 0). Find the velocity of the spheres. 17 e) Consider two spheres at (0,0,0) and (0, H, 0) in a Cartesian coordinate system; Ha. Each sphere is subject to an external force (Fo, 0, 0). Find the velocity of the spheres. T f) Discuss why these results differ from an isolated sphere subject to the same external force. Reper la 15 [2] [5] [5]