Consider a creeping flow where a needle is falling slowly at a constant velocity in a...

     

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Consider a creeping flow where a needle is falling slowly at a constant velocity in a highly viscous fluid. The needle is oriented with its axis at an angle 8 to the horizontal. Then the needle moving direction is not vertically down. Rather, it moves at an angle a to its axis as illustrated in the diagram. Use the force balance in conjunction with drag formulae for needle in creeping motion to prove that 2 tan a = cot. Show free-body diagram and the detailed derivation process. J Asymptotic formulas for a >> b (needle-like) 4л/b In(2a/b)-0.5 F₁ =C, μU,b, C, F=CuU b, C, ≈ 2C, T U a U₁&U₁ - F, & F n U Linearity of creeping motion means that normal and tangential flows may be superimposed without interaction. Vector split Non-horizontal needle falls at an angle 8: use force balance for a. Vector sum F DV Dt 0 -Vp + µV²V F Ө a Consider a creeping flow where a needle is falling slowly at a constant velocity in a highly viscous fluid. The needle is oriented with its axis at an angle 8 to the horizontal. Then the needle moving direction is not vertically down. Rather, it moves at an angle a to its axis as illustrated in the diagram. Use the force balance in conjunction with drag formulae for needle in creeping motion to prove that 2 tan a = cot. Show free-body diagram and the detailed derivation process. J Asymptotic formulas for a >> b (needle-like) 4л/b In(2a/b)-0.5 F₁ =C, μU,b, C, F=CuU b, C, ≈ 2C, T U a U₁&U₁ - F, & F n U Linearity of creeping motion means that normal and tangential flows may be superimposed without interaction. Vector split Non-horizontal needle falls at an angle 8: use force balance for a. Vector sum F DV Dt 0 -Vp + µV²V F Ө a

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