continuity equation

continuity equation

Introduction

The

continuity equation

[

1

]

is a mathematical statement about mass-preserving in a

continuum

material body

ℬ

. This principle indeed is but a partial re-statement of the

hypothesis

of

continuity

of motion

1

1

Cf. motion of continuum

, plus the

introduction

of a new physical constant

𝐌

=

𝐋

𝟑

⁢

𝐓

-

𝟐

. We shall

postulate

three properties for this

associate

positive quantity so-called

mass

,

m

=

m

⁢

(

ℬ

)

.

Concept of

mass density

In order to define mass

density

we postulate:

1. We require that mass be

additive

, so that the masses

d

⁢

m

of volume elements of the body

ℬ

add up to the total mass of the body , i.e.

m

⁢

(

ℬ

)

=

∫

𝒱

𝑑

m

,

where

𝒱

=

𝒱

⁢

(

𝐗

,

t

0

)

is the volume occupied by the body

ℬ

in an initial material reference

configuration

χ

⁢

(

ℬ

)

≡

χ

⁢

(

𝐗

,

t

0

)

,

t

0

≤

τ

≤

t

,

being

t

an arbitrary “present” instant.

2. We assume that mass is a

continuous function

of the volume. That is a

characteristic

assumption

in continuum mechanics; in the mathematical

language

we say that mass

m

⁢

(

ℬ

)

and volume

𝒱

⁢

(

ℬ

)

are in the

additive set functions.

3. In a field theory we must be able to describe the mass of a portion of a body. For this reason we introduce a point

function

called

mass density

, that we shall define in the following manner. Consider the

smooth mappings

ℬ

→

ℜ

given by

χ

:

𝒱

⁢

(

𝐗

,

t

0

)

→

𝔳

⁢

(

𝐱

,

τ

)

,

t

0

≤

τ

≤

t

,

(1)

where

𝐗

fix the position of any material particle

P

0

∈

ℬ

at

τ

=

t

0

, the initial reference configuration, and

𝐱

=

χ

⁢

(

𝐗

,

τ

)

fix the position of the same particle, at point

P

located in certain region

ℜ

embedded in the

Euclidean space

(

ℝ

3

,

∥

⋅

∥

)

, where the

volume spatial description

𝔳

⁢

(

𝐱

,

τ

)

of the body

ℬ

is defined. So, the indicated

coordinate

transformation

allows to define the

mapping

of

neighborhoods

N

⁢

(

𝐗

,

δ

)

↦

N

⁢

(

𝐱

,

ϵ

)

,

to which we can associate the pair of

(

m

δ

,

𝒱

δ

)

and

(

m

ϵ

,

𝔳

ϵ

)

, respectively. (

Arguments

are understood) Assuming now that all of these functions yield to zero as

(

δ

,

ϵ

)

→

0

, we define

2

2

Cf. Buck

[

3

]

, pp. 99-104.

ρ

0

≡

ρ

0

⁢

(

𝐗

,

t

⁢

o

)

:=

lim

δ

→

0

⁡

m

δ

𝒱

δ

,

ρ

≡

ρ

⁢

(

𝐱

,

τ

)

:=

lim

ϵ

→

0

⁡

m

ϵ

𝔳

ϵ

,

t

0

≤

τ

≤

t

,

(2)

if these limits exist. Moreover, we can see these densities as a mapping transformation from the material (Lagrangian) description defined by coordinates

X

α

in the initial material reference configuration at

τ

=

t

0

, to the spatial (Eulerian) description defined by coordinates

x

i

in any later spatial configuration at

τ

=

t

, i.e.

ρ

0

⁢

(

X

α

,

t

0

)

↦

ρ

⁢

(

x

i

,

t

)

,

ρ

0

=

J

⁢

ρ

,

J

=

|

∂

⁡

x

i

∂

⁡

X

α

|

.

(3)

Therefore the mapping

d

⁢

V

↦

d

⁢

v

between the volume elements

d

⁢

V

and

d

⁢

v

corresponding to the initial material configuration and any later space configuration will also be valid, i.e. for an arbitrary mass

element

d

⁢

m

in the body

ℬ

,

d

⁢

m

=

ρ

0

⁢

d

⁢

V

=

ρ

⁢

d

⁢

v

,

which is precisely the reason for the postulation of continuity equation.

Integral form of continuity equation

A set of particles

V

of positive mass

m

≡

m

(

ℬ

)

:=

∫

𝒱

ρ

0

d

V

≡

∫

𝔳

ρ

d

v

=

:

m

(

ℜ

)

,

(4)

is a body, and such invariance is valid also for any portion of continuum,

as we shall see in the differential form of that equation. By taking the material time derivative, we get

D

⁢

m

D

⁢

t

=

m

˙

=

∫

𝒱

ρ

0

⁢

𝑑

V

¯

˙

=

∫

𝔳

ρ

⁢

𝑑

v

¯

˙

=

0

.

(5)

Differential form of continuity equation

From the continuity and positiveness on the integrand of the last

integral

in Eq.(5), we obtain the following statement.

In regions where the density is

continuous

3

3

The mappings

𝐱

=

𝐱

⁢

(

𝐗

,

t

)

,

𝐗

=

𝐗

⁢

(

𝐱

,

t

)

,
are assumed to be

single-valued

and continuously differentiable with respect to each of their variables, except possibly at certain points, curves or surfaces.

the principle of mass conservation may be expressed in the

equivalent

forms

4

4

Cf. the remarks of Hilbert

[

2

]

,

pp. 100-108.

ρ

⁢

d

⁢

v

¯

˙

=

0

,

∇

x

⋅

𝐯

=

-

log

⁡

ρ

¯

˙

,

∂

⁡

ρ

∂

⁡

t

+

∇

x

⋅

(

ρ

⁢

𝐯

)

=

0

,

ρ

˙

+

ρ

⁢

∇

x

⋅

𝐯

=

0

,

(6)

the

derivatives

standing for material rate of change

5

5

Cf. time dilatation of a volume element

. These equations are essentialy due to Euler

[

4

]

. For isochoric motions

d

⁢

v

=

d

⁢

V

,

ρ

=

ρ

0

,

∇

x

⋅

𝐯

=

I

𝐝

=

0

,

I

𝐝

being the first

invariant

of Cauchy’s rate strain tensor

𝐝

.

Integral vs. differential approach

The basic laws that we apply in fluid mechanics and thermodynamics studies can be formulated in terms of finite (global) or

infinitesimal

(local) approach depending on the nature of the problem. In the former case we apply equations governing the gross

behavior

of the flow through an integral formulation which is usually easier to treat analytically. In the latter we deal with differential equations providing a means of determining the detailed behavior of the flow. Since the mentioned disciplines deal with the formulation of the basic laws in terms of finite systems, such formulations are the basis for deriving the control volume equations,

concept

that we shall develop to

continuation

.

Additional definitions

1. Control volume (CV).

Is an arbitrary but convenient volume spatial description

6

6

It is usual to consider fixed the control volume with respect to the chosen

coordinate system

.

through which the fluid flows. In fact , we are normally concerned with flows through devices such as pipelines,

turbines

, compressors, nozzles, etc. In these cases it is difficult to focus attention on a fixed identifiable quantity of mass. So it is much more convenient, for purpose of

analysis

, to focus attention on a fixed volume through which the fluid flows, a fact that plenty justifies to use the control volume approach. It will be denoted as

𝔳

⁢

(

𝐱

,

τ

)

. Since the mass is varying inside of the control volume , we have

∂

∂

⁡

t

⁢

∫

𝔳

ρ

⁢

(

𝐱

,

t

)

⁢

𝑑

v

=

∫

𝔳

(

∂

⁡

ρ

∂

⁡

t

)

⁢

𝑑

v

≠

0

.

In this case the material rate derivative must be considered as a local rate, because the spatial coordinates

x

i

are held

constant

for a fixed control volume.

2. Control surface (CS).

This is how we call the geometric

boundary

of the control volume. We shall denote it as

∂

⁡

𝔳

.

So, if we define a

unit vector

𝐧

always chosen to be the outward normal of the boundary

∂

⁡

𝔳

and

d

⁢

a

is an element of area on the control surface, we have

𝐝𝐚

=

𝐧

⁢

d

⁢

a

. Eventually this surface may be real or

imaginary

; it may be at rest or in motion. For

instance

, in pipeflow the lateral wall is a real physical boundary that comprises part of the control surface but the two

transversal

circular

portions of it are imaginary, i.e. there is no corresponding physical surface. These imaginary boundaries are selected arbitrarily for accounting purposes. Since the location of the control surface has a direct effect on the accounting procedure in applying the basic laws, it is extremely important that the control surface be clearly defined before beginning any form of analysis.

3. Mass flow rate.

7

7

So-called also

mass flux.

Let us suppose that a velocity field

𝐯

⁢

(

𝐱

,

t

)

is defined over a region

ℜ

of the space such that a control volume

𝔳

⁢

(

𝐱

,

t

)

⊂

ℜ

. Let

∂

⁡

𝔳

be its control surface. We define the mass flow rate across an area element

𝐝𝐚

on the control surface, as

d

⁢

𝔪

¯

˙

≡

d

⁢

𝔪

˙

:=

ρ

⁢

(

𝐱

,

t

)

⁢

𝐯

⁢

(

𝐱

,

t

)

⋅

𝐝𝐚

⁢

(

𝐱

,

t

)

=

ρ

⁢

(

𝐱

,

t

)

⁢

𝐯

⁢

(

𝐱

,

t

)

⋅

𝐧

⁢

(

𝐱

,

t

)

⁢

d

⁢

a

,

and the

net mass flow rate

or

efflux

across the control surface

∂

⁡

𝔳

, as

𝔪

˙

:=

∫

∂

⁡

𝔳

ρ

⁢

𝐯

⋅

𝐧

⁢

𝑑

a

,

(7)

where the variable arguments are understood. Let now

ϑ

=

ϑ

⁢

(

𝐱

,

t

)

be the angle between the velocity field

𝐯

(

tangent

to the flow streamlines) and the outward normal

𝐧

; we have

d

⁢

𝔪

˙

=

ρ

⁢

𝐯

⋅

𝐧

⁢

d

⁢

a

=

ρ

⁢

∥

𝐯

∥

⁢

d

⁢

a

⁢

cos

⁡

ϑ

=

{

an outflow,

if

⁢

0

≤

ϑ

<

π

/

2

an inflow,

if

⁢

π

/

2

<

ϑ

≤

π

.

It is clear that mass flow rate cannot cross the streamlines. The introduction of the definitions that we have done, allows to stablish the following useful

proposition

.

Continuity equation for a control volume.

Proposition 1

.

“The rate of mass change within a

stationary

control volume is equal to the net mass flow rate across its control surface”

.

Proof.

From Eqs.(5)-(6), and assuming diffeomorfic the coordinates transformation,

m

˙

=

0

=

∫

𝔳

ρ

⁢

𝑑

v

¯

˙

=

∫

𝔳

ρ

⁢

d

⁢

v

¯

˙

=

∫

𝒱

ρ

⁢

(

J

⁢

d

⁢

V

)

¯

˙

=

∫

𝒱

ρ

⁢

J

¯

˙

⁢

𝑑

V

=

∫

𝒱

(

ρ

˙

⁢

J

+

ρ

⁢

J

˙

)

⁢

𝑑

V

∫

𝒱

[

(

∂

⁡

ρ

∂

⁡

t

+

𝐯

⋅

∇

x

⁡

ρ

)

⁢

J

+

ρ

⁢

(

J

⁢

∇

x

⋅

𝐯

)

]

⁢

𝑑

V

=

∫

𝒱

∂

⁡

ρ

∂

⁡

t

⁢

(

J

⁢

d

⁢

V

)

+

∫

𝒱

(

𝐯

⋅

∇

x

⁡

ρ

+

ρ

⁢

∇

x

⋅

𝐯

)

⁢

(

J

⁢

d

⁢

V

)

,

∂

∂

⁡

t

⁢

∫

𝔳

ρ

⁢

𝑑

v

+

∫

𝔳

∇

x

⋅

(

ρ

⁢

𝐯

)

⁢

𝑑

v

=

0

,

and by applying the Gauss-Green

divergence theorem

, we obtain

∂

∂

⁡

t

⁢

∫

𝔳

ρ

⁢

𝑑

v

=

-

∫

∂

⁡

𝔳

ρ

⁢

𝐯

⋅

𝐧

⁢

𝑑

a

,

(8)

as desired.
∎

Remark.

In Eq.(8), the velocity field

𝐯

is relative to the CS. So, the

formula

is still valid for CV

rigid motions

(i.e. as a whole motions) as we may substitute the field

𝐯

by the relative velocity field

𝐰

=

𝐯

-

𝐯

C

⁢

V

.

Steady-state flow

In this case

8

8

Cf.

[

5

]

, for instance.

, the fluid properties are

independent

on time. Therefore Eq.(8) reduces to

∫

∂

⁡

𝔳

ρ

⁢

𝐯

⋅

𝐧

⁢

𝑑

a

=

0

,

(9)

thus the mass flow rate

𝔪

˙

i

⁢

n

, across the CS

∂

⁡

𝔳

i

⁢

n

, entering the CV is equal to the mass flow rate

𝔪

˙

o

⁢

u

⁢

t

, across the CS

∂

⁡

𝔳

o

⁢

u

⁢

t

, leaving the CV i.e.

𝔪

˙

=

𝔪

˙

i

⁢

n

:=

∫

∂

⁡

𝔳

i

⁢

n

ρ

|

𝐯

⋅

𝐧

|

d

a

=

∫

∂

⁡

𝔳

o

⁢

u

⁢

t

ρ

𝐯

⋅

𝐧

d

a

=

:

𝔪

˙

o

⁢

u

⁢

t

.

(10)

That equation is extensively used in applications such as internal combustion engines, reciprocating

machines

, turbomachinery, etc. On the other hand, if it is possible to define concrete inlet/outlet cross sections

A

on the CS, the flow is normal (i.e.

cos

⁡

ϑ

=

±

1

) and uniform (non-viscous flow, for example) and assuming constant density in each

section

(a reasonable supposition), we get

𝔪

˙

=

ρ

⁢

v

⁢

A

,

(

ρ

⁢

v

⁢

A

)

i

⁢

n

=

(

ρ

⁢

v

⁢

A

)

o

⁢

u

⁢

t

,

where

v

is the normal speed flow at the respective section. For multiple inlets and outlets, we have the obvious formulas

∑

𝔪

˙

i

⁢

n

=

∑

𝔪

˙

o

⁢

u

⁢

t

,

∑

(

ρ

⁢

v

⁢

A

)

i

⁢

n

=

∑

(

ρ

⁢

v

⁢

A

)

o

⁢

u

⁢

t

.

References

1

L. Euler,

Principes gÃÂ©nÃÂ©raux du mouvement des fluides,

Hist. Acad. Berlin

1755,

217-273, 1757.

2

D. Hilbert,

Mechanik der Continua,

lectures of 1906-1907; MS notes by A.R. Crathorne in Univ. Illinois Library, 1907.

3

R. C. Buck,

Advanced

Calculus

,

New York: McGraw-Hill Co., 1965.

4

L. Euler,

Sectio secunda de principiis motus fluidorum,

Novi Comm. Petrop.

14

(1769), 270-386, 1770.

5

R. W. Fox and A. T. McDonald,

Introduction to fluid mechanics,

John Wiley & Sons, Inc., 7-8, 1973.

Title

continuity equation

Canonical name

ContinuityEquation

Date of creation

2013-03-22 15:56:05

Last modified on

2013-03-22 15:56:05

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

8

Author

perucho (2192)

Entry type

Application

Classification

msc 53A45

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