Bernoulli's equation from an elemental fixed streamtube control volume

[Atouk]

Homework Statement

Demonstrate the Bernoulli equation by analyzing a volume control element

Relevant Equations

Reynolds transport theorem

Elemental fixed streamtube control volume from Professor White’s textbook “Fuid Mechanics”:

I was unable to develop the intermediate steps for the following approximations:

(continuity equation according to the book )

Where

and

(Momentum equation according to the book)

In the Conservation of Mass for the “elemental control volume”:

m(in) = ρVA

m(out) = (ρ+dρ)*(V+dV)*(A+dA)

But how to get this :

mout - min = dm

and

And I can not understand this :

d/dt (∫ V*ρ*d∀) ≈ (∂ρ / ∂t) * d∀

Thanks

 

[AvroArrow]

The first approximation is based on the continuity equation, which states that the rate of change of mass within a control volume is equal to the rate of mass flow into and out of the system. Mathematically, this can be represented as:m(in) - m(out) = dm/dtWhere m(in) and m(out) are the masses entering and leaving the control volume, respectively, and dm/dt is the rate of change of mass within the control volume. Expanding each of these terms and rearranging, we get:m(in) = ρVAm(out) = (ρ + dρ ) (V + dV) (A + dA)Substituting these into the equation above and simplifying, we get:dm/dt = (ρ + dρ ) (V + dV) (A + dA) - ρVAThe second approximation is based on the momentum equation, which states that the rate of change of momentum within a control volume is equal to the rate of net force acting on the control volume. Mathematically, this can be written as:d/dt (∫ V*ρ*d∀) ≈ (∂ρ / ∂t) * d∀Where d/dt is the rate of change with respect to time, V*ρ is the momentum, d∀ is the elemental area, and ∂ρ/∂t is the rate of change of density. This equation can be used to approximate the rate of change in momentum within the control volume.