p0222 - 152 CHAPTER 2. DIMENSIONAL ANALYSIS 2.22 Chapter 2,...

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152 CHAPTER 2. DIMENSIONAL ANALYSIS 2.22 Chapter 2, Problem 22 Problem: For laminar flow, Newton postulated that the shear stress is a function of viscosity, μ , and the velocity gradient, du/dy . Verify that the Buckingham Π Theorem implies there are no dimensionless groupings. Show that the indicial equations are linearly dependent, and (choosing the constant of proportionality to be 1) verify that τ = μdu/dy . Solution: The dimensional quantities and their dimensions are [ τ ]= M LT 2 , [ μ M LT , ^ du dy ± = L/T L = 1 T There are 3 dimensional quantities and 3 independent dimensions ( M,L,T ) , so that the number of dimensionless groupings is 0. The appropriate dimensional equation is [ τ ]=[ μ ] a 1 ^ du dy ± a 2 Substituting the dimensions for each quantity yields ML 1 T 2 = M a 1 L a 1 T a 1 T a 2 = M a 1 L a 1 T a 1 a 2 Thus, equating exponents, we arrive at the following three equations.
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p0222 - 152 CHAPTER 2. DIMENSIONAL ANALYSIS 2.22 Chapter 2,...

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