Hi, in this lecture we will focus on two important fundamental equations for the flow of fluids and gases. The continuity equation, which is very handy as it comes to the flow through channels, and the Euler equation, which is an equation an equation based on the conservation of momentum. Now first let us have a look at the continuity equation.
It is basically about the conservation of mass of a fluid flowing through a streamtube. Here we have such a streamtube, formed by streamlines. The flow comes in at the station 1 with the velocity v1 and comes out at the station 2 with velocity v2.
The areas are respectively A1 and A2. For steady flow the mass per second is equal to the mass per second. The second dead entry must be the same as the mass per second coming out at station 2. Now let us look at the column of fluid flowing through an area A with a steady velocity v in a time span dt. The mass in the column is the dead entry of the mass per second coming out at station 2. density rho times the volume.
The volume of this column is A times the length, which is V dT. The mass flow per unit of time is rho A V dT divided by dT. dT is both in the numerator and the denominator, so we found that the mass flow per second is rho times A times V. So the first fundamental equation for steady flow we have is rho A V is constant, or if you like, rho A V .
is rho2 A2 V2. This is known as the continuity equation. There is an assumption underlying this equation, and that is that rho and V are mean values over the area.
For incompressible flow, in which rho doesn't change, we can simplify the equation to Av is constant. Now let's move on to another fundamental equation, the Euler equation. Let's apply Newton's second law, forces mass times acceleration, to a flowing gas. Here we have a volume of air following a streamline.
The sides of the box have the length dx, dy and dz. At the left side the pressure P is working and at the opposite side we have P plus delta P. This delta P can be written as dp dx over dx, or in words dp is the pressure gradient over the volume times the length of the volume. In reality three forces act on the volume.
a pressure force, a friction force and a gravity force. For the next derivation we neglect the gravity force, since the streamlines do not deviate much in vertical direction and viscosity, so we have no friction forces. Now back to Newton's second law. If we concentrate on the force part at the left hand side of the equation, we can write for the resulting force on the volume in x direction p times dy dz minus between brackets It's p plus dp dx times dx, times dy dz or F is minus dp over dx times dx dy dz. This is the force on the fluid element due to pressure.
The force is positive in the flow direction. The mass times acceleration part on the right hand side of Newton's law. The mass m of the fluid element is rho times the volume, so rho times dx dy dz.
The acceleration is of course the Vr. over dt. This can also be written as dv over dx times dx over dt.
This last part is the velocity v. So for the acceleration we find a is dv over dx times v. Now if we put this all together we find minus dp over dx times dx dy dz is equal to rho dx dy dz times v times dv over dx. The volume is on both sides, so this cancels out. And finally we have found dp is minus rho V dv. Now keep in mind that in deriving it we neglected the gravitation and friction forces, and it is for steady flow. Since the density is still in, it also applies to compressible flow.
The relation we derived is known as the Euler equation. Leonhard Euler was a Swiss mathematician born in 1707 and he was a student of Johann Bernoulli, the father of Daniel. In the next lecture we will hear more about the Bernoulli family and Euler.