We
were discussing the basic concept of Lagrangian and Eulerian method, Types of fluid flow , Discharge or
flow rate and Continuity equation in three dimensions, in the subject of fluid
mechanics, in our recent posts.
Now
we will go ahead to understand the continuity equation in cylindrical polar
coordinates, in the field of fluid mechanics, with the help of this post.
Continuity equation in cylindrical polar coordinates
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. We can find the detailed information about the continuity
equation in our previous post.
Therefore,
for a flowing fluid through the pipe at every cross-section, the quantity of
fluid per second will be constant.
Let
us consider that we have one pipe through which fluid is flowing. Let us also consider
that type of flow is two dimensional and incompressible and for which polar
coordinates are r and θ.
We have following data from above figure
ABCD
is one fluid element between radius r and r + dr
dθ
is the angle made between the fluid element at the centre.
Ur
= velocity in radial direction
Uθ
= velocity in tangential direction
Continuity equation in cylindrical polar coordinates will be given by following equation.
We
will discuss now another important topic i.e. "Total acceleration in fluid mechanics" and "Velocity potential function", in the
subject of fluid mechanics, 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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