4.1 Conservation of mass
Every continuum body B possesses mass, denoted by m . It is a fundamental physical property
defined to be a measure of the amount of a material contained in the body B . In order to perform
a macroscopic study it is assumed that mass is continuously (or at least piecewise continuously)
distributed over an arbitrary region Ω (of physical space) with boundary surface ¶ Ω at time t .
The mass is a scalar measure (a positive number) which is invariant during a motion. Concentrated
masses such as those used in classical Newtonian mechanics is excluded.
4.1.1 Closed and open systems
A system is defined as a quantity of mass or a particular collection of matter in space. The
complement of a system, i.e., the mass or region outside the system is called the surroundings,
while the surface that separates the system from its surroundings is called the boundary or wall of
the system.
A closed system (or control mass) consists of a fixed amount of mass in a properly selected
region Ω in space with boundary surface ¶ Ω which depends on time t . No mass can cross (enter
or leave) its boundary, but energy, in the form of work or heat, can cross the boundary. The volume
of a closed system does not have to be fixed. If even energy does not interact between the system
and its surroundings, the boundary is said to be insulated. Such a system is called mechanically
and thermally isolated, which is an idealization for a physical system. There always exist
electromagnetic and other types of forces which permeate the space. Note that no physical system
is truly isolated.
An open system (or control volume) consists of a fixed amount of volume of a properly
selected region Ω C in space which is independent of time t . The enclosing boundary of a control
volume, over which both mass and energy can cross (enter or leave), is called the control surface
¶ ΩC .
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4.1-1
4.1.2 Conservation of mass
In non-relativistic physics mass cannot be produced or destroyed. It is assumed that during a
motion there are neither mass sources (reservoirs that supply mass) nor mass sinks (reservoirs that
absorb mass). So that mass m of a body is a conserved quantity. Hence, if a particle has a certain
mass in the reference configuration, it must stay the same during a motion. Considering a closed
system, obviously that holds for the total mass too. Hence,
m(W0 ) = m(W) > 0 ,
(4.1)
for all times t . Equation (4.1) is a statement of a fundamental mechanical law known as the
conservation of mass. The boundary surfaces in the reference and current configurations, with
volume V and v , are denoted by ¶W0 and ¶W , respectively. Note that the mass m is
independent of the motion and of the region occupied by the body. Hence, the material time
derivative of the mass m gives,
D
D
m(W0 ) =
m(W) = 0 .
Dt
Dt
(4.2)
The differential form of eqn (4.1) is,
d m ( X ) = dm ( x , t ) > 0 ,
(4.3)
with the infinitesimal mass element dm .
The mass of W0 and W is characterized by continuous (or at least piecewise continuous) scalar
fields, r0 = r0 ( X) > 0 and r = r ( x, t ) > 0 , respectively. They denote physical properties of the
same particle. Property r0 is called the reference mass density (or just density) and r is called
the spatial mass density during a motion x =  ( X, t ) . The spatial mass density, also known as
the density in the motion, depends on place x Î W and time t throughout the body. Note that r0
is time-independent and intrinsically associated with the reference configuration of the body.
Hence, r0 depends only on the position X in configuration W0 . If the density does not depend on
X Î W0 , i.e., if Grad r0 ( X) = 0 , the configuration is said to be homogeneous.
The mass densities of the points X and x are defined by,
r0 ( X ) =
lim
DV ( W0 ) 0
Dm(W0 )
,
DV (W0 )
r (x, t ) = lim
Dv ( W )  0
Dm(W)
,
Dv(W)
(4.4)
where Dm denotes a continuous function of incremental mass of an incremental volume element
in the reference and current configurations, denoted by DV and Dv , respectively.
Note that DV (W0 ) , Dv (W) , actually must not tend to zero since then the limit of r0 , r , would
show a discrete distribution according to the atomistic structure of matter. Therefore, to obtain
representative averages, DV (W0 ) and Dv (W) must be large in terms of an atomistic scale and
small in terms of a length scale of a certain physical problem. Usually the ratio of the length scale
of a physical problem and the length scale of an incremental volume element DV (W0 ) and Dv (W)
is of the order 103 or more.
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4.1-2
In the differential form eqn (4.4) is,
dm( X) = r0 ( X)dV ,
dm( x, t ) = r ( x, t )dv ,
(4.5)
with the standard infinitesimal volume elements dV and dv defined in the reference and current
configurations, respectively.
Substituting eqn (4.5) into (4.3),
r0 ( X)dV = r (x, t )dv > 0 ,
local form of conservation
of mass
(4.6)
which means that the volume increases when density decreases. By integrating the infinitesimal
mass over the entire region, the total mass m of that region can be determined. Hence, an alternate
expression for eqn (4.6) is,
m = ò r0 ( X)dV = ò r (x, t )dv = const > 0 ,
W0
global form of conservation
of mass
(4.7)
W
for all times t , which implies the rate form,
m =
Dm D
=
r (x, t )dv = 0 .
Dt
Dt òW
(4.8)
Hence, conservation of mass requires that the material time derivative of m is zero for all regions
Ω of a body which change with time (mass remains unchanged during the motion of Ω ).
An equation which holds at every point of a continuum and for all times, for example, eqn
(4.6), is referred as the local (or differential) form of that equation (local means pointwise). An
equation in which physical quantities over a certain region of space are integrated is referred to as
the global (or integral) form of that equation, for example, eqn (4.7).
In general, local forms are ideally suited for approximation techniques such as the finite
difference method, while global forms are the best to start with when the finite element method is
employed.
4.1.3 Continuity mass equation
A relationship between the reference mass density r0 ( X) Î W0 and the spatial mass density
r ( x, t ) Î W is determined.
Change the variable of integration in eqn (4.7) from x =  ( X, t ) to X by using eqn (2.51) i.e.,
dv = J (X, t )dV ,
ò [r ( X) - r (  ( X, t ), t ) J ( X, t )]dV = 0 .
0
(4.9)
W0
By assuming that V is an arbitrary volume of region W0 , it can be concluded that the integrand in
eqn (4.9) must vanish everywhere. Hence,
r0 ( X) = r (  ( X, t ), t ) J ( X, t ) ,
December 20, 2015
(4.10)
4.1-3
holds for all X Î W0 . It represents the continuity mass equation (continuity stands for constancy
of mass) in the material (or Lagrangian) description, which is the appropriate description in solid
mechanics.
Since the reference mass density r0 is independent of time, from eqn (4.10),
¶r0 ( X)
= r 0 ( X) = 0 ,
¶t
(4.11)
which is the rate form of eqn (4.10) in the material description.
Example 4.1 Show how the spatial mass density r = r ( x, t ) changes with time. In particular,
derive the rate form of continuity mass equation in the spatial (or Eulerian) description, which is
(expressed in terms of velocity components)
r ( x, t ) + r ( x, t )divv ( x, t ) = 0
or
r + r
¶va
= 0,
¶xa
(4.12)
or in the two equivalent forms,
¶r ( x , t )
+ gradr (x, t ) ⋅ v ( x, t ) + r ( x, t )divv ( x, t ) = 0 ,
¶t
(4.13)
¶r (x, t )
+ div(r (x, t ) v (x, t )) = 0 ,
¶t
(4.14)
for all x Î W and for all times t .
Solution:
Since r 0 = 0 , from eqn (4.10),
D
(r J ) = r J = 0 .
Dt
·
(4.15)
In order to express eqn (4.15) in terms of spatial velocity components, use eqn (2.177.6)

J = J div v ,
·

r J = r J + r J = J (r + r div v ) = 0 ,
(4.16)
where the material time derivative of the spatial density function r is, from eqn (2.25),
r =
Dr ¶r
=
+ gradr ⋅ v .
¶t
Dt
(4.17)
Since J > 0 from eqn (4.16) the desired result eqn (4.12) can be deduced, which is the
corresponding local form of eqn (4.8.2).
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4.1-4
With the material time derivative of the spatial density function eqn (4.17) and from identity
(1.288) i.e., div(Fu) = F divu + u ⋅ gradF , can obtain from eqn (4.12), the two equivalent forms
eqns (4.13) and (4.14). ■
------------------------------------------------------If the density of a continuum body is constant at any particle, then from eqn (4.12), with r = 0
can obtain a kinematical restriction which characterizes an isochoric (volume preserving) motion,
i.e., divv = 0 (compare also with eqn (2.179.5)). A continuum body is imcompressible if every
motion it undergoes satisfies r = 0 .
The rate forms of continuity mass eqns (4.12) – (4.14) show how the spatial mass density r
changes as time changes. These equations represent the continuity mass equation in the spatial
description, which is the appropriate description in fluid mechanics.
4.1.4 Conservation of mass for an open system
Sometimes it is necessary to work with an open system given by a region Ω C and boundary surface
¶Ω C .
At a certain time t a control volume contains the mass m(t ) = ò r ( x, t )dv . Since the region
WC
of integration Ω C does not depend on t , integration and differentiation commute, and it is possible
to write,
m (t ) =
D
¶r (x, t )
r (x, t )dv = ò
dv .
ò
Dt W
¶t
W
C
(4.18)
C
Applying the divergence theorem for a fixed amount of volume Ω C , using eqn (1.299),
ò div(r(x, t ) v(x, t ))dv = ò
WC
r (x, t ) v(x, t )) ⋅ nd s ,
(4.19)
¶WC
where n denotes the outward unit vector field perpendicular to the boundary control surface ¶ΩC .
The term
ò r v ⋅ nds determines the flux of r v out of Ω
C
across ¶Ω C .
¶WC
Integrating the continuity mass equation in the form of eqn (4.14) over a certain region Ω C
and using eqns (4.18) and (4.19),
D
r (x, t )dv = - ò r (x, t ) v(x, t )) ⋅ nd s .
Dt ò
W
¶W
C
C
Conservation of mass for a
control volume in the global
form
(4.20)
Equation (4.20) asserts that the material time derivative of the mass inside a control volume
Ω C is equal to the flux of r v entering Ω C across ¶Ω C . The global form eqn (4.20) is widely
used in fluid dynamics.
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