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p0264 - 2.64 CHAPTER 2 PROBLEM 64 235 2.64 Chapter 2...

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2.64. CHAPTER 2, PROBLEM 64 235 2.64 Chapter 2, Problem 64 Problem: Experimental studies are planned for a rotational axisymmetric flowfield. The vorticity of the flow, ω , has dimensions 1 /T and depends on initial circulation, Γ o (dimensions L 2 /T ), radius, r , time, t , and kinematic viscosity of the fluid, ν . How many dimensionless groupings are there? What are the dimensionless groupings? Solution: The dimensional quantities and their dimensions are [ ω ] = 1 T , [ Γ o ] = L 2 T , [ r ] = L, [ t ] = T, [ ν ] = L 2 T There are 5 dimensional quantities and 2 independent dimensions ( L, T ) , so that the number of dimensionless groupings is 3. The appropriate dimensional equation is [ ω ] = [ Γ o ] a 1 [ r ] a 2 [ t ] a 3 [ ν ] a 4 Substituting the dimensions for each quantity yields T 1 = L 2 a 1 T a 1 L a 2 T a 3 L 2 a 4 T a 4 = L 2 a 1 + a 2 +2 a 4 T a 1 + a 3 a 4 Thus, equating exponents, we arrive at the following two equations. 0 = 2 a 1 + a 2 + 2 a 4 1 = a 1 + a 3 a 4 We can solve immediately for a 1 from the second equation, viz., a 1 = 1 + a 3 a 4 Substituting into the first equation yields
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