Find the relevant dimensionless ? terms for the drag force FD on a sphere with d

Question

Find the relevant dimensionless ? terms for the drag force FD on a sphere with diameter D that is moving with velocity V through a fluid with density ? and dynamic viscosity, ?. I.e., FD = function (D, V, ?, ?). Use the FLT system. The reference dimensions for ? = FL-4T2 and ? = FL-2T. Identify the number of variables, write out all reference dimensions, identify the number of ? terms, select repeating variables, and write out the final form ?1 = function(? terms).

Explanation / Answer

FD = function (D, V, ?, µ) Let FD = D^a*V^b*?^c*u^d =>F = L^a * L^b * T^-b * F^c * L^-4c * T^2c * F^d * L^-2d * T^d =>F = F^(c+d) * L^(a+b-4c-2d) * T^(-b+2c+d) =>c+d = 1, a+b-4c-2d = 0, -b+2c+d = 0 =>c = 1-d, b = 2c+d = 2-2d+d = 2-d, a = 4c+2d-b = 4-4d+2d-2+d = 2-d =>a = 2-d, b = 2-d, c = 1-d =>FD = D^(2-d) * V^(2-d) * ?^(1-d) * u^d = (D^2V^2/?)*(u/?VD)^d Number of pi terms = 1 = ?VD/u.......Inverse of reynolds number

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