We
were discussing the basic concept of Lagrangian
and Eulerian method, types
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.
ρ1 A1
V1 = ρ2 A2 V2
A1 V1 =
A2 V2
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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