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Wednesday, 21 March 2018

CONTINUITY EQUATION IN CYLINDRICAL COORDINATES

We were discussing the basic concept of Lagrangian and Eulerian methodTypes 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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