One of the most important equations in fluids is the continuity equation or conservation
of mass, and here we are going to look at in its differential form. So this is drho/dt
plus d(rho*u)/dx, plus the d(rho*v/dy), plus d(rho*w)/dz, and in order for conservation
of mass to be satisfied that has to be equal to 0, and rho is equals our density. We can
also write it in this form, were drho/dt, plus del dot rho V, equals 0. Lets suppose
we have a steady flow then this reduces to del dot rho V equals 0. Since there is no
time dependence. Now lets assume we have in-compressible flow. Since we have in-compressible flow density
does not change, and so we can pull it out of the equation and since the equation equals
0, we can get rid of density, and we are left with del dot V, equals 0, or written in our
differential form du/dx, plus dv/dy, plus dw/dz equals 0. This not only satisfies continuity
or conservation of mass equation. It also means that the volume dilatation rate is equal
to 0. We can also look at it in cylindrical coordinates, and simplify it as necessary.
So we start again with d(rho)/dt, plus 1 over 2 times d(r*rho Vr)/dr, plus 1 over r d(rhoVtheta)/dtheta,
plus d(rhoVz)/dz, and this is has to equal 0. So again we can look at it as steady which
means our drho/dt term equals 0, and finally we can look at it in-compressible flow, and
we end up with 1 over r d(rVr)/dr, plus 1 over r d(Vtheta)/dtheta, plus d(Vz)/dz, and
that equals 0. So if we have a steady in-compressible plane 2-D flow or continuity equation simplifies.
So plane 2D, steady state, in-compressible. To du/dx, plus dv/dy, equals 0. What this
does it allows us to define what is known as a stream function. So this is defined in
terms of u, and v. Such that u equals the derivative of the stream function with respect
to y, and v equals negative the derivative of the stream function with respect to x.
The first thing we are going to do here is check for continuity. So lets look d/dx, of
now we are going to replace our u with our stream function, plus d/dy times negative
now the derivative of the stream function with respect to x. This just equals d^2 times
stream function/dxdy, minus d^2 stream function/dydx, which is the same thing, and that equals 0.
We can finally do for cylindrical coordinates, which is a little bit more complicated. So
Lets start by defining the stream function in cylindrical coordinates. So our Vr equals
1 over r, the stream function with respect to theta, and our Vtheta equals minus the
derivative of the stream function with respect to r. So again we check for continuity. d/dr
of r, Vr, times 1 over r, plus 1 over r d(Vtheta)dtheta, and we have to figure out if it equals 0.
So we substitute our Vr, and our Vtheta into our equations with respect to our stream function.
So we have 1 over r, d/dr, there is our stream function. Plus 1 over r, d/dtheta, now minus
the stream function. The derivative of the stream function with respect to r. Does that
equal 0. So we have 1 over r, second derivative of the stream function, drdtheta, minus 1
over r, times the second derivative if the stream function dthetadr, and that equals
0.