So, I was trying to obtain the point form of the conservation of linear momentum equation in integra

Mohammad Cannon

Mohammad Cannon

Answered question

2022-06-20

So, I was trying to obtain the point form of the conservation of linear momentum equation in integral form, namely:
Ω V ρ V d S + Ω p d S = 0
According to the Gauss theorem for a closed surface S:
S A d S = V A d V
But if I apply that to the above equation I get
Ω V ρ V d S = Ω ( V ρ V ) d V = Ω ( V ) ρ V d V
Which can't be right, since for an incompressible flow V = 0.
Isn't the dot product supposed to be commutative? What am I missing?
I apologize for any misuse of mathematical notation, let me know of any mistakes.

Answer & Explanation

massetereqe

massetereqe

Beginner2022-06-21Added 21 answers

The correct identity is
( ρ v v ) = ρ v v + ( v ) ρ v ,
where only the second term on the RHS vanishes in incompressible flow.
To check, this should all be consistent with the Navier-Stokes equations for steady flow (minus viscous terms that you have excluded):
ρ v v = p ,
which follows when v = 0 from
0 = Ω ρ v v d S + Ω p d S = Ω ( ρ v v ) d V + Ω p d V = Ω { ρ v v + ( v ) ρ v + p } d V = Ω { ρ v v + p } d V

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