p0296 - 246 CHAPTER 2. DIMENSIONAL ANALYSIS 2.96 Chapter 2,...

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246 CHAPTER 2. DIMENSIONAL ANALYSIS 2.96 Chapter 2, Problem 96 2.96(a): The dimensional quantities and their dimensions are [ U ]= L T , [ L L, [ c L, [ ω 1 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. 2.96(b): Using the indicial method, the appropriate dimensional equation is [ U ]=[ L ] a 1 [ c ] a 2 [ ω ] a 3 [ ν ] a 4 Substituting the dimensions for each quantity yields LT 1 = L a 1 L a 2 T a 3 L 2 a 4 T a 4 = L a 1 + a 2 +2 a 4 T a 3 a 4 Thus, equating exponents, we arrive at the following two equations: 1= a 1 + a 2 +2 a 4 a 3 a 4 We can solve immediately for a 1 and a 3 ,v iz . , a 1 =1 a 2 2 a 4 and a 3 a 4 Substituting back into the dimensional equation, we have [ U L ] 1 a 2 2 a 4 [ c ] a 2 [ ω ] 1 a 4 [ ν ] a 4 which simplifies to [ U ω L ] } c L ] a 2 } ν ω L 2 ] a 4 Therefore, the dimensionless groupings are U ω L , c L , ν ω L 2 Using E. S. Taylor’s method, we have the following.
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p0296 - 246 CHAPTER 2. DIMENSIONAL ANALYSIS 2.96 Chapter 2,...

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