Establish the Reynolds number for a fluid flowing through the pipe in Fig. 8–2 using dimensional analysis, realizing that the flow is a function of the density ρ and viscosity μ of the fluid, along with its velocity V and the pipe’s diameter D.

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Accepted Answer

Define the Physical Variables. Here n = 4. Using the M-L-T system in Table 8–1, we have

[latex] \begin{array}{ll} ext { Density, } ho & M L^{-3} \ ext { Viscosity, } \mu & M L^{-1} T^{-1} \ ext { Velocity, } V & L T^{-1} \ ext { Diameter, } D & L \end{array} [/latex]

Since all three primary dimensions (M, L, T) are used here, then m = 3. Thus, there is (n - m) = (4 - 3) = 1 Π term.

Select the Repeating Variables. We will choose ρ, μ, V as the m = 3 repeating variables, because the collection of their dimensions contains the m = 3 primary dimensions. (Of course, another selection, such as μ, V, D, can also be made.)

Π Term, q = D. Since D has not been selected, it becomes the q variable, which will be set to the first power (exponent one). And so the [latex] \Pi ext { term is } \Pi= ho^a \mu^b V^c D [/latex].

Dimensional Analysis. The dimensions for this Π term are

[latex] \begin{aligned} \Pi & = ho^a \mu^b V^c D \ & =\left(M^a L^{-3 a} ight)\left(M^b L^{-b} T^{-b} ight)\left(L^c T^{-c} ight) L=M^{a+b} L^{-3 a-b+c+1} T^{-b-c} \end{aligned} [/latex]

Since Π is to be dimensionless, the exponents on each dimension must be set equal to zero, i.e.,

For M: 0 = a + b
For L: 0 = -3a - b + c + 1
For T: 0 = -b - c
Solving yields

a = 1, b = -1, c = 1

And so,

[latex] \Pi_{ Re }= ho^1 \mu^{-1} V^1 D=\frac{ ho V D}{\mu} [/latex]

This result is the Reynolds number, Eq. 8–3, where the “characteristic length” L is the pipe diameter D. Later in the book we will give a more thorough discussion on using the Reynolds number, and will show how viscous and inertia forces play a predominant role in defining whether the flow through the pipe is laminar or transitions to turbulent.

[latex] \operatorname{Re}=\frac{ ext { inertia force }}{ ext { viscous force }}=\frac{ ho V L}{\mu} [/latex] (8-3)

Question 8.1: Establish the Reynolds number for a fluid flowing through th...

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