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.

image001

Assumption:- width of flow perpendicular to paper = unity

Free body diagram of an element

image002

Apply equilibrium condition

image003

But we know that image004

So, image005

image006 are independent form y

 

By integrating equation (1)

image007

⇒ Again integrate with respect to y

image008

Apply boundary condition

image009

  1. At y = 0, u = 0
    C2 = 0
  2. At y=H, u=0
    image010

So, image011

image012

Maximum velocity

image013

Mean velocity

Mass flow rate = image014

Mass flow rate, when considering average velocity.

image015

By equating both terms {from eq. (a) and (b)} and putting the expression of image016

image017

From expression of image018

image019

Shear stress distribution:

By Newton’s Law of viscosity

image020

image021

Pressure difference b/w two points along the flow

Consider average velocity expression

image022

image023

Laminar flow through pipe: (circular)

image024

Consider a fluid element having radius r and length dx

Free body diagram of an element

image025

Apply horizontal equilibrium equation

Internal flow:-

image026

According to Newton’s Law of Viscosity

image027

from first figure in this section

 image028

put the value of image029 in eq. (a)

image030

By integrating it

image031

At image032 ---no slip condition

image033

Maximum Velocity:

image034

So from the expression of u, put r=0

image035

Mean Velocity:

Mass flow rate is constant throughout the pipe

image036

From the expression of image018

image037

Pressure distribution:

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

image038

By Rearranging

image040

image041

Velocity and shear stress profile in a circular pipe

image042

Head Loss

  • Flow between Flat plate
    We know that image043
    Head Loss 
    image044
  • Circular pipe flow
    We know that image045
    Head Loss image046
    image047

NOTE:

  • Dependency of flow on Reynolds number
    Reynolds number image048
    image049 density of fluid
    image050 Velocity of flow
    image051 Dynamic Viscosity
    image052 Characteristic dimension
    For Laminar flow image053
    If, image054 Turbulent flow
  • If, 2000 image055 transition flow

  • Head Loss equation by Darcy’s
    image056
    Here image057 Friction factor
    image058 Velocity (average)
    image059 Length of pipe
    image052 Diameter of pipe
    This equation is valid for both turbulent and laminar flow.
  • Friction factor for circular pipe flow
    In circular pipe
    image060
    But Darey’s equation
    image061

By equating both expressions

image062

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