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: ∫∫∫=V

dVuP~~

ρ

∫∫∫=V

dVm ρ

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

= + ⋅∫∫∫ ∫∫∫ ∫∫

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 that“ mass is neither created not destroyed in a closed

system”

4

Conservation of Mass

( )~0

M MV V

D dV u dVDt t

∂ρ ρ = = +∇⋅ ρ ∂ ∫∫∫ ∫∫∫

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

t∂ρ

⇒ +∇⋅ ρ =∂

Process: We have taken an integral conservation law and used it to produce a differential balance for mass at any point

( )

0

~

=∂∂

+∂∂

+∂∂

∴

∂∂

+∂∂

=∂∂

≡⋅∇

ii

i

i

ii

i

ii

i

xu

xu

t

xu

xuu

xu

ρρρ

ρρρρHowever,

andi

i

D uDt t xρ ∂ρ ∂ρ= +∂ ∂

~

1 D uDtρ

⇒ − = ∇⋅ρ

Thus if the density of fluid particles changes, the velocity field must be divergent. Conversely, if fluid densities remain constant,

~0u∇⋅ =

Let where is an intensive property (amount/mass)F f f= ρ

dVxfuu

xf

tf

tf

dVfux

ft

dVfDtD

M

MM

V iii

i

Vi

iV

∫

∫∫

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

=∴

ρρρρ

ρρρ

)(

)()(

dVDtDf

dVxfu

tfu

xt

M

M

V

V iii

i

∫

∫

=

∂∂

+∂∂

⇒=∂∂

+∂∂

ρ

ρρρ 0)(But

dVDtDfdVf

DtD

MM VV∫∫ =∴ ρρ

Any other fluid property (scalar, vector,.. also drop triple integral)

Why is this important/useful?

Because Newton’s 2nd law:

dVDt

uDdVu

DtD

MM VV∫∫ = ~

~ρρ

dVFdVuDtD

MM VV∫∫ =

~~ρ

Dt

uDF ~~

ρ=∴

But from above:

Rate of Change of Momentum = Net Applied Force

Net Applied Force = Mass Acceleration×Independent of volume type!

Some Observations

1. Incompressible~

1 uDtD

⋅∇=−ρ

ρ

01=−

DtDρ

ρ

[ No volumetric dilatation, fluid particle density conserved]

0~

=∂∂

=⋅∇∴i

i

xuu

Differential form of “Continuity”

2. Slightly Compressible

• Typically found in stratified conditions where

0( , , , ) ( ) '( , , , )x y z t z x y z tρ = ρ = ρ + ρ +ρ

• Boussinesq Approximation- Vertical scale of mean motion << scale height- or

0

' 1ρ +ρ<<

ρ

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 “Proof”

2dPc dP c dd

= → = ρρ

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.ρd

dP

• If pressure in fluid is “hydrostatic”

2dP d ggdz dz c

ρ ρ= −ρ ∴ = −

Now

and

d d dzdt dz dtρ ρ=

wdtdz

≈ [ Streamline curvature small]

2cgw

dtd ρρ −

=∴

21

cgw

dtd

=−ρ

ρ

Typically: g ≈10 m/s2 ; c ≈ 1500 m/s ; w ~ 0.1m/s

0~≅⋅∇∴ 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

𝐹𝐹. 𝑡𝑡 = 𝑚𝑚𝑚𝑚 𝑡𝑡= 𝑚𝑚 𝑚𝑚𝑡𝑡= 𝑚𝑚𝑚𝑚

𝐹𝐹 𝑑𝑑𝑡𝑡 = 𝑑𝑑 𝑚𝑚𝑚𝑚𝐹𝐹 = 𝑑𝑑(𝑚𝑚𝑚𝑚)

𝑑𝑑𝑑𝑑

13

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 – surfactants important- very important for multiphase flows (bubbles, droplets,. free

surfaces!)

16

Stresses at a point Is specified by 9 components

Where 𝜎𝜎𝑥𝑥𝑥𝑥,𝜎𝜎𝑦𝑦𝑦𝑦,𝜎𝜎𝑧𝑧𝑧𝑧 are called normal stressesthe rest is called shear stresses

𝜎𝜎𝑥𝑥𝑥𝑥 𝜏𝜏𝑥𝑥𝑦𝑦 𝜏𝜏𝑥𝑥𝑧𝑧𝜏𝜏𝑦𝑦𝑥𝑥 𝜎𝜎𝑦𝑦𝑦𝑦 𝜏𝜏𝑦𝑦𝑧𝑧𝜏𝜏𝑧𝑧𝑥𝑥 𝜏𝜏𝑧𝑧𝑦𝑦 𝜎𝜎𝑧𝑧𝑧𝑧

17

Newton's 2nd law for the object is given by:

bodysurfaceV

FFdVudtdtP

dtdF ∑∑∫∫∫∑ +===

~~~~~)( ρ

Surface forcese.g. Friction

Body forcese.g. Gravity

Example: Block of mass m pushed with force Falong surface with friction coefficient b:

=−=

+−=−++−=

−++−=

+=

∫∫∫∫∫∫

∑∑∫∫∫

0

3

~

3

~

1

~

1

~

~

3

~

3

~

1

~

1

~~

~~~

mgNdtdwm

Fbudtdum

amgaNaFabudt

udm

dVagaNaFabudVudtd

FFdVudtd

Body

VSurface

V

bodysurfaceV

ρρ

ρF

m

N

g

u

Fixed volume–VF : Flow of fluid through system boundary (control surface) is non zero, but velocity of boundary is zero. For this case we get

Material Volume–VM : 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

∂=

∂∫∫∫ ∫∫∫

~ ~ ~( , )

M MV V A

D FF x t dV dV F u n dADt t

∂= + ⋅

∂∫∫∫ ∫∫∫ ∫∫

( )~ ~ ~MA V

F u n dA F u dV⋅ = ∇ ⋅∫∫ ∫∫∫

( )~ ~( , )

M MV V

D FF x t dV F u dVDt t

∂ = +∇ ⋅ ∂ ∫∫∫ ∫∫∫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.

∫∫∫∫∫∫∫∫∫∫ ⋅+⋅+∂∂

=A

rA

bVV

dAnuFdAnuFdVtFdVtxF

DtD

~~~~~),(

bu~

bruuu~~~

−=

~

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

22

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)

23

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,𝜌𝜌 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡

𝒖𝒖 = 𝑚𝑚 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡 , 𝑣𝑣 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡 ,𝑤𝑤 𝑥𝑥,𝑦𝑦, 𝑧𝑧, 𝑡𝑡

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

25

Conservation of Momentum – Navier-Stokes

We have:

~ ~

~ ~

Du uF u u

Dt t

∂ ∑ = ρ = ρ + ⋅∇ ∂

Two kinds of forces:• Body forces • Surface forces

Two kinds of acceleration:• Unsteady • Advective (convective/nonlinear)

Two kinds of surface forces: • Those due to pressure• 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 – remember difference in nomenclature,i.e. τij ← σij)

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

1θ

2θ

11σ

12σ

11σ12σ

21σ

21σ

22σ

22σ

11σ

12σ

21σ22σ

n

1n

2n

1dx

2dx ds

dF

cut away

1 11 2 21 1= = +F dx dxσ σforce component in x1 direction

1 2 11 11 21

11 1 21 2

11 1 21 2 ~

cos cos

∴ = = σ +σ

=σ θ +σ θ=σ +σ

dF dx dxfds ds ds

n n n[ has magnitude of 1]

Defining the stress tensor to be ijσ

11 12 13

21 22 23

31 32 33

σ σ σ σ ≡ σ σ σ σ σ σ

And in general

jjjj nfandnf 2211 σσ ==∴

jjii nf σ=

d

But [see Kundu p90]ijji σσ =

or

or (3D)

= σ =

= σ ⋅ =

= σ ⋅ = ⇒ = σ ⋅∫∫

ii ij j

totalCS

dFf nds

dFf nds

dFf n F n dAdA

“ Surface force per unit area”(note this is a 2D area)

Total, or net, force due to surface stresses

Conservation of momentum

2),( 1

1

112111

dxx

xx∂∂

+σσ

2),( 1

1

112111

dxx

xx∂∂

−σσ

2),( 2

2

212121

dxx

xx∂∂

−σσ

2),( 2

2

212121

dxx

xx∂∂

+σσ

1x

2x

3x

2),( 3

3

312131

dxx

xx∂∂

−σσ

2),( 3

3

312131

dxx

xx∂∂

+σσ

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

∂σ ∂σ= σ + −σ + ∂ ∂

∂σ ∂σ+ σ + −σ + ∂ ∂ ∂σ ∂σ

+ σ + −σ + ∂ ∂ ∂σ∂σ ∂σ

= + + ∂ ∂ ∂ ∂σ

=∂

Defining i component of surface force per unit volume to be i

VF∴

~For body forces we use gravity = ig gρ = ρ

~gF

DtDu

Vi ρρ +=∴

In general : ∂

= σ = ∇⋅σ∂

iV ij

j

Fx

j

iji

i

xg

DtDu

∂

∂+=

σρρ “Cauchy’s equation

of motion”

Force = divergence of stress tensor

3

Note that usually -g g e=

Important Note: This can also be derived from the IntegralFrom of Newton’s 2nd Law for a MaterialVolume VM

∫∫∫ +=A jijV iV i dAdVgdVu

DtD

MM

σρρ

∫∫ =MM V

iV i dV

DtDudVu

DtD ρρBut

and [Gauss' Theorem]M

ijij jA V

j

dA dVx

∂σσ =

∂∫ ∫

0M

ijiiV

j

Du g dVDt x

∂σ∴ ρ −ρ − = ∂ ∫

ijii

j

Du gDt x

∂σ∴ ρ = ρ +

∂

Constitutive relation for a Newtonian fluid“Equation that linearly relates the stress to the rate of

strain in a Newtonian Fluid Medium”(i) Static Fluid: - By definition cannot support a shear stress

- still feels thermodynamic pressure (in compression)

ij ijp∴ σ = − δ

(ii) Moving Fluid: - develops additional components of stress (due to viscosity)

ij ij ijp∴ σ = − δ + τ Hypothesis

Note difference from Kundu !Deviatoric stress tensor [Viscous stress tensor]=ijτ

If medium is isotropic and stress tensor is symmetric

only 2 non-zero elements of

ij ij mm ij2 e eτ = µ + λ δ

Assume mnijmnij eK=τ

ijmnK = 4th order tensor (81 components!) that depend on thermodynamic state of medium

⇒ K

which gives

or

ijijij eup µδµσ 2)ˆ32( +⋅∇+−=

See derivation of λ in Kundu, p 100

Special cases

(i) Incompressible 0ˆ =⋅∇→ u

ijijij ep µδσ 2+−=∴

(ii) Static 0=→ ijeijij pδσ −=∴

In summary

Cauchy's equation

Constitutive relation fora compressible, Newtonian fluid.

ijijij eupii µδµσ 2)ˆ32()( +⋅∇+−=

( ) ijii

j

Dui gDt x

∂σρ = ρ +

∂

Navier-Stokes equation

2

2

ii ij ij

j

i iji j

Du g p eDt x

pg ex x

∂ ρ = ρ + − δ + µ ∂

∂ ∂ =ρ − + µ ∂ ∂

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:

+

+−

∂∂

+= ijijkki

ii eep

xg

DtDu µδµρρ 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

∂∂ρ = − +ρ + µ

∂ ∂

∂∂∂ ∂= − +ρ + µ + ∂ ∂ ∂ ∂

∂∂∂= − +ρ +µ +µ

∂ ∂ ∂ ∂ ∂

∂= − +ρ +µ∇

∂

),,,( tzyxf≠µAssuming

23

2

22

2

21

222

xu

xu

xu

xxuu iii

jj

ii ∂

∂+

∂∂

+∂∂

=∂∂

∂=∇where

~

2

~

~ ugpDt

uD∇++−∇= µρρ

If “Inviscid” 0≈µ

~

~ gpDt

uDρρ +−∇= Euler Equation

Or in vector notation

Inertia Pressure gradient

Gravity (buoyancy)

Divergence of viscous stress (friction)

Equation of Motion Viscid Flow (Navier-Stokes Equation)𝜕𝜕𝜌𝜌𝜕𝜕𝑑𝑑

+ 𝜕𝜕 𝑢𝑢𝜌𝜌𝜕𝜕𝑥𝑥

+ 𝜕𝜕 𝑣𝑣𝜌𝜌𝜕𝜕𝑦𝑦

+ 𝜕𝜕 𝑤𝑤𝜌𝜌𝜕𝜕𝑧𝑧

= 0 ……….. Continuity eqaution

𝜌𝜌 𝜕𝜕𝑢𝑢𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑢𝑢𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑢𝑢𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑢𝑢𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑢𝑢𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑢𝑢𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑢𝑢𝜕𝜕𝑧𝑧2

𝜌𝜌 𝜕𝜕𝑣𝑣𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑣𝑣𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑣𝑣𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑣𝑣𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑣𝑣𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑣𝑣𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑣𝑣𝜕𝜕𝑧𝑧2

𝜌𝜌 𝜕𝜕𝑤𝑤𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑤𝑤𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑤𝑤𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑤𝑤𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑤𝑤𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑤𝑤𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑤𝑤𝜕𝜕𝑧𝑧2

− ρ𝑔𝑔Independent variable: x, y, z, tDependent variables: u, v, w…… velocity in x, y, zρ, P… density, pressureµ…. viscosity

42

Equation of Motion inviscid Flow (Euler Equation)𝜕𝜕𝜌𝜌𝜕𝜕𝑑𝑑

+ 𝜕𝜕 𝑢𝑢𝜌𝜌𝜕𝜕𝑥𝑥

+ 𝜕𝜕 𝑣𝑣𝜌𝜌𝜕𝜕𝑦𝑦

+ 𝜕𝜕 𝑤𝑤𝜌𝜌𝜕𝜕𝑧𝑧

= 0

𝜌𝜌 𝜕𝜕𝑢𝑢𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑢𝑢𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑢𝑢𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑢𝑢𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑢𝑢𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑢𝑢𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑢𝑢𝜕𝜕𝑧𝑧2

𝜌𝜌 𝜕𝜕𝑣𝑣𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑣𝑣𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑣𝑣𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑣𝑣𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑣𝑣𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑣𝑣𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑣𝑣𝜕𝜕𝑧𝑧2

𝜌𝜌 𝜕𝜕𝑤𝑤𝜕𝜕𝑑𝑑

+ 𝑚𝑚 𝜕𝜕𝑤𝑤𝜕𝜕𝑥𝑥

+ 𝑣𝑣 𝜕𝜕𝑤𝑤𝜕𝜕𝑦𝑦

+ 𝑤𝑤 𝜕𝜕𝑤𝑤𝜕𝜕𝑧𝑧

= 𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

+ 𝜇𝜇 𝜕𝜕2𝑤𝑤𝜕𝜕𝑥𝑥2

+ 𝜕𝜕2𝑤𝑤𝜕𝜕𝑦𝑦2

+ 𝜕𝜕2𝑤𝑤𝜕𝜕𝑧𝑧2

− ρ𝑔𝑔Independent variable: x, y, z, tDependent variables: u, v, w…… velocity in x, y, zρ, P… density, pressureµ…. viscosity

= 0

= 0= 0

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Types of fluid Flow

1. Real and Ideal Flow:

Friction = 0Ideal Flow ( μ =0)Energy loss =0

Friction = oReal Flow ( μ ≠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

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2. Steady and Unsteady Flow

H=constant

V=constant

Steady Flow with respect to time•Velocity is constant at certain position w.r.t. time

Unsteady Flow with respect to time•Velocity changes at certain position w.r.t. time

H ≠ constant

V ≠ constant

Steady flow occurs when conditions of a point in a flow field don’t change with respect to time ( v, p, H…..changes w.r.t. time

( )( )( )( ) 0

0

≠∂∂

=∂∂

t

tsteady

unsteady

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Uniform Flow means that the velocity is constant at certain time in different positions(doesn’t 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

≠∂∂

=∂∂

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)

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4. Flow Dimensionality (cont.)• A flow field is best characterized by its velocity distribution.

• A flow is said to be one-, two-, or three-dimensional if the flow velocity varies in one, two, or three dimensions, respectively.

• 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).

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5. Laminar and Turbulent Flow:

In Laminar Flow:•Fluid flows in separate layers•No mass mixing between fluid layers•Friction mainly between fluid layers•Reynolds’ Number (RN ) < 2000•Vmax.= 2Vmean

In Turbulent Flow:•No separate layers•Continuous mass mixing •Friction mainly between fluid and pipe walls•Reynolds’ Number (RN ) > 4000•Vmax.= 1.2 Vmean

VmaxVmean

VmaxVmean

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5. Laminar and Turbulent Flow (cont.):

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6.Rotational and irrotational flowsr⊥

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 ≠ 0.

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Irrotational flow

rotational flow

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Types of motion or deformation of fluid element

Linear translation

Rotational translation

Linear deformation

angular deformation

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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 , ϕ) the velocity u = (uz , ur , u') is independent of the azimuthal coordinate ϕ, and u' = 0.

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