table? To classify the smoothness of a flowing fluid, Osborne Reynolds developed

Question

table? To classify the smoothness of a flowing fluid, Osborne Reynolds developed the now famous dimensionless quantity of Reynolds number (All of the dimensions should cancel). His theory stated that the smoothness or roughness (a lot of eddies or swirling) of a fluid depended upon: How fast the fluid was moving (velocity) v [] m/s The density of the fluid [s] kg/m3 The diameter of the pipe D [] m How hard it was to move the fluid (viscosity) [s] g/(cm s) Determine the equation for the Reynolds number is using the method of dimensional analysis we learned in class. (Hint: NRe-f(v, p, D, )) 8.

Explanation / Answer

First of all the dimension given are

Velocity(v) — L / T

Diameter(D) — L

Density() — M / L3

Viscosity() — M / LT  

Let us assume the product as  ^a v^b D^c ^d.

The exponents a through d have to be adjusted so that the M, L, and T dimensions of the product cancel out. One of the exponents can be chosen arbitrarily, say d = 1, but then a, b, and c have to be adjusted by solving three equations, one each for M, L, and T, expressing the condition that the product be dimensionless.

This will give three linear equations

-3a +b +c -1 = 0 (for L)

+a +1 = 0 (for M)

-b -1 = 0 (for T)

Solving these three equations will give us

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

This will give us the formulae that is opposite of reynolds number snd if we take d=-1 we will get reynolds number.

let d=-1

a=1,b=1,c=1

we will get reynolds formulae.

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