equations of continuity. outline 1.time derivatives & vector notation 2.differential equations...

Equations of Continuity

Outline

1.Time Derivatives & Vector Notation

2.Differential Equations of Continuity

3.Momentum Transfer Equations

Introduction

FLUID

In order to calculate forces exerted by a moving fluid as well as the consequent transport effects, the dynamics of flow must be described mathematically (kinematics).

Continuousmedium

Infinitesimal pieces of fluid

Eulerian Perspective – the flow as seen at fixed locations in space, or over fixed volumes of space (the perspective of most analysis)

Lagrangian Perspective – the flow as seen by the fluid material (the perspective of the laws of motion)

Control volume: finite fixed region of space (Eulerian)

Coordinate: fixed point in space (Eulerian)

Fluid system: finite piece of the fluid material (Lagrangian)

Fluid particle: differentially small finite piece of the fluid material (Lagrangian)

Perspectives of Fluid Motion

Lagrangian Perspective

The motion of a fluid particle is relative to a specific initial position in space at an initial time.

1 , 2 , 3 ,

position:

( , , ) ( , , ) ( , , ) p p p p p p p p px f x y z t y f x y z t z f x y z t


x

y

zLagrangian coordinate system

pathline

position vector

p p px y zr i j k

1 ,

2 ,

3 ,

( , , )velocity:

( , , )

( , , )

p p px

p p py

p p pz

f x y z tv

tf x y z t

vt

f x y z tv

t

partial (local) time derivatives


x

y

z

t = t1 t = t2y

xz

1 1 1, 1

2 1 1, 1

3 1 1, 1

position 1:

( , , )

( , , )

( , , )

p p p

p p p

p p p

x f x y z t

y f x y z t

z f x y z t

1 2 2, 2

2 2 2, 2

3 2 2, 2

position 2:

( , , )

( , , )

( , , )

p p p

p p p

p p p

x f x y z t

y f x y z t

z f x y z t

Consider a small fluid element with a mass concentration moving through Cartesian space:


x

y

z

t = t1 t = t2y

xz

Consider a small fluid element with a mass concentration moving through Cartesian space:

1 1 1, 1

concentration 1:

( , , ) p p px y z t 2 2 2, 2

concentration 2:

( , , ) p p px y z t


Total change in the mass concentration with respect to time:

2 1 2 1 2 1

2 1 2 1 2 1

2 1 2 1

2 1 2 1

t t x x

t t t t t x t t

y y z z

y t t z t t

If the timeframe is infinitesimally small:

2 1 2 1

2 1 2 1

2 1 2 1

2 1 2 1

2 1 2 1

2 1 2 1

lim lim

lim lim

t t t t

t t t t

x x

t t t x t t

y y z z

y t t z t t


Total Time Derivative

d x y z

dt t x t y t z t

x y z

Dv v v

Dt t x y z

local derivative convective derivative

Substantial Time Derivative


x y z

Dv v v

Dt t x y z

D

Dt tv

x y zv v v

x y z

v i j k

i j k

vector notation

stream velocity

gradient


Problem with the Lagrangian Perspective

The concept is pretty straightforward but very difficult to implement (since to describe the whole fluid motion, kinematics must be applied to ALL of the moving particles), often would produce more information than necessary, and is not often applicable to systems defined in fluid mechanics.

D

Dt tv

Eulerian Perspective

flow

x

y

z

Motion of a fluid as a continuum

Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z).


flow

x

y

z

Motion of a fluid as a continuum

Velocity expressed as a function of time t and spatial position (x, y, z)

Eulerian coordinate system


Difference from the Lagrangian approach:

Eulerian Lagrangian


Difference from the Lagrangian approach:

Eulerian

Lagrangian

Outline




Equation of Continuity

differential control volume:

Differential Mass Balance

Rate of Rate of Rate of

accumulation mass in mass out

mass balance:

Rate of

mass in

x y zx zyv y z v x z v x y

Rate of

mass out

x y zx x z zy yv y z v x z v x y

Rate of mass

accumulationx y z

t

Differential Mass Balance

Substituting:

x y zx zy

x y zx x z zy y

x y z v y z v x z v x yt

v y z v x z v x y

Rearranging:

x xx x x

y yy y y

z zz z z

x y z v v y zt

v v x z

v v x y

Differential Equation of Continuity

Dividing everything by ΔV:

Taking the limit as ∆x, ∆y and ∆z 0:

y yx x z zy y yx x x z z zv vv v v v

t x y z

yx zvv v

t x y z


yx zvv v

t x y z

v

divergence of mass velocity vector (v)

Partial differentiation:

yx zx y z

vv vv v v

t x y z x y z


Rearranging:

yx zx y z

vv vv v v

t x y z x y z

substantial time derivative

yx zvv vD

Dt x y zv

If fluid is incompressible: 0 v


In cylindrical coordinates:

1 10

r zrv v vd

dt r r r z

2 2 1where , tan y

r x yx

If fluid is incompressible:

10

r r zvv v v

r r r z

Outline




Differential Equations of Motion

Control Volume

For 1D fluid flow, momentum transport occurs in 3 directions

Fluid is flowing in 3 directions

Momentum transport is fully defined by 3 equations of motion

Momentum Balance

Sum of forcesRate of Rate of Rate of

acting in accumulation momentum in momentum out

the systemx x xx

convective

Rate of Rate of

momentum in momentum out

Rate of Rate of


Rate of

momentum in

x x

x x

molecular

Rate of

momentum outx x

Consider the x-component of the momentum transport:

Momentum Balance

convective

Rate of Rate of


x x x xx x

x x

xv v v v y z

y x y xy y y

z x z xz z z

v v v v x z

v v v v x y

Due to convective transport:

Momentum Balance

molecular

Rate of Rate of


xx xxx x x

yx yx y

x

y

x

y z

y

zx zxz z z

x z

x y

Due to molecular transport:

Momentum Balance



the systemx x xx

Sum of forces

acting in

the systemx x x x

x

p p y z g x y z


Momentum Balance



the systemx x xx

Rate of

accumulationx

x

vx y z

t


Differential Momentum Balance

Substituting:

x x x xx x x

y x y xy y y

z x

xx xxx x

x

z

x

xz z z

y z

vx y z

tv v v v y z

v v v v x z

v v v v x y

yx yxy y y

z

x x x x

x zxz z z

x z

x

p p y z g x y z

y


Dividing everything by ΔV:

y x y xx x x x y y yx x x

z x xx xxx x xz

yx yx zx zxy y y z z z

x x

xz z

x

z

xx

p pg

x

x

y

v v v vv v v v

x y

v v

z

t

v v

z

v

Differential Equation of Motion

Taking the limit as ∆x, ∆y and ∆z 0:

y xx x

yxxx z

z x

xx

x

x y z

pg

x

v vv v v v

x y z

v

t

Rearranging:

y xx x

x

yxxx x

z

z

xxv vv v v vv

t

x yg

x y

p

z

z x


For the convective terms:

y xx x z x

yx zx x xx y z x

v vv v v v

x y z

vv vv v vv v v v

x y z x y z

For the accumulation term:

x xx

yx x zx x y z

v vv

t t t

vv v vv v v v

t x y z x y z


Substituting:

x x x

yx x z

x

x x

y z

yx z

yxxx z

y

x

z

xx

vv v

v v vv v v

x y z

vv vv

x y z

vv v v v

t x y z x

x y z

z

g

y

p

x


x xx xx y

yxxx zxx

z

v v vv v v

x y z

v

gx y x

t

z

p

EQUATION OF MOTION FOR THE x-COMPONENT

Substituting:


Substituting:

EQUATION OF MOTION FOR THE y-COMPONENT

y yy yx y

xy yy zy

y

z

v v vv v v

x y z

xg

v

y z

t

p

y


Substituting:

EQUATION OF MOTION FOR THE z-COMPONENT

z zz zx y

yzxz zzz

z

v v vv v v

x y z

v

gx y z

t

z

p


Substantial time derivatives:

yxxx zxxx

xy yy zyyy

yzxz zzzz

Dv pg

Dt x y z x

Dv pg

Dt x y z y

Dv pg

Dt x y z z


In vector-matrix notation:

yxxx zx

x xxy yy zy

y y

z z

yzxz zz

px y z

xv gD pv g

Dt x y z yv g

p

zx y z

Dp

Dt

vg


Cauchy momentum equation

• Equation of motion for a pure fluid

• Valid for any continuous medium (Eulerian)

• In order to determine velocity distributions, shear stress must be expressed in terms of velocity gradients and fluid properties (e.g. Newton’s law)

Dp

Dt

vτ g

Cauchy Stress Tensor

Stress distribution: normal stresses

shear stresses

xx

yy

zz

xy yx

xz zx

yz zy

Cauchy Stress Tensor

Stokes relations (based on Stokes’ hypothesis)

22

3

22

3

2= 2

3

xxx

yyy

zzz

v

xv

y

v

z

v

v

v

yxxy yx

x zxz zx

y zyz zy

vv

y x

v v

z x

v v

z y

where yx zvv v

x y z

v

Navier-Stokes Equations

Assumptions

1. Newtonian fluid

2. Obeys Stokes’ hypothesis

3. Continuum

4. Isotropic viscosity

5. Constant density

Divergence of the stream velocity is zero


Applying the Stokes relations per component:

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

yxxx zx x x x

xy yy zy y y y

yzxz zz z z z

v v v

x y z x y z

v v v

x y z x y z

v v v

x y z x y z


Navier-Stokes equations in rectangular coordinates2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

x x x xx

y y y yy

z z z zz

Dv v v v pg

Dt x y z x

Dv v v v pg

Dt x y z y

Dv v v v pg

Dt x y z z

2Dp

Dt

vg v

Cylindrical Coordinates

2

2 2

2 2 2 2

2

1 1 2

1

r r r rr z

r r r rr

r r r rr z

v vv v v vv v

t r r r z

rv v v vpg

r r r r r r z

v vv v v vv v

t r r r z

p

r

2 2

2 2 2 2

2

2 2

2 2 2

1 1 2

1 1

r r r rr z

z z zz

rv v v vg

r r r r r z

v vv v v vv v

t r r r z

v v vpg r

z r r r r z

Applications of Navier-Stokes Equations

Application

The Navier-Stokes equations may be reduced using the following simplifying assumptions:

1. Steady state flow

0

0 all components

tv

t

Application


2. Unidirectional flow

2 2 2 2

2 2 2 2

flow along -direction only

x x x xv v v v

x y z x

x

Application


2. Unidirectional flow

2 2

2 2 2 2

2

2

1 1 2

flow along -direction only

rv v v v

r r r r r z

vz

z

Application


3. Constant fluid properties

• Isotropy (independent of position/direction)

• Independent with temperature and pressure

incompressible fluid:

0 from the equation of continuity v

Application


4. No viscous dissipation (INVISCID FLOW)

0 0

Dp

Dt

τ

vg Euler’s Equation

Application


5. No external forces acting on the system

0 no external pressure gradient

0 no gravity effects

p

g

Application


6. Laminar flow

x x x x xx y z

Dv v v v vv v v

Dt t x y z

x xDv v

Dt t

Example

Derive the equation giving the velocity distribution at steady state for laminar flow of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates. The velocity profile desired is at a point far from the inlet or outlet of the channel. The two plates will be considered to be fixed and of infinite width, with flow driven by the pressure gradient in the x-direction.

Example

Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a vertical pipe of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates.

equations of continuity. outline 1.time derivatives & vector notation 2.differential equations...

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