Laminar and turbulent flow

In laminar flow the streak-lines are straight lines.

The fluid flows smoothly down the pipe.

Module 15 Lesson 24 LAMINAR FLOW

In turbulent flow the streak-lines show wiggles and vortices. The fluid does not flow smoothly down the pipe.

Module 15 Lesson 24 TURBULENT FLOW

What happens when fluid enters a pipe?

Module 15 Lesson 24 1.1 TURBULENT FLOW and LAMINAR FLOW

The fluid adjacent to the wall sticks to the wall due to friction effects. This is the no-slip condition and occurs for all liquids.

• This boundary layer grows until it reaches all parts of the pipe.

• Inside the inviscid core, viscosity effects are not important.

• The entrance region for laminar flow is given by

Module 15 Lesson24 1.1

Past here the flow is fully developed.

Laminar flow analysis

Assumptions, outside entrance region:

Module 15 Lesson24 1.2

And steady flow. Horizontal flow



Apply F = ma to a cylinder.

The cylinder becomes distorted as t → t + ∂t

• The pressure is constant along the vertical direction.

• The pressure along horizontal direction does change. Δp = p2 − p1 < 0

• There is a viscous shear stress acting along the surface cylinder and The shear stress is a function of the radius of the cylinder.

Application of F = ma

Module 15 Lesson 24 APPLICATION OF F = MA

 Module 15 Lesson 24 1.1 APPLICATION OF F = MA

Module 15 Lesson 24 1.2 APPLICATION OF F = MA

Module 15 Lesson 24 1.7 APPLICATION OF F = MA

Module 15 Lesson 24 1.8 APPLICATION OF F = MA

The shear stress is largest at the walls

Laminar velocity profile

To determine the laminar velocity profile, assume we have a Newtonian fluid, so

Module 15 Lesson 24 1.9 APPLICATION OF F = MA

The flow rate is parabolic, with largest velocity in  middle of pipe and zero velocity at wall.

 Module 15 Lesson 24 1.10 APPLICATION OF F = MA

Laminar flow rate

Just need to integrate the laminar velocity profile over the cross sectional area. Divide cross section into thin annular strips

 Module 15 Lesson 24 1.13 APPLICATION OF F = MA

Pouiseuille’s Law and Interpretation

The fundamental result

 Module 15 Lesson 24 1.14 APPLICATION OF F = MA

is usually called Poiseuille’s Law. Laminar flow in pipes is sometimes termed Hagen-Poiseuille’s flow.

Flow along a pipe is driven by a pressure difference.

The viscosity acts to retard the passage of the fluid along the pipe through the no-slip condition at the wall. The flow rate

Module 15 Lesson 24 1.15 APPLICATION OF F = MA

Last modified: Saturday, 31 August 2013, 10:50 AM