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 crosssection, 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 Xdirection
Gain
of mass in Xdirection = ρ u dy.dz – [(ρ u dy.dz) + (∂ / ∂x) (ρ u dy.dz) dx]
Gain
of mass in Xdirection =  (∂ / ∂x) (ρ u dx.dy.dz)
Similarly,
we will have following equations for mass gain in Ydirection and Zdirection
Gain
of mass in Ydirection =  (∂ / ∂y) (ρ v dx.dy.dz)
Gain
of mass in Zdirection =  (∂ / ∂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 coordinates in its most
general form. This equation will be applicable to following types of fluid flow.
1.
Steady and unsteady flow
2.
Uniform and nonuniform 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