Equation of continuity

Equation of continuity:

Consider a steady flow of an incompressible fluid as shown in the figure. For a steady flow, the velocity of a particle remains constant at a given point but it can vary from point to point. For example, consider section A1 and A2. Section A1 has larger cross sectional area than the section A2. Let v1 and v2 be the velocities of the fluid at sections A1 and A2 respectively. This is because, a particle has to move faster in the narrower section (where there is less space) to accommodate particles behind it hence its velocity increases. When a particle enters a wider section, it slows down because there is more space. Because the fluid is incompressible, the particles moves faster through a narrow section and slow down while moving through wider section. If the fluid does not move faster in a narrow regain, it will be compressed to fit into the narrow space.

Consider a tube of flow as shown in the figure. All the fluid that passes through a tube of flow must pass through any cross section that cuts the tube of flow. We know that all the fluid is confined to the tube of flow. Fluid can not leave the tube or enter the tube. Consider section A1 and A2 located at points A and B respectively. Matter is neither created nor destroyed within the tube enclosed between section A1 and A2. Therefore, the mass of the fluid within this region is constant over time. That means, if mass m of the fluid enters the section A1 then equal mas of fluid should leave the section A2.

Let the speed of the fluid which crosses the section EFGH at point A in time interval Δt be v1 . Thus, the mass of the fluid entering the tube through the cross section at point A is ρA1v1Δt. Similarly, let the speed of the fluid be v2 at point B. The fluid crosses the section PQRS of area A2 in time interval Δt. Thus, the mass of the fluid leaving the tube through the cross section at B is ρA2v2Δt. As fluid is incompressible, the mass of the fluid entering the tube at point A is the same as the mass leaving the tube at B. Mass of the fluid in section EFGH = mass of fluid in section PQRS ρA1v1Δt = ρA2v2Δt

A1v1 = A2v2 or, Av = constant --- eqn 1

Av is the volume rate of flow of a fluid, i.e., Av = dV/dt . The quantity dV/dt is the volume of a fluid per unit time passing through any cross section of the tube of flow. It is called the volume flux. Similarly, ρdV/dt = dm/dt is called mass flux. Equation 1 is called the equation of continuity in fluid dynamics. The continuity equation says that the volume rate of flow of an incompressible fluid for a steady flow is the same throughout the flow.