Equation of Continuity

· Because liquids are incompressible, the rate of flow into an area must equal the rate of flow out of an area. This is known as the equation of continuity.

· The equation of continuity can show how much the speed of a liquid increases if it is forced to flow through a smaller area. For example, if the area of a pipe is halved, the velocity of the fluid will double.

· Although gases often behave as fluids, they are not incompressible the way liquids are and so the continuity equation does not apply.



TERMS


· continuity

Lack of interruption or disconnection; the quality of being continuous in space or time.

· incompressible
Unable to be compressed or condensed.

Equation of Continuity can be expressed as:

                         

m = ρ_i1 v_i1 A_i1 + ρ_i2 v_i2 A_i2 +..+ ρ_in v_in A_im

                             = ρ_o1 v_o1 A_o1 + ρ_o2 v_o2 A_o2 +..+ ρ_om v_om A_om             (1)

where

m = mass flow rate (kg/s)

ρ = density (kg/m³)

v = speed (m/s)

A = area (m²)

With uniform density equation (1) can be modified to

q = v_i1 A_i1 + v_i2 A_i2 +..+ v_in A_im

    = v_o1 A_o1 + v_o2 A_o2 +..+ v_om A_om         (2)

where

q = flow rate (m³/s)

ρ_i1 = ρ_i2 = . . = ρ_in = ρ_o1 = ρ_o2 = . .= ρ_om

Example







Water runs through a water main of cross-sectional area 0.4
m2 with a velocity of 6 m/s. Calculate the velocity of the
water in the pipe when the pipe tapers down to a cross-
sectional area of 0.3 m2.
Answer: 





Water enters a typical garden hose of diameter 1.6 cm with
a velocity of 3 m/s. Calculate the exit velocity of water from
the garden hose when a nozzle of diameter 0.5 cm is
attached to the end of the hose.

Answer: 

First, find the cross-sectional areas of the entry(A1)
and exit (A2) sides of the hose.



Next, apply the continuity equation for fluids to solve for the
water velocity as it exits the hose (v2).

Sources

http://www.aplusphysics.com/courses/honors/fluids/continuity.html

https://www.boundless.com/physics/fluids/fluids-in-motion/flow-rate-and-the-equation-of-continuity/

http://www.hunter.cuny.edu/catalyst/catalyst-seminar-documents/fall-2010-catalyst-seminar/BasicEquationsofFluidMechanics.pdf