Question

The pressure difference, , across a partial blockage in an artery (called a stenosis) is approximated by the equation where is the blood velocity, the blood viscosity , the blood density , the artery diameter, the area of the unobstructed artery, and the area of the stenosis. Determine the dimensions of the constants and . Would this equation be valid in any system of units

Answer 1

The question is incomplete. The complete question is  :

The pressure difference, Δp, ac$K_u$ross a partial blockage in an artery (called a stenosis) is approximated by the equation :

$\Delta p=K_v\frac{\mu V}{D}+K_u\left(\frac{A_0}{A_1}-1\right)^2 \rho V^2$

Where V is the blood velocity, μ the blood viscosity {FT/L2}, ρ the blood density {M/L3}, D the  artery diameter, $A_0$ the area of the unobstructed artery, and A1 the area of the stenosis.  Determine the dimensions of the constants $K_v$ and $K_u$. Would this equation be valid in any  system of units?

Solution :

From the dimension homogeneity, we require :

$\Delta p=K_v\frac{\mu V}{D}+K_u\left(\frac{A_0}{A_1}-1\right)^2 \rho V^2$

Here, x means dimension of x. i.e.

$[ML^{-1}T^{-2}]=\frac{[K_v][ML^{-1}T^{-1}][LT^{-1}]}{[L]}+[K_u][1][ML^{-3}][L^2T^{-2}]$

                    $=[K_v][ML^{-1}T^{-2}]+[K_u][ML^{-1}T^{-2}]$

So, $[K_u]=[K_v]=[1 ]=$ dimensionless

So, $K_u$ and $K_v$  are dimensionless constants.

This equation will be working in any system of units. The constants $K_u$ and $K_v$ will be different for different system of units.