LECTURE 3
ECTURE
L 3
GENERAL FRAMEWORK FOR CONSERVATION
G
ENERAL FRAMEWORK FOR CONSERVATION
LAWS IN FLUID MECHANICS. THE PRINCIPLE OF
LAWS IN FLUID MECHANICS HE PRINCIPLE OF
. T
MASS CONSERVATION AND RELATED EQUATIONS.
MASS CONSERVATION AND RELATED EQUATIONS
.
Fundamental Principles of Mechanics tell us what happens with:
·ð mass
·ð linear momentum
·ð angular momentum
·ð energy
during a motion of a fluid medium.
Basic equations of the Fluid Mechanics are derived from these principles.
Additionally, the reference to the 2nd Principle of Thermodynamics may be necessary in
order to recognized physically feasible solutions.
CONSERVATION LAWS  GENERAL FRAMEWORK
CONSERVATION LAWS  GENERAL FRAMEWORK
Consider an extensive physical quantity H. The spatial distribution of this quantity can be
characterized by means of its mass-specific density h. AT this point we do not precise if the
field h is scalar, vector or tensor.
Consider the finite control (not fluid!) volume Wð embedded in the fluid. The total amount of
the quantity characterized by the density field h is expressed by the volume integral
H (t) =ð
òðrðh dV
Wð
where rð denotes the mass density of the fluid. We ask the fundamental question: what is the
rate of the temporal change of H? The general answer is
dH d dH dH
ºð +ð
òðrðh dV =ð
dt dt dt dt
production flow through Å›ðWð
Wð
i.e., The total rate is the sum of two contributions:
·ð change rate due to the production/destruction of the quantity H,
·ð change rate due to the transport of H by the fluid entering/leaving Wð through the
boundary Å›ðWð.
Note that the second contribution can be expressed by the following surface integral (see
figure)
dH
=ð -ð rðhuðn dS
òð
dt
flow through Å›ðWð0
Å›ðWð
where uðn Å›ðWð =ð v ×ðn denotes the normal
Å›ðWð
component of the fluid velocity at the boundary.
The sign in the formula is due to the fact that the
normal vector n point outwards, so the negative
value of uðn corresponds to the incoming flow
(positive  for the outflow).
The general principle of conservation (or rather variation!) of the quantity H can be cast into
the following form
dH
=ð E
sources
dt
production
where E stands for the  source terms which describe time-specific production or
sources
destruction of the quantity H in the volume Wð.
The particular character of the source terms depends on the quantity H:
1. Mass of fluid
Then h ºð 1 and H (t) =ð M (t) =ð
òðrð dV
Wð
In this case E ºð 0 since mass cannot be produced or created!
sources
2. Linear momentum
Then h ºð v and P(t) =ð
òðrðv dV
Wð
In this case the source term is the sum of all external forces acting on the fluid contained in Wð
E ºð FS +ð FV =ð ÃdS +ð
sources
òð òðrðf dS
{ð {ð
Å›ðWð Wð
surface volumetric
forces on Å›ðWð forces in Wð
where à denotes the stress vector at the boundary Å›ðWð.
3. Angular momentum
Then h ºð x ´ð v and K(t) =ð ´ð rðv dV
òðx
Wð
In this case, the source term is the sum of all external moments of forces acting on the fluid
contained in Wð
E ºð MS +ð MV =ð x ´ð à dS +ð ´ð rðf dS
sources
òð òðx
{ð {ð
Å›ðWð Wð
surface volumetric
moment on Å›ðWð moment in Wð
4. Energy
Here we mean total energy which is the sum of internal and kinetic energy of the fluid.
1 1
Then h =ð u +ð v ×ð v =ð u +ð v2 and
2 2
1
E(t) =ð
2
òðrð(u +ð v2 )dV
Wð
where u denotes the mass-specific internal energy of the fluid and v is the magnitude of the
fluid velocity.
The source terms include:
·ð work performed per one time unit (power) by surface and volumetric forces
·ð conductive heat transfer through the boundary Å›ðWð
·ð heat production by internal processes and /or by absorbed radiation.
We can write
E(t) =ð PS +ð PV +ð QÅ›ðWð +ð QWð
{ð {ð
1ð2ð3ð
internal heat
conduction of
power of external
sources
heat through Å›ðWð
forces
where the mechanical power terms are
PS =ð à ×ð v dS , PS =ð
òð òðrðf ×ð v dV
Å›ðWð Wð
and the heat terms are
QÅ›ðWð =ð -ð qh ×ðndS , QWð =ð
h
òð òðrðgð dV
Å›ðWð Wð
In the above, the symbol qh denotes the vector of conductive heat flux through the boundary
Å›ðWð (we will see later that it can be expressed by the temperature gradient) and the symbol gð
h
stands for the mass-specific density of internal heat sources in the fluid.
EQUATION OF MASS CONSERVATION
EQUATION OF MASS CONSERVATION
We have already mentioned that for the mass the source terms are absent. Thus, we have
dM dM dM
=ð -ð =ð 0
dt dt dt
production flow through Å›ðWð
or, equivalently
d
rðv ×ð ndS) =ð 0
dt
òðrð dV -ð (-ð òð
Wð Å›ðWð
Since the volume Wð is fixed we can change order of the volume integration and time
differentiation. We can also apply the GGO Theorem to the surface integral to transform it to
the volume one. This is what we get
Å›ð
Å›ðt
òð[ rð +ð Ńð ×ð(rðv)]dV =ð 0
Wð
Finally, since the volume Wð can be chosen as arbitrary part of the whole flow domain then 
assuming sufficient regularity of the integrated expression  we conclude that
Å›ð
rð +ð Ńð×ð(rðv) =ð 0
Å›ðt
at each point of the fluid domain. We have derived the differential equation of mass
conservation!
The obtained form of this equations is called conservative (sic!). However, other equivalent
forms can be obtained by using standard manipulations with differential operators:
D
Å›ð Å›ð
0 =ð rð +ð Ńð×ð(rðv) =ð rð +ð v×ðŃðrð +ð rðŃð×ð v =ð rð +ð rðŃð×ðv
Å›ðt Å›ðt
1ð4ð2ð4ð4ð
4ð 3ð
Dt
D
rð
Dt
In the index notation
D
Å›ð Å›ð Å›ð Å›ð Å›ð Å›ð
0 =ð rð +ð (rðuð ) =ð rð +ðuð rð +ð rð uð =ð rð +ð rð uð
Å›ðx j j Å›ðx Å›ðx j Å›ðx j
Å›ðt Å›ðt
j j j j
Dt
1ð4ð 3ð
4ð2ð4ð4ð
D
rð
Dt
Note that:
1. If the flow is stationary, i.e. none of the parameters is explicitly time-dependent, then the
equation of mass conservation reduces to the form
Ńð×ð(rðv) =ð v ×ðŃðrð +ð rðŃð×ðv =ð 0
2. If rð ºð const then the mass conservation equation reduces to the particularly simple form
(the continuity equation)
Ńð×ð v =ð 0
In words: the divergence of the velocity field of the constant-density fluid (liquid)
vanishes identically in the whole flow domain. Note that this condition is the geometric
constrain imposed on the class of admissible vector fields rather than evolutionary equation.