I was trying to derive the following equation from my knowledge of aerodynamics:
$$\frac{dp}{dR}=\rho \frac{v^2}{R}$$
(Where $\rho$ is the fluid density, $p$ is the pressure and $R$ is the radius of the circular path the fluid has taken)
In the end, I managed to reach the final equation; but I'm not sure if what I did is mathematically correct, or even allowed. (I'm just a high school student; however I've studied this on my own.) The following is my workings, and I've put a number in square brackets indicating the expressions I'm not sure are mathematically correct or make sense:
In wikipedia, it said that pressure is defined as follows:
$\large p=-\frac{F}{A}\qquad$ where F is the magnitude of the normal force and p is the local pressure
I didn't agree with the minus sign so I changed the above equation to a version I thought was better:
$\large p=\frac{\mathbf{|F|}}{|d\mathbf{A}|}\qquad$ [ 1 ] $\qquad$ where $dA$ is a small surface element such that $:\mathbf{F}=-dA \times {\hat {n}} $
where $\hat {n}$ is a unit vector perpendicular to the surface area.
Then, I used the following diagram to continue my workings:
$\mathbf {|F|}\approx {\delta {m}} \large \frac{v^2}{\mathbf{|R|}} \qquad$ where $\delta {m}$ is the mass of the small element shown in the image.
Therefore, we have:
$\large p=\frac{\mathbf{|F|}}{|d\mathbf{A}|} \approx {\frac{\delta {m}}{|d\mathbf {A}|} \large \frac{v^2}{\mathbf{|R|}}} \qquad$ where $\mathbf{R}$ is in the outwards direction
We also have: $\qquad \delta {m}=\rho \delta {V} \approx {\rho |d\mathbf{A}|\ |\delta {\mathbf{R}}|}\ \rightarrow \ p \approx{\rho} |\delta {\mathbf{R}}| \large \frac{v^2}{\mathbf{|R|}} \ \rightarrow \ \frac {\large p}{|\delta {\mathbf{R}}|} \approx {ρ \frac{v^2}{R}}$
$\lim_{|\delta {\mathbf{R}}| \to \ 0} (\frac {\large p}{|\delta {\mathbf{R}}|})=\frac{dp}{d \mathbf {R}}=ρ \frac{v^2}{R} \qquad$ [ 2 ]
I wasn't myself satisfied with this working, I'm not even sure if $\ \lim_{|\delta {\mathbf{R}}| \to \ 0} (\frac {\large p}{|\delta {\mathbf{R}}|})=\frac{dp}{d \mathbf {R}}$. So, if you could direct me to the proper proof I'd really appreciate it.
Thank you