INSTRUCTOR: In this screen cast, we're going to discuss
the Blasius solution for boundary layer flow.
So the Navier-Stokes equations as shown here
for steady two dimensional, laminar, incompressible flow
past the flat plate, with negligible gravitation effects.
We're going to assume that v, which is the y component
of velocity, is very much less than u,
which is the x component of velocity
and d/dx is very much less than d/dy.
And we can make these assumptions
since the boundary layer is thin
because then the velocity normal
to the plate will be very much less than that
parallel to the plate.
And the rate of change of any parameter
across the boundary layer, in other words,
through its thickness,
is gonna be much greater than that along the flow direction.
In addition, there is no pressure gradient
within the boundary layer.
And so we're left with that u du/dx plus v du/dy
is equal to the kinematic viscosity d squared u
d y squared.
So our boundary conditions for the boundary layer
are that u is equal to v which is equal to zero
at y equals zero.
So if we draw a picture of the velocity profile
along this plate.
So this is x and this is y.
We get something that looks like this, which is parabolic.
You can see that u and v are zero at the plate.
In addition, so here's our free stream velocity
as y approaches infinity, our u, or our velocity profile
is going to approach big U, or the free strain velocity.
So this boundary layer equation
is a non linear partial differential equation,
but by applying a coordinate transformation
and change of variables, the velocity dissolution reduces
the PDEs to ordinary differential equation.
And this is done by introducing what's known
as a similarity variable.
And that's gonna be eta, which is gonna be equal
to the free strain velocity U divided by the kinematic
viscosity times x to the 1/2.
And by manipulating the boundary layer equations
and applying the boundary conditions,
when we integrate the resulting equation on a computer,
then we can find this similarity variable as a function
of little u over big U as shown in the table below.
So note that little u over big U is approximately 0.99
when eta is equal to five.
And the reason that that's important is because
that's the definition of the boundary layer thickness.
In other words, at y equals del.
Now that's the boundary layer thickness.
Little u over big U is equal to 0.99.
So we go over here to our table
and you see that 0.99 is when eta is about equal to five.
So our eta is equal to del, so we replace the y
with our boundary layer thickness,
U over the kinematic viscosity times x
raised to the 1/2.
So if we want to actually find the boundary layer thickness,
we just rewrite this as del is equal to five
times the square root of the kinematic viscosity
times x, divided by big U, or the free spring velocity.
And, as Reynold's number can be written
as U times x divided by nu, or the kinematic viscosity,
then we can write a
relationship for del with respect to the Reynold's number,
and this is equal to five divided by the square root
of the Reynold's number based on x.