Proof of Continuity Equation in Polar Coordinates

This article is especially for the Equation of Continuity in the 3D cartesian co-ordinate system and the Derivation of continuity equation in cylindrical coordinates. But let us describe some basic knowledge about fluid which makes it easier to understand the integral form of the continuity equation.

What is equation of continuity?

Continuity equation in 1D

Equation of continuity states that the mass of fluid entering is equal to the sum of the mass of fluid leaving and change in mass.

Mass of fluid entering = Mass of fluid leaving + Change in mass

Change in mass is zero for steady flow. In such a case,

Mass of fluid entering = Mass of fluid leaving.

Consider cross-section of pipe at 1-1 and 2-2.

Let,

be the density, velocity, and cross-sectional area at 1-1 and

be the density, velocity, and cross-sectional area at 2-2.

Mass of fluid entering section 1-1 per unit time,

Mass of fluid entering section 2-2 per unit time,

According to the principle of conservation of mass,

For constant density,

or, A₁ * V₁ = A₂ * V₂

or, Q₁ = Q₂

Which is the required equation of continuity.

One, two and three dimensional flow

In one dimensional (1D) flow, fluid properties are a function of time and one space coordinate. In other words, conditions vary only in the direction of flow, not across the cross-section.

V = f(x,t)

The streamlines in 1D flow are Straight and parallel e.g flow in the pipe.

In two-dimensional (2D) flows, fluid properties are a function of time and two space coordinate. In other words, conditions vary in the direction of flow and in one direction at a right angle to this.

V = f(x,y,t)

The streamlines in 2D flow are curves, e.g flow in the mainstream of a wide river, flow between parallel plates.

In three-dimensional (3D) flows, fluid properties are a function of time and three space coordinates. In other words, conditions vary in the direction of flow, across the cross-section, and across the depth of flow.

V = f(x,y,z,t)

The streamline in 3D flow is space curves e.g flood flow in the river, flow in converging or diverging pipe.

In general fluid flow is three-dimensional. In many cases, the greatest changes only occur in two directions or even only in one. The changes in the other direction can be effectively ignored making analysis much simpler.

Discharge and mean velocity of flow

The total quantity of fluid flowing per unit time through a particular cross-section is called the discharge of flow rate. If the discharge is measured in terms of mass, it is called mass flow rate. If it is measured in terms of volume, it is called volumetric flow rate. It is represented by Q.

Unit :- m³/s or litres/s (volumetric flow rate), Kg/s (mass flow rate).

In many cases, the variation of velocity over the cross-section can be neglected. The velocity is assumed to be constant and A is the cross-sectional area, then discharge = (Q) is,

Q = A * V

In terms of mass flow rate

Different types of flow lines

1) Path line

The path line traced by a single fluid particle in motion over a period of time in a flow field is known as a path line. It indicates the direction of the velocity of the same particle at successive intervals of time. Path lines can intersect themselves.

2) Stream line

Equation of Continuity | Derivation of continuity equation in cylindrical coordinates

Streamline is an imaginary curve drawn through the flow field in such a way that the tangent to it at any point indicates the direction of velocity at that point. As streamlines join points of equal velocity, these are velocity contours. It is useful to visualize the flow pattern.

Properties

Close to a solid boundary, streamlines are parallel to that boundary.
Fluid cannot cross streamline. (as the velocity component perpendicular to it is zero)
Streamlines can not cross each other.
Any particles starting on one streamline will stay on that same streamline.
Streamline spacing varies inversely with velocity. Series of streamline represent flow pattern at an instant.
In unsteady flow, position of streamlines can change with time.
In steady flow, the position of streamline does not change.

3) Stream tube

An imaginary tube is formed by a group of streamlines through a small closed stream tube e.g pipes, nozzles.

Properties

The walls of a stream tube are streamlines.
It has finite dimensions
Fluid cannot flow across a streamline, so fluid cannot cross a stream tube wall.
A stream tube is not like a pipe. Its walls move with the fluid.
In unsteady flow stream tubes can change position with time.
In steady flow, the position of stream tubes does not change.

4) Steak Line

A steak line is a line made by a series of fluid particles passing through a fixed point in the flow field. E.g path taken by smoke coming out of the chimney, movement of particles after the dye is injected.

Basic principle of fluid flows

1) Principle of conservation of mass

It states that mass can neither be created nor destroyed, i.e total mass of a system remains constant. The continuity equation is derived from this equation.

2) Principle of conservation of energy

It states that energy can neither be created nor destroyed, i.e total energy of a system remains constant. The energy equation is derived from this principle.

3) Principle of conservation of momentum

It states that the change in momentum of a body is equal to the product of force and time increment during which it acts. The momentum equation is derived from this principle.

Continuity equation in 3D cartesian co-ordinate system

Consider a fluid element of length dx, dy, and dz in X, Y, and Z direction respectively. Let u, v, and w be the velocity in the X, Y, and Z directions respectively.