Step 1. Find the dimensions of the left-hand side of the equation.
In this part of the equation what we have is:
[tex]\frac{\Delta P}{L}[/tex]Here Δ represents a change in P, in the end, ΔP and P have the same units
And we are indicated that the units of P and L are:
[tex]\begin{gathered} P=\lbrack N\cdot m^2\rbrack \\ L=\lbrack m\rbrack \end{gathered}[/tex]Note: we use [ ] to represent that they are units.
Thus in this left part of the equation, the units are:
[tex]\frac{\lbrack N\cdot m^2\rbrack}{\lbrack m\rbrack}[/tex]Simplifying the m:
[tex]\lbrack N\cdot m\rbrack[/tex]Step 2. We have found that the units on the left-hand side are N*m
Now, we have to find the units on the right-hand side.
On the right-hand side of the equation we have:
[tex]\frac{300(1-\phi^2)\mu U}{2D^2\phi^3}+\frac{175\rho U^2}{100D\phi^3}[/tex]The units of the variables are:
[tex]\begin{gathered} \phi=Dimensionless\text{ (no units)} \\ \mu=\lbrack Pa\cdot s\rbrack \\ U=\lbrack\frac{m}{s}\rbrack \\ D=\lbrack m\rbrack \\ \rho=\lbrack\frac{\operatorname{kg}}{m^3}\rbrack \end{gathered}[/tex]Substituting the units (we can ignore the numbers and the dimensionless terms):
[tex]\frac{300(1-\phi^2)\mu U}{2D^2\phi^3}+\frac{175\rho U^2}{100D\phi^3}=\frac{\lbrack Pa\cdot s\rbrack\lbrack\frac{m}{s}\rbrack}{\lbrack m^2\rbrack}+\frac{\lbrack\frac{kg}{m^3}\rbrack\lbrack\frac{m^2}{s^2}\rbrack}{\lbrack m\rbrack}[/tex]Simplifying the divisions and multiplications between the units:
[tex]\frac{\lbrack Pa\cdot m\rbrack}{\lbrack m^2\rbrack}+\frac{\lbrack\frac{kg}{ms^2}\rbrack}{\lbrack m\rbrack}[/tex]Simplifying further:
[tex]\lbrack\frac{Pa}{m}\rbrack+\lbrack\frac{kg}{m^2s^2}\rbrack[/tex]Since the units of Pascals can be also represented as:
[tex]Pa=\lbrack\frac{kg}{m\cdot s^2}\rbrack[/tex]The second term can also be expressed as Pa/m:
[tex]\lbrack\frac{Pa}{m}\rbrack+\lbrack\frac{Pa}{m}\rbrack[/tex]The addition of two terms with the same units does not change the units, thus, the units on the right-hand side are:
[tex]\lbrack\frac{Pa}{m}\rbrack[/tex]Step 3. Compare.
The units on the left-hand side are:
[tex]\lbrack N\cdot m\rbrack[/tex]And the units on the right-hand side are:
[tex]\lbrack\frac{Pa}{m}\rbrack[/tex]To compare them, let's convert the units of the left-hand side to Pascals using the following known relation between Newtons and Pascals:
[tex]\lbrack N\rbrack=\lbrack\frac{Pa}{m^2}\rbrack[/tex]Using this, the units of the left-hand side are:
[tex]\lbrack N\cdot m\rbrack=\lbrack\frac{Pa}{m^2}\cdot m\rbrack=\lbrack\frac{Pa}{m}\rbrack[/tex]As you can see, The units of both sides are Pa/m, thus we have proven that the two sides are dimensionally consistent.
