Prof. Mohammad Saud Afzal

Today we will explore the laminar boundary layer. Can anyone tell me what a boundary layer is?

Is it the layer of fluid in contact with a surface where velocity changes?

Exactly! The boundary layer is crucial in understanding how fluids behave near surfaces. Now, what happens to the speed of the fluid as it approaches a surface?

The velocity decreases to zero at the surface because of the no-slip condition.

Right! This decrease in speed forms the boundary layer, where the velocity transitions from zero at the surface to a free stream velocity. This is crucial in hydraulic engineering.

Let’s look at the velocity profile given by $\frac{u}{U} = 2\frac{y}{\delta} - \left(\frac{y}{\delta}\right)^2$. What do you think each variable represents?

I think $u$ is the velocity at a certain point in the boundary layer, and $U$ is the free stream velocity?

Correct! And $\delta$ represents the boundary layer thickness. How would you approach finding the thickness from this profile?

We would need to integrate the profile to find metrics like the momentum thickness.

Exactly! Integrating gives us the momentum thickness, which reflects the energy losses due to viscous effects within the layer. Remember the formula for momentum thickness we derived?

Now, using the von-Karman momentum integral equation, can anyone remind us how we derive the wall shear stress?

We substitute the formula for momentum thickness and take the derivative to find $\tau_0$.

Correct! And what is the formula we ended with for $\tau_0$?

$\tau_0 = \frac{2}{15}\rho U^2 \frac{d\delta}{dx}$?

Perfect! This equation links the shear stress to how the boundary layer thickness changes with respect to distance.

Let’s calculate the boundary layer thickness for a plate 1 meter long submerged in water. What is our first step?

We need the velocity in consistent units and the viscosity of water.

Exactly! If we use 150 mm/s and the viscosity as 0.01 poise, what do we convert that into for calculation?

We convert it to meters per second for uniformity.

Correct! Once we have all parameters set, we can use the derived equations to find $\delta$.