Poiseuille and Couette Flow

We are going to discuss about poiseuille and Couette flow. Couette flow refers to the laminar flow of viscous fluid in the space between two parallel plates, one of which is moving relative to other in the fluid dynamics. The flow is driven by the good quality of viscous drag force acting on the fluid and applied pressure gradient parallel to plates. Poiseuille flow is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe in fluid dynamics. The flow is laminar viscous and incompressible and the flow through a constant circular cross section is considerably longer than its diameter. According to Hagen – Poiseuille equation, the fluid flow will be turbulent for velocities and pipe diameter above a threshold, and thus leading to the larger pressure drops than expected. This is said to be poiseuille and Couette flow.

Poiseuille flow through pipe

• A tube showing the imaginary lamina.
• The lamina moving at different speeds in a cross section of the tube. Those closest to the edge of the tube are moving slowly while those near the center are moving quickly.

In a tube, due to friction the liquid in the center is moving faster while the liquid touching the walls of the tube is stationary. If there are a bunch of circular layers (lamina) of liquid, by their radial distance from the centre of the tube, each velocity can be determined.

To find out the motion of the liquid, we must need to know all forces acting on each lamina:

• The force pushing the liquid through the tube is change in pressure and is multiplied by the area:  This force is in the direction of the motion of the liquid – the negative sign comes from the conventional way. We define ΔP = P_end − P_top < 0.
• The pull from the faster lamina immediately closer to the centre of the tube and the drag from the slower lamina immediately closer to the walls of the tube.

From the definition of pressure, the first of these forces comes and the other two forces require us to modify the equations above that we have for viscosity.

Couette flow with pressure gradient

When the pressure gradient is imposed in a direction parallel to the plate's couette flow situation arises. The Navier strokes equations simplify to where dp/dx is the pressure gradient parallel to the plates and µ is fluid viscosity.

The pressure gradient can be positive or negative. Thus the poiseuille and Couette flow.

Questions:

• Explain poiseuille flow?
• Explain Couette flow?