EQUATION OF CONTINUITY

In a tube of flow shown in the figure, the velocity of a particle

In a non-viscous flow, all particles in a given cross-section have

the same velocity. Let the velocity of the fluid at cross-section P,

of area A1, and at cross-section, of area A2, is v1 and v2

respectively. Let ρ1 and ρ2 represent density of the

fluid at P and Q respectively. Then, as the fluid can not pass through the wall and can neither be created or destroyed, the mass flow rate (also called mass flux) at P and Q will be equal and is given by

dm = ρ1 A1 v1 = ρ2 A2 v2

dt

This equation is known as the law of conservation of mass in fluid dynamics. For liquids, which are almost incompressible, ρ1 = ρ2.

 A1 v1 = A2 v2 … … (1) which implies Av = constant … … (2) or, v a 1/A

Equations (1) and (2) are known as the equations of continuity in liquid flow. The product of area of cross-section, A and the velocity of the fluid, v at this cross section, i.e., Av, is known as the volume flow rate or the volume flux.

Thus, velocity of liquid is larger in narrower cross-section and vice versa. In the narrower cross-section of the tube, the streamlines are closer thus increasing the liquid velocity. Thus, widely spaced streamlines indicate regions of low speed and closely spaced streamlines indicate regions of high speed.

TUBE OF FLOW:

The tubular region made up of a bundle of

surface is called a tube of flow.

A wall made surrounds the tube of flow

of streamlines. As the streamlines do not

intersect, a particle of fluid cannot cross this

wall. Hence the tube behaves somewhat like a pipe of the same shape.

 

VENTURIE METER (principle)

flow rate of liquid in a pipe. One end of a manometer is

connected to the broad end of the venturie meter and the

other end to the throat. The cross-sectional area of the

broad end and velocity and pressure of liquid there are

A, v1 and P1 respectively. The cross-sectional area of

the throat and velocity and pressure of liquid there are

a, v2 and P2 respectively. The densities of the liquid

flowing through the venturie meter and of the manometer

liquid ( mercury ) are ρ and ρ’ respectively.

 THE CHANGE IN PRESSURE WITH DEPTH: -

The expression for hydrostatic pressure in a stationary liquid can be obtained as a special case of Bernoulli’s equation taking v1 = v2 = 0. Taking point 1 at a depth of h from the surface of liquid and point 2 on the surface where pressure is atmospheric pressure, Pa, and noting that h = y2 - y1, we get 

P1 = Pa + ρgh.