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Unlabelled CONTINUITY EQUATION IN THREE DIMENSIONS

CONTINUITY EQUATION IN THREE DIMENSIONS

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 in three dimensions with the help of this post.

Continuity equation in three dimensions

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 we have one pipe through which fluid is flowing. Let us consider a fluid element of having length dx, dy and dz in the direction of X, Y and Z respectively.
Mass of fluid entering the face ABCD per second = Density x velocity in x direction x Area ABCD
Mass of fluid entering the face ABCD per second = ρ x u x dy.dz = ρ u dy.dz
Mass of fluid leaving the face EFGH per second = (ρ u dy.dz) + (∂ / ∂x) (ρ u dy.dz) dx

Gain of mass = Mass flow rate in to the system – Mass for rate out of the system

Gain of mass = Mass of fluid entering the face ABCD per second - Mass of fluid leaving the face EFGH per second

Therefore, we will have following equation for mass gain in X-direction
Gain of mass in X-direction = ρ u dy.dz – [(ρ u dy.dz) + (∂ / ∂x) (ρ u dy.dz) dx]
Gain of mass in X-direction = - (∂ / ∂x) (ρ u dx.dy.dz)

Similarly, we will have following equations for mass gain in Y-direction and Z-direction
Gain of mass in Y-direction = - (∂ / ∂y) (ρ v dx.dy.dz)
Gain of mass in Z-direction = - (∂ / ∂z) (ρ w dx.dy.dz)

Net gain of masses = - [∂ / ∂x (ρ u) + ∂ / ∂y (ρ v) + ∂ / ∂z (ρ w)] dx.dy.dz

As per the principle of conservation of mass, mass could not be created or destroyed in the fluid element. Therefore, net increase of mass per unit time in the fluid element should be equal to the rate of increase of mass in the fluid element.

Mass of fluid in the fluid element = ρ dx.dy.dz

Rate of increase of mass in the fluid element = ∂ρ/∂t. dx.dy.dz

Therefore,
- [∂ / ∂x (ρ u) + ∂ / ∂y (ρ v) + ∂ / ∂z (ρ w)] dx.dy.dz = ∂ρ/∂t. dx.dy.dz
- [∂ / ∂x (ρ u) + ∂ / ∂y (ρ v) + ∂ / ∂z (ρ w)] = ∂ρ/∂t

∂ρ/∂t + ∂ / ∂x (ρ u) + ∂ / ∂y (ρ v) + ∂ / ∂z (ρ w) = 0

Above equation is the continuity equation in Cartesian co-ordinates in its most general form. This equation will be applicable to following types of fluid flow.

1. Steady and un-steady flow
2. Uniform and non-uniform flow
3. Compressible and incompressible flow

For steady flow, continuity equation will be as mentioned here

For steady flow, ∂ρ/∂t = 0

∂ / ∂x (ρ u) + ∂ / ∂y (ρ v) + ∂ / ∂z (ρ w) = 0

For incompressible flow, continuity equation will be as mentioned here

For incompressible flow, ρ will be constant and we will have following continuity equation

∂u / ∂x + ∂v / ∂y + ∂w / ∂z = 0

Above equation is the continuity equation in three dimensions

For two dimensional continuity equation, w = 0

 ∂u / ∂x + ∂v / ∂y = 0

We will now go ahead to discuss the expression of continuity equation in cylindrical polar coordinates 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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