This screencast we will be solving an equation in which are are trying to find a velocity
as a function of both position and time by using the continuity equation in its differential
form. We also know the continuity equation, the conservation of mass. If we take a look
at this continuity equation. In its mathematical operator form here. We can also expand this
gradient operator. Remember the gradient operator dot product of density times velocity vector
can be expanded to me equal to the following. Note that V is a velocity vector. We are taking
the dot product of the operator with respect to all the perpendicular terms drop out. So
this the simplified left over form that we see here. So now we can rewrite our continuity
equation. It looks like the following. So now there is a couple of things we can do.
First off we have a steady state process. So our fluid is flowing at steady state, no
accumulation or depreciation of mass happens in our control volume. So no change in mass
with time. Hence this first term is 0. So the continuity equation becomes the following.
Lastly for steady state flow and in-compressible fluids the density is not a function of position.
It is a constant. The density is a constant you can pull it out of the differentials and
if we divide everything by density it just cancels out. So we are left with the following.
Now going back to our mathematical operator we can also write this form as the following.
So now we have 3 different ways of writing the continuity equation depending on the assumptions
we are making. So now lets take a look at a problem were we want to determine the velocity
of the fluid, which we use the differential form of this continuity equation. We have
a 1-d flow is found to have a non-uniform density as represented by rho is equal to
e^(cos(t)). With an initial velocity equal to 1 for time t=pi/2, and x equal to 0. We
want to determine the velocity of this flow as a function of position and time. Lets tart
by writing down our governing equation. The overall differential form for continuity equation.
So unfortunately we do not have steady state, and we do know that the density is a function
of position. It is non-uniform. So we cannot simplify to that easy form we saw before.
So lets take this equation and break it down into the individual terms using the chain
rule. Applying the change rule we get the following. Now lets just recall the chain
rule. If I take this part of the equation and we are trying to break it down. Remember
we are applying the differential that is what is in the parentheses with respect to x. So
the chain rule says that this is the density times the partial of u with respect to x,
plus u times the partial density with respect to x. So now each term becomes 2 individual
terms. So what can we cross out from now. Now we know density is a function of time.
So we cannot get rid of this first term. We are only looking at 1-d flow. Which means
the flow is only in the x direction. So we can get rid of any terms that have v, or w
as they represent a y velocity, or a z velocity. So this is gone, and this is gone. Plus this
term is gone and that term is gone. Those are all equal to 0. We also see that density
is not a function of x, because we are looking at our equation and we see there is no x as
part of that function. So that means that this differential is going to be equal to
0. That means that we are left with 2 terms. The first term density with respect to time,
and this third term. The velocity in the x-direction, with respect to x. So at this point we just
use our nice mathematical skills and we solve, but what are we solving for? We are solving
for velocity in the x-direction. So we want u as a function of x and t. So we need to
get du/dx equal to something. So we can separate variables and go from there. First we get
to evaluate drho/dt. We have our equation up here with respect to t. So if we take the
differential with respect to t. We get negative sin(t) e^cos(t). Now the second part of the
equation rho times du/dt. Is just equal to e^cost*du/dx. So now we can combine these
and set them equal to 0. We can pull e^(cos(t)) out of our equation that is equal to 0. So
now we can cross out the exponential term. We are left with du/dx is equal to the sin(t).
So we are almost there. Now we just need to evaluate this integral. Separate variables,
and integrate both sides. We get u is equal to the sin(t) times x, plus some constant.
So now we need to solve for this constant, and luckily we have initial conditions that
we can use. So our velocity is equal to 1 for a time of pi/2. The sine of pi/2 is 1,
and x equals 0. So really we see that our constant is equal to 1. So now we plug that
back in to our equation. We get our final answer. Our velocity as a function of x and
t. Is equal to xsin(t) plus 1. So hopefully this gives you an idea on how to use the continuity
equation in a differential form. When given types of problems for fluid flow.