Saturday, 17 March 2018

CONTINUITY EQUATION IN FLUID MECHANICS

CONTINUITY EQUATION IN FLUID MECHANICS

We were discussing the basic concept of Lagrangian and Eulerian methodtypes of fluid flow and discharge or flow rate in the subject of fluid mechanics in our recent posts. Now we will start a new topic in the field of fluid mechanics i.e. continuity equation with the help of this post.

Continuity equation

When fluid flow through a full pipe, the volume of fluid entering in to the pipe must be equal to the volume of the fluid leaving the pipe, even if the diameter of the pipe vary.

 
Therefore we can define the continuity equation as the equation based on the principle of conservation of mass.

Therefore, for a flowing fluid through the pipe at every cross-section, the quantity of fluid per second will be constant.

Let us consider we have one pipe through which fluid is flowing. Let us consider two section 1-1 and 2-2 as displayed here in following figure.
Where,
V1 = Average velocity of flowing fluid at cross-section 1-1
ρ1= Density of flowing fluid at cross-section 1-1
A1 = Area of cross-section of pipe at cross-section 1-1
V2 = Average velocity of flowing fluid at cross-section 2-2
ρ2= Density of flowing fluid at cross-section 2-2
A2 = Area of cross-section of pipe at cross-section 2-2
Flow rate at section 1-1 = ρ1 A1 V1
Flow rate at section 2-2 = ρ2 A2 V2

 
Recall the principle of conservation of mass, we will have
Flow rate at section 1-1 = Flow rate at section 2-2

ρ1 A1 V1 = ρ2 A2 V2

Above equation will be termed as continuity equation and this equation will be applicable for compressible and incompressible fluid.

If we want to secure the continuity equation for only incompressible fluid, we will recall the basic definition of incompressible fluid and we will have ρ1 = ρ2

Therefore, continuity equation for incompressible fluid will be given by following equation as mentioned here.

A1 V1 = A2 V2

We will now go ahead to discuss the concept of continuity equation in three dimensions in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google

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