[MUSIC]
In this capstone course,
you will do a project in Computational Fluid Dynamics, CFD.
The governing equations for CFD are the Navier-Stokes equations, so that
we need to discuss where these equations come from and what they look like.
If you take a course in Fluid Mechanics, you'll derive these equations.
But that will take a couple of weeks.
So we won't do that here.
So the first equation is called the continuity equation, it's del.u.
The divergence of the velocity field equals zero.
The continuity equation expresses conservation of mass and
it takes this form when the density of the fluid is everywhere constant.
The Navier-Stokes equation
is du/dt plus u.del u,
this is called the material acceleration.
It's the Navier-Stokes equation is F equals MA.
So this is the MA part of the equation, where we have divided through by the mass.
But the problem with F equals MA is that you follow the particle as it moves.
But this equation, we're at a fixed point in the fluid and
the fluid element is moving past our fixed point.
So the acceleration then has to use the material acceleration.
And that's equal to the force is divided by the mass.
The first term here is the pressure forces is one over the density,
which is a constant times the gradient of the pressure.
And the second term of the forces is the viscosity, the frictional forces.
So that's the Kinematic viscosity, mu the Greek symbol mu,
times the Laplace del squared time the velocity field.
So this is the Navier-Stokes equation, and
this is what's called an in compressible fluid.
So that means the density of the fluid is constant.
Some students have trouble understanding the vector notation in this equation.
So here I would like to break this up into components, so that you can see
what the meaning is in terms of the components of the velocity field.
So if we write the velocity field, which is a vector in terms of its components,
the velocity in the x direction, we're going to call scalar u.
So that's a function of position and that points in the x direction.
So I'll use this i vector as the unit vector in the x direction.
It's also a function of time, but I'll drop the time dependence here,
plus the velocity component in the y direction, I'll call v.
So v(y, z) equals jay the unit vector in the y direction,
plus the component in the z direction, which I call w.
So w(x, y, z) times the component in the z direction, which is k.
So if this is the velocity field,
then we can write these equations in terms of the components,
the continuity equation, del.u equals zero is the divergence of u.
So we have du/dx plus dv/dy
plus dw/dz equals zero,
so that's a scalar equation.
Del.u equals zero, written in terms of the components.
Now the Navier-Stokes equations is a vector equation.
So it has its equivalent to three equations.
If we separated up into its components,
the first equation then will be the x component of u dt.
So the first equation is du scalar dt,
that's the x component plus u.del.
So u.del now is the first component of u dotted into the first component of del.
So it's u and the first component of del is ddx.
And then u vector means the first
component of u, so u du/dx.
And then u.del, the second component will be v and
then the first component of u here still,
so du/dy plus the third component of u and
the third derivative dd/dz.
So plus w du/dz.
So that's this u.del u term, and
this is equal to the first component of the gradient of p.
So minus one overall,
[COUGH] dp/dx plus mu del squared u, so plus mu.
And then del squared u is the Laplace in operator
times the acting on the first component of u.
So that will be d squared u dx
squared plus d squared u dy
squared plus d squared u dz squared.
Okay, so this vector equation then has three components.
This is the first component.
So the second component is not so hard to see.
All you do is replace you by v, this u d dx stays the same and
you replace this u by v, we replace v d dy stays the same.
We replace this u by v and w d dz stays the same and replace this v by v.
Then the pressure term becomes dp/dy and
the viscous term then you just replace you by v that's the second component.
The third component, then, you replace u by w, so
w here and that will be u dw/dx plus v dw/dy plus w dw/dz.
And then it will be a dp/dz and then a d square w dx
squared plus d squared w dy squared plus d squared w dz squared.
So there's two more components.
So plus two more equations.
Okay, so that's an introduction then to the Navier-Stokes equations.
We won't derive them in this course, but
we need to solve these equations to solve a problem in computational fluid dynamics,
which will be the flow around the cylinder.
So let me summarize our governing equations in three
dimensions is the continuity equation, del.u equals zero.
And the Navier-Stokes equations partial of u with
respect to t plus u.del u minus one over rho,
the gradient of the pressure plus mu times del squared u.
So this is four equations.
And in three dimensions, very computational e expensive to
solve them if you have a three-dimensional problem.
But in this course, we'll only be considering a two-dimensional problem.
I'm Jeff Chazanov.
Thanks for watching, and I'll see you in the next video.