In kinematics of flow, the study is only focused on the parameters that cause the motion of the fluid and not the forces that cause the motion of a fluid particle. The discharge and continuity equation are topics that are used to study the flow of a fluid through a pipe or a channel.

Rate of Flow or Discharge (Q)

Discharge or rate of flow (Q) is defined as the fluid flowing per second through a channel or section of a pipe. This rate of flow is expressed in terms of “volume” when the fluid flow is incompressible and is taken in terms of “weight” when the flow is compressible.

For liquid flow, discharge (Q) is expressed in litres/sec or m3/s and for gaseous flow, the discharge is expressed in kgf/s or Newton/s.

If the liquid flowing through a pipe of area of cross-section ‘A’ has a velocity of ‘V’, then discharge,

Q = A x V          Eq.1

Continuity Equation in Fluid Mechanics

The continuity equation is developed based on the principle of conservation of mass. The continuity equation states that the rate of fluid flow through the pipe is constant at all cross-sections. That is, the quantity of fluid per second is constant throughout the pipe section.

Consider a fluid, flowing through a pipe with varying cross-sectional areas, as shown in figure-1 below. Consider two sections 1-1 and 2-2 as shown.

Continuity Equation in Fluid Mechanics
Fig.1. Continuity Equation in Fluid Mechanics

The area of the cross-section in sections 1 and 2 be A1 and A2 respectively. The velocity and density of fluid at section 1-1 be V1 and J1 & that of section-2-2 be V2 and J2. Then, from Equation-1,

Rate of Flow or Discharge at Section 1-1 , Q1 = J1A1V1

Rate of flow or discharge at Section 2-2 ,Q2 = J2A2V2

Based on the Continuity equation, the rate of flow of fluid in section 1-1 is equal to the rate of flow of fluid in section 2-2. Then,

Q1 = Q2

J1A1V1 =J2A2V2

The above equation is applicable to compressible flow (The fluid flow in which the density varies with time). For incompressible flow, the continuity equation is given by the equation,

                                A1V1 =A2V2

Continuity Equation for 3D and 2D

The continuity equation for three-dimensional and two-dimensional flow can be expressed either in cartesian coordinates or in polar coordinates.

Continuity equation in Cartesian Coordinates (3D)

The continuity equation in cartesian coordinates can be applicable for:

1. Steady and Unsteady Fluid Flow

2. Uniform and Non-Uniform Fluid Flow

3. Compressible and Incompressible Fluid Flow

Continuity Equation for Steady Flow

Continuity Equation for Steady Flow

Continuity Equation for Incompressible Flow

Continuity equation in Cartesian Coordinates (2D)

Continuity equation in Polar Coordinates

Also Read: What is Open Channel Flow? Types of Flow in Open Channels