Question

Find the primary dimensions of each of the following terms. a. (pV2)/2 (kinetic pressure), where p is fluid density and V is velocity b. T (torque) c. P (power) d. (PV2L)/ (Weber number), where p is fluid density, V is velocity, L is length, and is surface tension.

Solution

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a)\textbf{a)}

In order to determine the primary dimension of the term under the section a)\textbf{a)} we will consider the base units for every variable. Since the unit used for the density is:

kgm3\mathrm{\frac{kg}{m^3}}

Therefore, we can write the primary dimension for the density, using the relation between the signs and the dimensions of given base units as:

[kgm3]massvolumeML3[\mathrm{\frac{kg}{m^3}}]-\mathrm{\frac{mass}{volume}}-\mathrm{\frac{M}{L^3}}

The unit used for the velocity is:

ms\mathrm{\frac{m}{s}}

We are going to write the primary dimension for the velocity, using the relation between the signs and the dimensions of given base units as:

mslengthtimeLT\mathrm{\frac{m}{s}}-\mathrm{\frac{length}{time}}-\mathrm{\frac{L}{T}}

Finally we are able to determine the primary dimension of the kinetic pressure as:

ρV22ML3(LT)2ML3L2T2MLT2\begin{align*} \frac{\rho\cdot{V^2}}{2}&-\mathrm{\frac{M}{L^3}\cdot\left(\frac{L}{T}\right)^2}\\ &-\mathrm{\frac{M}{L^3}\cdot\frac{L^2}{T^2}}\\ &-\mathrm{\frac{M}{L\cdot{T^2}}} \end{align*}

MLT2\boxed{\mathrm{\frac{M}{L\cdot{T^2}}}}

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