Solution_HW1

# Solution_HW1 - Solutions for homework set#1 Problem#1...

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Solutions for homework set #1: Problem #1: Consider a spherical particle falling freely in a large fluid body. It is found experimentally that the terminal settling velocity u of the particle is dependent on: (1) particle diameter D; (2) the buoyant weight of the particle W (the difference between the gravity and buoyant forces acting on the particle); (3) fluid density ρ , and (4) fluid viscosity μ . It is assumed that the velocity u can be mathematically expressed as products of powers of the above variables (i.e. similar to the example of dimensional analysis given in the previous lectures). a b c d u CD W ρ µ = Obtain the relationship for u using dimensional analysis in which the set of linear equations is solved in terms of the exponent c . Show that the respective dimensionless groups obtained by solving the set of linear equations in terms of the exponent b and c are correlated. Answer: If a b c d u CD W ρ µ = ( ) ( ) ( ) ( ) b c d a -1 -2 -3 1 -1 L t L MLt ML M L t C = Equating dimensions: M 0 L 1 3 t -1 2 b c d a b c d b

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