In our study of inviscid, incompressible flow, I would be remise if I did not talk about the work of Johann and Daniel Bernoulli as well as Leonhard Euler. They made an important contribution to the field of fluid dynamics by creating what is known as Bernoulli's equation.
Bernoulli's Equation can be applied to inviscid, incompressible flows and can be written as
[ 1 ]
$$ P = \frac{1}{2} \rho v^2 = const $$
where $ P $ is the pressure, $ \rho $ is the density, and $ v $ is the velocity.
As a side note/recap, inviscid, incompressible flow means that the density of the air is not changing (incompressible / not compressible) and air has no friction and viscosity is negligible (inviscid / not viscous).
Eq. (1) is the result of a mild-derivation to get Euler's equation, found below:
[ 2 ]
$$ dP = -\rho V dV $$
Euler derived Eq. (2) from the substantial derivative of an x component's momentum in an inviscid flow with no body forces. This equation relates to the change in velocity along a streamline to a change in pressure in the same streamline. We can also take Eq. (2) and integrate it between two points along a streamline. Let's try it!
[ 3 ]
$$ \int_{P_1}^{P_2} dP = -\rho \int_{v_1}^{v_2} v dv $$ $$ \Rightarrow P_2 - P_1 = -\rho (\frac{v_2^2}{2} - \frac{v_1^2}{2}) $$ $$ \Rightarrow P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 $$
This result in Eq. (3) is what we have for Eq. (1), and it is important to note that Eq. (1) holds true along a streamline.
Eq. (1) and Eq. (3) will hold along a streamline for both rotational flow and irrotational flow. However, it is important to note that in a rotational flow, the constant value will change from streamline to streamline, but in an irrotational flow, the constant will remain the same between any two points.