Reynolds transport theorem

Reynolds transport theorem

Introduction

Reynolds transport theorem

[

1

]

is a fundamental theorem used in formulating the basic laws of fluid mechanics. We will enunciate and demonstrate in this entry the referred

theorem

. For our purpose, let us consider a fluid flow, characterized by its streamlines, in the

Euclidean vector space

(

ℝ

3

,

∥

⋅

∥

)

and embedded on it we consider, a

continuum

body

ℬ

occupying a volume

𝒱

whose particles are fixed by their material (Lagrangian)

coordinates

𝐗

, and a region

ℜ

where a control volume

𝔳

is defined whose points are fixed by it spatial (Eulerian) coordinates

𝐱

and bounded by the control surface

∂

⁡

𝔳

. An arbitrary

tensor field

of any rank is defined over the fluid flow according to the following

definition

.

Definition 1

.

We call an

extensive tensor

property

to the

expression

Ψ

⁢

(

𝐱

,

t

)

:=

∫

𝔳

ψ

⁢

(

𝐱

,

t

)

⁢

ρ

⁢

(

𝐱

,

t

)

⁢

𝑑

v

,

(1)

where

ψ

⁢

(

x

,

t

)

is the respective

intensive tensor property

.

Theorem’s hypothesis

The kinematics of the continuum can be described by a

diffeomorphism

χ

which, at any given instant

t

∈

[

0

,

∞

)

⊂

ℝ

, gives the spatial coordinates

𝐱

of the material particle

𝐗

,

𝒱

×

[

0

,

∞

)

→

𝔳

×

[

0

,

∞

)

,

t

↦

t

,

𝐗

↦

𝐱

=

χ

⁢

(

𝐗

,

t

)

.

Indeed the above sentence corresponds to a

change of coordinates

which must verify

J

=

|

∂

⁡

x

i

∂

⁡

X

j

|

≡

|

F

i

⁢

j

|

≠

0

,

F

i

⁢

j

:=

∂

⁡

x

i

∂

⁡

X

j

,

J

being the

Jacobian

of transformation and

F

i

⁢

j

the Cartesian components of the so-called

strain

gradient

tensor

𝐅

.

Reynolds transport theorem 1

.

The material rate of an extensive tensor property associate to a continuum body

B

is equal to the local rate of such property in a control volume

v

plus the efflux of the respective intensive property across its control surface

∂

⁡

v

.

Proof.

By taking on Eq.(1) the material time derivative,

D

⁢

Ψ

D

⁢

t

=

Ψ

˙

=

∫

𝔳

ψ

⁢

ρ

⁢

𝑑

v

¯

˙

=

∫

𝒱

ψ

⁢

ρ

⁢

J

⁢

𝑑

V

¯

˙

=

∫

𝒱

ψ

⁢

ρ

⁢

J

¯

˙

⁢

𝑑

V

=

∫

𝒱

(

ψ

⁢

ρ

¯

˙

⁢

J

+

ψ

⁢

ρ

⁢

J

˙

)

⁢

𝑑

V

=

∫

𝒱

{

J

⁢

[

∂

∂

⁡

t

⁢

(

ψ

⁢

ρ

)

+

𝐯

⋅

∇

x

⁡

(

ψ

⁢

ρ

)

]

+

ψ

⁢

ρ

⁢

(

J

⁢

∇

x

⋅

𝐯

)

}

⁢

𝑑

V

=

∫

𝒱

{

[

∂

∂

⁡

t

⁢

(

ψ

⁢

ρ

)

]

+

[

𝐯

⋅

∇

x

⁡

(

ψ

⁢

ρ

)

+

(

ψ

⁢

ρ

)

⁢

∇

x

⋅

𝐯

]

}

⁢

(

J

⁢

d

⁢

V

)

=

∫

𝔳

∂

∂

⁡

t

⁢

(

ψ

⁢

ρ

)

⁢

𝑑

v

+

∫

𝔳

∇

x

⋅

(

ψ

⁢

ρ

⁢

𝐯

)

⁢

𝑑

v

=

∂

∂

⁡

t

⁢

∫

𝔳

ψ

⁢

ρ

⁢

𝑑

v

+

∫

∂

⁡

𝔳

ψ

⁢

ρ

⁢

𝐯

⋅

𝐧

⁢

𝑑

a

,

since

∂

t

⁡

(

d

⁢

v

)

=

0

(

𝐱

fixed) on the first integral and by applying the Gauss-Green

divergence theorem

on the second

integral

at the left-hand side. Finally, by substituting Eq.(1) on the first integral at the right-hand side, we obtain

Ψ

˙

=

∂

⁡

Ψ

∂

⁡

t

+

∫

∂

⁡

𝔳

ψ

⁢

ρ

⁢

𝐯

⋅

𝐧

⁢

𝑑

a

,

(2)

endorsing the theorem statement.
∎

References

1

O. Reynolds,

Papers on mechanical and physical subjects-the sub-mechanics of the Universe,

Collected Work, Volume III, Cambridge University Press, 1903.

Title

Reynolds transport theorem

Canonical

name

ReynoldsTransportTheorem

Date of creation

2013-03-22 16:00:48

Last modified on

2013-03-22 16:00:48

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

7

Author

perucho (2192)

Entry type

Theorem

Classification

msc 53A45

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