## Continuity of Flow

### May 14, 2009

Matter is neither created nor destroyed. This principle conservation of mass can be applied to a flowing fluid.

Considering any fixed region the flow constituting a control volume.** **

Mass of fluid entering per unit time= Mass of fluid leaving per unit time + Increasing/decreasing of mass of fluid in the control volume per unit time

For steady flow, the mass of fluid in the control volume remains constant and relation reduces to

Mass of fluid entering per unit time = Mass of fluid leaving per unit time

*Applying this principle to steady flow in a stream tube having a cross sectional area small enough for the velocity to be considered as constant over any given cross-section, for the region between sections I and 2, since there can be no flow through the walls of a stream tube; *

*Mass entering per unit time at section 1 = Mass leaving per unit time-at section 2*

*Suppose that at section 1 the area of the stream tube is *δ*A _{1}, the velocity of the fluid u_{1} and its density *r

*δ*

_{1}while at section 2 the corresponding values are*A*ρ

_{2}, u_{2}and

_{2}whileMass entering per unit time at 1 = δA_{1} u_{1} ρ_{1}

Mass leaving per unit time at 2 = δA_{2} u_{2} ρ_{2}

δA_{1} u_{1} ρ_{1}= δA_{2} u_{2} ρ_{2 = }Constant

*This is the equation of continuity for the flow of a compressible fluid through a any fixed region in stream tube, u _{1} and u_{2 }being the velocities measured at right angles to the cross-sectional areas *δ

*A*δ

_{1}and*A*

_{2}.For the flow of a real fluid through a pipe or other conduit, the velocity will vary from wall to wall. However, using the mean velocity, the equation of continuity for steady flow can be written as

δA_{1} u_{1} ρ_{1}= δA_{2} u_{2} ρ_{2 = }m dot.

where δA_{1} and δA2 are the cross-sectional areas and m is the mass rate of flow.

If the fluid can be considered as incompressible, so that ρ_{1} =ρ_{2} equation reduce to,

δA_{1} u_{1} = δA_{2} u_{2 = }Q dot

The continuity equation can also be applied to determine the relation between the flows into and out of a junction. In Figure, for steady conditions,

Total inflow to junction = Total outflow from junction,

ρ_{1}Q_{1} =ρ_{2}Q_{2 }+ρ_{3}Q_{3}

For an incompressible fluid, ρ1 =ρ2 =ρ3 so that

Q_{1} =Q_{2} +Q_{3}

A_{1}`v_{1}= A_{2}`v_{2}+ A_{3}v_{3}

In general, if we consider flow towards the junction as positive and flow away from the junction as negative, then for steady flow at any junction the algebraic sum of all the mass flows must be zero:

∑ρQ=0