Question

obtain the equation of continuity for the incompressible non viscous fluid having steady flow through a pipe.

Answer 1

Answer:

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Answer 2

The equation of continuity for the incompressible non-viscous fluid having steady flow through a pipe.

consider that the fluid flows in the pipe for a short period of time say Δt. The distance that the fluid will cover is Δx1 moving with the velocity of v1.

therefore, the distance covered by the fluid is Δx1 = v1Δt

the volume of the fluid that will flow in the pipe is,

V = (A1 )(Δx1) = (A1)(v1)(Δt), where A1 is the area of the pipe.

mass(m) = density(ρ)x volume(V)

m= ρ1A1v1Δt                    (1)

now, mass flux has to be calculated at the lower end.

Mass flux is defined as the mass of the fluid per unit time passing through any cross-sectional area.

for the lower end with cross-sectional area A1, the mass flux will be,

Δm1/Δt = ρ1A1v1                 (2)

similarly, the mass flux at the upper end will be,

Δm2/Δt =ρ2A2v2              (3)

here, v2 is the velocity of the fluid through the upper end of the pipe.

A2 is the cross-sectional area of the upper end of the pipe.

as the flow of the fluid is steady, therefore, the density of the fluid between the lower end and the upper end of the pipe remains the same with time.

so, the flux at the lower end and upper end of the pipe is equal.

using equations (2) and (3)

ρ1A1v1= ρ2A2v2                 (4)

which means ρAv= constant

this equation proves the law of conservation of mass in fluid dynamics. Also, if the fluid is incompressible, the density will remain constant for steady flow.

so, ρ1=ρ2

equation (4) becomes,

A1v1 = A2v2

which means Av=constant          (5)

now, if R is the volume flow rate, then equation (5) can be expressed as

R=Av =constant

Hence the equation of continuity for the incompressible non-viscous fluid having steady flow through a pipe is R=Av =constant.