Internal Flows
In this notes sheet...
An internal flow is any flow that is fully enclosed by fixed surfaces, and is predominantly determined by the viscous forces at the surfaces. The most common example is pipe flow.
As flow enters a pipe, you can see the effect of the viscous force coming into play:
Initially, before the fluid enters the pipe, the flow is inviscid: the velocity field is uniform.
Throughout the entrance length, the velocity decreases closer to the surfaces.
The flow is fully developed in a constant velocity field, with zero relative velocity at the boundaries (no-slip condition).
The entrance length is defined as the length of pipe for the flow to fully develop.
Fully Developed Laminar Flow
The most general form of this is between two flat, parallel, infinite plates:
In laminar flow, all velocities are parallel to the net direction of flow.
Taking the infinitely small fluid particle above, we can apply conservation of momentum to the horizontal forces acting on in:
The left and right forces in grey are pressure forces
The top and bottom forces in red are viscous forces
We do not know the directions of these, only that they point in opposite directions
The Following Steps show how to derive the velocity profile for fully developed laminar flow between two infinite plates, but the same steps should be followed to derive any similar velocity profile.
Pressure Forces
Pressure force on the left:
Difference in pressure over distance δx:
Pressure force on the right:
Viscous Forces
We know that the viscous force is given as:
Therefore, bottom viscous force is given as:
The velocity on the top surface is given as:
So, the top viscous force is:
We know that the top viscous force must be positive, and te bottom force negative (see notes sheet on force in fluids).
Newton’s First Law & Conservation of Momentum
Since the velocity is independent of x-position in laminar flow, there is no change in velocity in this direction. This means there is no net horizontal force acting on the particle, so all the forces mus