# Laminar Flow

By Deepanshu Rastogi|Updated : November 24th, 2021

This article contains basic notes on the "Laminar Flow"  topic of the "Fluid Mechanics & Hydraulics" subject.

## Laminar Flow

### Laminar Flow

• Laminar flow is also known as viscous flow
• In laminar flow, viscous force is highly is highly predominant.

Case–I

#### The laminar flow between 2- parallel plates

Consider a fluid element in the flow field. An element has thickness dy, length dx and y distance from the bottom plate.

Assumption:- width of flow perpendicular to paper = unity

Free body diagram of an element

Apply equilibrium condition

But we know that

So,

are independent form y

By integrating equation (1)

⇒ Again integrate with respect to y

Apply boundary condition

1. At y = 0, u = 0
C2 = 0
2. At y=H, u=0

So,

Maximum velocity

Mean velocity

Mass flow rate =

Mass flow rate, when considering average velocity.

By equating both terms {from eq. (a) and (b)} and putting the expression of

From expression of

#### Shear stress distribution:

By Newton’s Law of viscosity

#### Pressure difference b/w two points along the flow

Consider average velocity expression

Laminar flow through pipe: (circular)

Consider a fluid element having radius r and length dx

Free body diagram of an element

Apply horizontal equilibrium equation

Internal flow:-

According to Newton’s Law of Viscosity

from first figure in this section

put the value of  in eq. (a)

By integrating it

At  ---no slip condition

#### Maximum Velocity:

So from the expression of u, put r=0

#### Mean Velocity:

Mass flow rate is constant throughout the pipe

From the expression of

Pressure distribution:

In the calculation of pressure difference always consider average velocity So.

By Rearranging

#### Velocity and shear stress profile in a circular pipe

• Flow between Flat plate
We know that
• Circular pipe flow
We know that

NOTE:

• Dependency of flow on Reynolds number
Reynolds number
density of fluid
Velocity of flow
Dynamic Viscosity
Characteristic dimension
For Laminar flow
If,  Turbulent flow
• If, 2000  transition flow

• Head Loss equation by Darcy’s

Here  Friction factor
Velocity (average)
Length of pipe
Diameter of pipe
This equation is valid for both turbulent and laminar flow.
• Friction factor for circular pipe flow
In circular pipe

But Darey’s equation

By equating both expressions

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