Conservation Laws
1
Conservation Laws Conservation laws describe the conservation of certain quantities
and are based on fundamental physical laws. When taken together, they establish relations between the various
parameters / quantities of the system Velocity, Pressure, Density,
The commonest conserved quantities are Mass Momentum, energy
Forms Differential: applicable at a point Integral: applicable to an extended region
VA
Mass of object:
Momentum of object: β«β«β«=V
dVuP~~
Ο
β«β«β«=V
dVm Ο
2
n
dA
( )A t( )V t
The general form of all conservation laws that we will use is:
Rate of change of F = effects of volume sources + effects of surface sourcesM MV V A
D F dV D dV C n dADt
= + β β«β«β« β«β«β« β«β«
Quantity (F) Volume sources (D) Surface sources (C)Momentum1 Gravity Stresses
(pressure/viscous)Heat Dissipation Diffusion/radiationSalt (scalar) None DiffusionAlgae (e.g.) Growth Diffusion
1 Note: momentum is a vector quantity
The Law of Conservation of Mass Within some problem domain defined by a control volume,
the net mass of fluid passing from outside to inside through the control surfaces equals the net increase of mass in the control volume.
In its most general form: the law stat thatβ mass is neither created not destroyed in a closed
systemβ
4
Conservation of Mass
( )~0
M MV V
D dV u dVDt t
βΟ Ο = = +ββ Ο β β«β«β« β«β«β«
Mass is conserved (non-relativistic fluid mechanics)
For any arbitrary material volume
Since integral is zero for any volume, the integrand must be zero
( )~0u
tβΟ
β +ββ Ο =β
Process: We have taken an integral conservation law and used it to produce a differential balance for mass at any point
( )
0
~
=ββ
+ββ
+ββ
β΄
ββ
+ββ
=ββ
β‘β β
ii
i
i
ii
i
ii
i
xu
xu
t
xu
xuu
xu
ΟΟΟ
ΟΟΟΟHowever,
andi
i
D uDt t xΟ βΟ βΟ= +β β
~
1 D uDtΟ
β β = ββ Ο
Thus if the density of fluid particles changes, the velocity field must be divergent. Conversely, if fluid densities remain constant,
~0uββ =
Let where is an intensive property (amount/mass)F f f= Ο
dVxfuu
xf
tf
tf
dVfux
ft
dVfDtD
M
MM
V iii
i
Vi
iV
β«
β«β«
ββ
+ββ
+ββ
+ββ
=
ββ
+ββ
=β΄
ΟΟΟΟ
ΟΟΟ
)(
)()(
dVDtDf
dVxfu
tfu
xt
M
M
V
V iii
i
β«
β«
=
ββ
+ββ
β=ββ
+ββ
Ο
ΟΟΟ 0)(But
dVDtDfdVf
DtD
MM VVβ«β« =β΄ ΟΟ
Any other fluid property (scalar, vector,.. also drop triple integral)
Why is this important/useful?
Because Newtonβs 2nd law:
dVDt
uDdVu
DtD
MM VVβ«β« = ~
~ΟΟ
dVFdVuDtD
MM VVβ«β« =
~~Ο
Dt
uDF ~~
Ο=β΄
But from above:
Rate of Change of Momentum = Net Applied Force
Net Applied Force = Mass AccelerationΓIndependent of volume type!
Some Observations
1. Incompressible~
1 uDtD
β β=βΟ
Ο
01=β
DtDΟ
Ο
[ No volumetric dilatation, fluid particle density conserved]
0~
=ββ
=β ββ΄i
i
xuu
Differential form of βContinuityβ
2. Slightly Compressible
β’ Typically found in stratified conditions where
0( , , , ) ( ) '( , , , )x y z t z x y z tΟ = Ο = Ο + Ο +Ο
β’ Boussinesq Approximation- Vertical scale of mean motion << scale height- or
0
' 1Ο +Ο<<
Ο
Allows us to treat fluid as if it were slightly incompressible
Note: Sound and shock waves are not included !
Reference density (1000 kg/m3 for water)
Background variation (typ. 1-10 kg/m3 for water)
Perturbation density due to motion (typ. 0.1-10 kg/m3 for water)
Informal βProofβ
2dPc dP c dd
= β = ΟΟ
If a fluid is slightly compressible then a small disturbancecaused by a change in pressure, , will cause a change indensity . This disturbance will propagate at celerity, c.Οd
dP
β’ If pressure in fluid is βhydrostaticβ
2dP d ggdz dz c
Ο Ο= βΟ β΄ = β
Now
and
d d dzdt dz dtΟ Ο=
wdtdz
β [ Streamline curvature small]
2cgw
dtd ΟΟ β
=β΄
21
cgw
dtd
=βΟ
Ο
Typically: g β10 m/s2 ; c β 1500 m/s ; w ~ 0.1m/s
0~β β ββ΄ u
The Law of the Conservation of Momentum Momentum is one description of the state of motion of a
system of masses. The physical system responsible for the transformation of the
state of motion of a mass system from the initial state to the final state is effected by a system of FORCES acting over TIME
πΉπΉ. π‘π‘ = ππππ π‘π‘= ππ πππ‘π‘= ππππ
πΉπΉ πππ‘π‘ = ππ πππππΉπΉ = ππ(ππππ)
ππππ
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Stress Field A fluid subjected to two types of forces
Surface forces, and Body forces Interfacial forces
1. Surface forces Forces that are acting on the surfaces or boundaries of a fluid
element/ control volume through direct contact, A force per unit surface area is STRESS. The concept of stress provides a convenient means to describe the
manner in which forces are acting on the boundaries of the medium are transmitted through the medium.
Stress developed depends position of the molecules on the surface the average relative motion of the molecules
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2. Body forces Body forces are all forces that are
developed with out physical contact with the fluid element, and are distributed through out the element.
e.g., gravitational forces The magnitude and direction of the gravity
force is given by the product of the mass of the fluid element times the local acceleration due to gravity.
Two types of stresses Pressure (P) Viscous stress
When a fluid has no motion, the only stress is the first kind. This stress component is normal to the surface.
When the fluid moves, there will be a viscous stress.
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3. Interfacial forces: - act at fluid interfaces, esp. phase discontinuities (air/water)- do not appear directly in equations of motion (appear as
boundary conditions only) - e.g. surface tension β surfactants important- very important for multiphase flows (bubbles, droplets,. free
surfaces!)
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Stresses at a point Is specified by 9 components
Where πππ₯π₯π₯π₯,πππ¦π¦π¦π¦,πππ§π§π§π§ are called normal stressesthe rest is called shear stresses
πππ₯π₯π₯π₯ πππ₯π₯π¦π¦ πππ₯π₯π§π§πππ¦π¦π₯π₯ πππ¦π¦π¦π¦ πππ¦π¦π§π§πππ§π§π₯π₯ πππ§π§π¦π¦ πππ§π§π§π§
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Newton's 2nd law for the object is given by:
bodysurfaceV
FFdVudtdtP
dtdF βββ«β«β«β +===
~~~~~)( Ο
Surface forcese.g. Friction
Body forcese.g. Gravity
Example: Block of mass m pushed with force Falong surface with friction coefficient b:
=β=
+β=β++β=
β++β=
+=
β«β«β«β«β«β«
βββ«β«β«
0
3
~
3
~
1
~
1
~
~
3
~
3
~
1
~
1
~~
~~~
mgNdtdwm
Fbudtdum
amgaNaFabudt
udm
dVagaNaFabudVudtd
FFdVudtd
Body
VSurface
V
bodysurfaceV
ΟΟ
ΟF
m
N
g
u
Fixed volumeβVF : Flow of fluid through system boundary (control surface) is non zero, but velocity of boundary is zero. For this case we get
Material VolumeβVM : Consists of same fluid particles and thus the bounding surface moves with the fluid velocity. Thus, the second term from the Leibnitz rule is now non-zero, so
Using Gauss' theorem:
~( , )
F FV V
d FF x t dV dVdt t
β=
ββ«β«β« β«β«β«
~ ~ ~( , )
M MV V A
D FF x t dV dV F u n dADt t
β= + β
ββ«β«β« β«β«β« β«β«
( )~ ~ ~MA V
F u n dA F u dVβ = β β β«β« β«β«β«
( )~ ~( , )
M MV V
D FF x t dV F u dVDt t
β = +β β β β«β«β« β«β«β«This is Reynolds transport theorem, where D/Dt is the same as d/dtbut implies a material volume.
~~
Note that the Reynolds transport theorem is often written in themore general form which does not assume that the control volumeis bounded by a material surface. Instead, the control volume isassumed to move at some velocity and that of the fluid is defined as relative to the control volume, such that
In this case, V is not necessarily a material surface. If ur=0, thenub=u and we revert to the form on the previous page.
β«β«β«β«β«β«β«β«β«β« β +β +ββ
=A
rA
bVV
dAnuFdAnuFdVtFdVtxF
DtD
~~~~~),(
bu~
bruuu~~~
β=
~
Methods of Analysis Fluid flow problems can be analyzed in one of the three
approaches: Control volume of integral approach Infinitesimal or differential approach Experimental approach
In fluid dynamics system A quantity of matter or a finite region in space is choosen for
study Control Volume / open system Control mass /closed system/ system
21
Control volume or integral approach A control volume (CV) is an arbitrary finite volume of an
arbitrary shape that is chosen from a fluid region for analysis. The boundaries of a control volume are referred as control surfaces (CS).
A control volume is an open system mass and energy enter and leave the control volume through
the control surfaces. A control volume is fixed in space
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Control Mass / Closed System/ System A quantity of matter of fixed identity is chosen for a study
i.e., fixed mass of fluidIn a closed system, mass is not allowed to enter or leave.
Unlike a control volume, a control mass moves with the fluid since we are dealing with the same fluid elements (mass)
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Infinitesimal or differential formulation Control volume / Integral formulations are useful when we are
interested in the gross behavior of a flow field and its effect on devices.
It does not provide a detailed (i.e., point by point) knowledge of the flow field.
To obtain this detailed knowledge, we must formulate the conservation equations in differential forms.
The analysis will be in terms of infinitesimal system. Recall that the fluid properties are continuous in both spatial
coordinates and time,ππ π₯π₯,π¦π¦, π§π§, π‘π‘
ππ = ππ π₯π₯,π¦π¦, π§π§, π‘π‘ , π£π£ π₯π₯,π¦π¦, π§π§, π‘π‘ ,π€π€ π₯π₯,π¦π¦, π§π§, π‘π‘
24
Experimental Approach Analytical solutions exists for simple cases Simple geometry Simple initial and boundary conditions
A need for experimental and laboratory based approaches Full and model scales Dimensional Analysis and similarity
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Conservation of Momentum β Navier-Stokes
We have:
~ ~
~ ~
Du uF u u
Dt t
β β = Ο = Ο + β β β
Two kinds of forces:β’ Body forces β’ Surface forces
Two kinds of acceleration:β’ Unsteady β’ Advective (convective/nonlinear)
Two kinds of surface forces: β’ Those due to pressureβ’ Those due to viscous stresses Divergence of Stress Tensor
Plan for derivation of the Navier Stokes equation
1. Determine fluid accelerations from velocities etc. 2. Decide on forces 3. Determine how surface forces work : stress tensor4. Split stress tensor into pressure part and viscous part5. Convert surface forces to volume effect (Gauss' theorem)6. Use integral theorem to get pointwise variable p.d.e.7. Use constitutive relation to connect viscous stress tensor to strain
rate tensor8. Compute divergence of viscous stress tensor (incompressible fluid)9. Result = Incompressible Navier Stokes equation
Stress at a point (From K&C β remember difference in nomenclature,i.e. Οij β Οij)
What is the force vector I need to apply at a face defined by theunit normal vector to equal that of the internal stresses?
~n
Consider a small (differential) 2-D element
1ΞΈ
2ΞΈ
11Ο
12Ο
11Ο12Ο
21Ο
21Ο
22Ο
22Ο
11Ο
12Ο
21Ο22Ο
n
1n
2n
1dx
2dx ds
dF
cut away
1 11 2 21 1= = +F dx dxΟ Οforce component in x1 direction
1 2 11 11 21
11 1 21 2
11 1 21 2 ~
cos cos
β΄ = = Ο +Ο
=Ο ΞΈ +Ο ΞΈ=Ο +Ο
dF dx dxfds ds ds
n n n[ has magnitude of 1]
Defining the stress tensor to be ijΟ
11 12 13
21 22 23
31 32 33
Ο Ο Ο Ο β‘ Ο Ο Ο Ο Ο Ο
And in general
jjjj nfandnf 2211 ΟΟ ==β΄
jjii nf Ο=
d
But [see Kundu p90]ijji ΟΟ =
or
or (3D)
= Ο =
= Ο β =
= Ο β = β = Ο β β«β«
ii ij j
totalCS
dFf nds
dFf nds
dFf n F n dAdA
β Surface force per unit areaβ(note this is a 2D area)
Total, or net, force due to surface stresses
Conservation of momentum
2),( 1
1
112111
dxx
xxββ
+ΟΟ
2),( 1
1
112111
dxx
xxββ
βΟΟ
2),( 2
2
212121
dxx
xxββ
βΟΟ
2),( 2
2
212121
dxx
xxββ
+ΟΟ
1x
2x
3x
2),( 3
3
312131
dxx
xxββ
βΟΟ
2),( 3
3
312131
dxx
xxββ
+ΟΟ
Dimensions:dx1 . dx2 . dx3
Sum of surface forces in x1 direction:
11 1 11 111 11 2 3
1 1
21 2 21 221 21 1 3
2 2
31 3 31 331 31 1 2
3 3
3111 211 2 3
1 2 3
2 2
2 2
2 2
ji
j
dx dx dx dxx x
dx dx dx dxx x
dx dx dx dxx x
dx dx dxx x x
dVx
βΟ βΟ= Ο + βΟ + β β
βΟ βΟ+ Ο + βΟ + β β βΟ βΟ
+ Ο + βΟ + β β βΟβΟ βΟ
= + + β β β βΟ
=β
Defining i component of surface force per unit volume to be i
VFβ΄
~For body forces we use gravity = ig gΟ = Ο
~gF
DtDu
Vi ΟΟ +=β΄
In general : β
= Ο = ββ Οβ
iV ij
j
Fx
j
iji
i
xg
DtDu
β
β+=
ΟΟΟ βCauchyβs equation
of motionβ
Force = divergence of stress tensor
3
Note that usually -g g e=
Important Note: This can also be derived from the IntegralFrom of Newtonβs 2nd Law for a MaterialVolume VM
β«β«β« +=A jijV iV i dAdVgdVu
DtD
MM
ΟΟΟ
β«β« =MM V
iV i dV
DtDudVu
DtD ΟΟBut
and [Gauss' Theorem]M
ijij jA V
j
dA dVx
βΟΟ =
ββ« β«
0M
ijiiV
j
Du g dVDt x
βΟβ΄ Ο βΟ β = β β«
ijii
j
Du gDt x
βΟβ΄ Ο = Ο +
β
Constitutive relation for a Newtonian fluidβEquation that linearly relates the stress to the rate of
strain in a Newtonian Fluid Mediumβ(i) Static Fluid: - By definition cannot support a shear stress
- still feels thermodynamic pressure (in compression)
ij ijpβ΄ Ο = β Ξ΄
(ii) Moving Fluid: - develops additional components of stress (due to viscosity)
ij ij ijpβ΄ Ο = β Ξ΄ + Ο Hypothesis
Note difference from Kundu !Deviatoric stress tensor [Viscous stress tensor]=ijΟ
If medium is isotropic and stress tensor is symmetric
only 2 non-zero elements of
ij ij mm ij2 e eΟ = Β΅ + Ξ» Ξ΄
Assume mnijmnij eK=Ο
ijmnK = 4th order tensor (81 components!) that depend on thermodynamic state of medium
β K
which gives
or
ijijij eup Β΅Ξ΄Β΅Ο 2)Λ32( +β β+β=
See derivation of Ξ» in Kundu, p 100
Special cases
(i) Incompressible 0Λ =β ββ u
ijijij ep Β΅Ξ΄Ο 2+β=β΄
(ii) Static 0=β ijeijij pΞ΄Ο β=β΄
In summary
Cauchy's equation
Constitutive relation fora compressible, Newtonian fluid.
ijijij eupii Β΅Ξ΄Β΅Ο 2)Λ32()( +β β+β=
( ) ijii
j
Dui gDt x
βΟΟ = Ο +
β
Navier-Stokes equation
2
2
ii ij ij
j
i iji j
Du g p eDt x
pg ex x
β Ο = Ο + β Ξ΄ + Β΅ β
β β =Ο β + Β΅ β β
The general form of the Navier-Stokes equation is given by substitutionof the constitutive equation for a Newtonian fluid into the Cauchy equation of motion:
+
+β
ββ
+= ijijkki
ii eep
xg
DtDu ¡δ¡ΟΟ 2
32
Incompressible form (ekk=0):
22
2
2
122
ijii
i j
jii
i j j i
jii
i j j i j
i ii
eDu p gDt x x
uup gx x x x
uup gx x x x xp g ux
ββΟ = β +Ο + Β΅
β β
βββ β= β +Ο + Β΅ + β β β β
βββ= β +Ο +Β΅ +Β΅
β β β β β
β= β +Ο +Β΅β
β
),,,( tzyxfβ Β΅Assuming
23
2
22
2
21
222
xu
xu
xu
xxuu iii
jj
ii β
β+
ββ
+ββ
=ββ
β=βwhere
~
2
~
~ ugpDt
uDβ++ββ= Β΅ΟΟ
If βInviscidβ 0βΒ΅
~
~ gpDt
uDΟΟ +ββ= Euler Equation
Or in vector notation
Inertia Pressure gradient
Gravity (buoyancy)
Divergence of viscous stress (friction)
Equation of Motion Viscid Flow (Navier-Stokes Equation)ππππππππ
+ ππ π’π’πππππ₯π₯
+ ππ π£π£πππππ¦π¦
+ ππ π€π€πππππ§π§
= 0 β¦β¦β¦.. Continuity eqaution
ππ πππ’π’ππππ
+ ππ πππ’π’πππ₯π₯
+ π£π£ πππ’π’πππ¦π¦
+ π€π€ πππ’π’πππ§π§
= πππππππ₯π₯
+ ππ ππ2π’π’πππ₯π₯2
+ ππ2π’π’πππ¦π¦2
+ ππ2π’π’πππ§π§2
ππ πππ£π£ππππ
+ ππ πππ£π£πππ₯π₯
+ π£π£ πππ£π£πππ¦π¦
+ π€π€ πππ£π£πππ§π§
= πππππππ₯π₯
+ ππ ππ2π£π£πππ₯π₯2
+ ππ2π£π£πππ¦π¦2
+ ππ2π£π£πππ§π§2
ππ πππ€π€ππππ
+ ππ πππ€π€πππ₯π₯
+ π£π£ πππ€π€πππ¦π¦
+ π€π€ πππ€π€πππ§π§
= πππππππ₯π₯
+ ππ ππ2π€π€πππ₯π₯2
+ ππ2π€π€πππ¦π¦2
+ ππ2π€π€πππ§π§2
β ΟππIndependent variable: x, y, z, tDependent variables: u, v, wβ¦β¦ velocity in x, y, zΟ, Pβ¦ density, pressureΒ΅β¦. viscosity
42
Equation of Motion inviscid Flow (Euler Equation)ππππππππ
+ ππ π’π’πππππ₯π₯
+ ππ π£π£πππππ¦π¦
+ ππ π€π€πππππ§π§
= 0
ππ πππ’π’ππππ
+ ππ πππ’π’πππ₯π₯
+ π£π£ πππ’π’πππ¦π¦
+ π€π€ πππ’π’πππ§π§
= πππππππ₯π₯
+ ππ ππ2π’π’πππ₯π₯2
+ ππ2π’π’πππ¦π¦2
+ ππ2π’π’πππ§π§2
ππ πππ£π£ππππ
+ ππ πππ£π£πππ₯π₯
+ π£π£ πππ£π£πππ¦π¦
+ π€π€ πππ£π£πππ§π§
= πππππππ₯π₯
+ ππ ππ2π£π£πππ₯π₯2
+ ππ2π£π£πππ¦π¦2
+ ππ2π£π£πππ§π§2
ππ πππ€π€ππππ
+ ππ πππ€π€πππ₯π₯
+ π£π£ πππ€π€πππ¦π¦
+ π€π€ πππ€π€πππ§π§
= πππππππ₯π₯
+ ππ ππ2π€π€πππ₯π₯2
+ ππ2π€π€πππ¦π¦2
+ ππ2π€π€πππ§π§2
β ΟππIndependent variable: x, y, z, tDependent variables: u, v, wβ¦β¦ velocity in x, y, zΟ, Pβ¦ density, pressureΒ΅β¦. viscosity
= 0
= 0= 0
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Types of fluid Flow
1. Real and Ideal Flow:
Friction = 0Ideal Flow ( ΞΌ =0)Energy loss =0
Friction = oReal Flow ( ΞΌ β 0)Energy loss = 0
Ideal Real
If the fluid is considered frictionless with zero viscosity it is called ideal.In real fluids the viscosity is considered and shear stresses occur causing conversion of mechanical energy into thermal energy
44
2. Steady and Unsteady Flow
H=constant
V=constant
Steady Flow with respect to timeβ’Velocity is constant at certain position w.r.t. time
Unsteady Flow with respect to timeβ’Velocity changes at certain position w.r.t. time
H β constant
V β constant
Steady flow occurs when conditions of a point in a flow field donβt change with respect to time ( v, p, Hβ¦..changes w.r.t. time
( )( )( )( ) 0
0
β ββ
=ββ
t
tsteady
unsteady
45
Uniform Flow means that the velocity is constant at certain time in different positions(doesnβt depend on any dimension x or y or z)
3. Uniform and Non uniform Flow
Non- uniform Flow means velocity changes at certain time in different positions ( depends on dimension x or y or z)
YY
x x
( )( )( )( ) 0
0
β ββ
=ββ
x
xuniform
Non-uniform46
4. Flow dimensionality Generally, flow is 3-dimensional Fluid properties vary in three directions The most complete description is given by three dimensional analysis
Under some conditions, flow field can be reduced to 2-d or 1-d The relative variation of the fluid properties with the directions may
vary
Flow dimensionality; (a) 1-D flow between horizontal plates, (b) 2-D flow in a 3-Dbox, (c) 3-D flow in a 3-D box. (source: Lecture in Elementary Fluid Mechanics J. M. McDonough)
47
4. Flow Dimensionality (cont.)β’ A flow field is best characterized by its velocity distribution.
β’ A flow is said to be one-, two-, or three-dimensional if the flow velocity varies in one, two, or three dimensions, respectively.
β’ However, the variation of velocity in certain directions can be small relative to the variation in other directions and can be ignored.
The development of the velocity profile in a circular pipe. V =V(r, z) and thus the flow is two-dimensional in the entrance region, and becomes one-dimensional downstream whenthe velocity profile fully develops and remains unchanged in the flow direction, V =V(r).
48
5. Laminar and Turbulent Flow:
In Laminar Flow:β’Fluid flows in separate layersβ’No mass mixing between fluid layersβ’Friction mainly between fluid layersβ’Reynoldsβ Number (RN ) < 2000β’Vmax.= 2Vmean
In Turbulent Flow:β’No separate layersβ’Continuous mass mixing β’Friction mainly between fluid and pipe wallsβ’Reynoldsβ Number (RN ) > 4000β’Vmax.= 1.2 Vmean
VmaxVmean
VmaxVmean
49
5. Laminar and Turbulent Flow (cont.):
50
6.Rotational and irrotational flowsrβ₯
A rotational flow is one in which fluid elements moving in the flow field will undergo rotation. The rotation is given by the angular velocity of any two mutually perpendicular line elements of the element Mathematically, this is given by curl u β 0.
51
Irrotational flow
rotational flow
52
Types of motion or deformation of fluid element
Linear translation
Rotational translation
Linear deformation
angular deformation
53
Plane flowA flow is said to be plane or two-dimensional if it is everywhere orthogonal to one direction and independent of translations along such direction.In a plane flow it is therefore possible to choose a system of Cartesian coordinates (x1, x2, x3) so that u has the form u = (u1, u2, 0), and u1 and u2 do not depend on x3.
Axisymmetric flowA flow is said to be axisymmetric if, chosen a proper system of cylindrical coordinates (z, r , Ο) the velocity u = (uz , ur , u') is independent of the azimuthal coordinate Ο, and u' = 0.
54