CH.1. DESCRIPTION OF MOTION Continuum Mechanics Course (MMC)

Overview

1.1. Definition of the Continuous Medium 1.1.1. Concept of Continuum

1.1.2. Continuous Medium or Continuum

1.2. Equations of Motion 1.2.1 Configurations of the Continuous Medium

1.2.2. Material and Spatial Coordinates

1.2.3. Equation of Motion and Inverse Equation of Motion

1.2.4. Mathematical Restrictions

1.2.5. Example

1.3. Descriptions of Motion 1.3.1. Material or Lagrangian Description

1.3.2. Spatial or Eulerian Description

1.3.3. Example

2

Lecture 5

Lecture 1

Lecture 2

Lecture 4

Lecture 3

Overview (cont’d)

1.4. Time Derivatives 1.4.1. Material and Local Derivatives

1.4.2. Convective Rate of Change

1.4.3. Example

1.5. Velocity and Acceleration 1.5.1. Velocity

1.5.2. Acceleration

1.5.3. Example

1.6. Stationarity and Uniformity 1.6.1. Stationary Properties

1.6.2. Uniform Properties

3

Lecture 6

Lecture 7

Lecture 10

Lecture 9

Lecture 8

Overview (cont’d)

1.7. Trajectory or Pathline 1.7.1. Equation of the Trajectories 1.7.2. Example

1.8. Streamlines 1.8.1. Equation of the Streamlines 1.8.2. Trajectories and Streamlines 1.8.3. Example 1.8.4. Streamtubes

1.9. Control and Material Surfaces 1.9.1. Control Surface

1.9.2. Material Surface

1.9.3. Control Volume

1.9.4. Material Volume

Summary

4

Lecture 12

Lecture 11

Ch.1. Description of Motion

1.1 Definition of the Continuous Medium

5

The Concept of Continuum

Microscopic scale: Matter is made of atoms which may be grouped in

molecules. Matter has gaps and spaces.

Macroscopic scale: Atomic and molecular discontinuities are disregarded. Matter is assumed to be continuous.

6

Continuous Medium or Continuum

Matter is studied at a macroscopic scale: it completely fills the space, there exist no gaps or empty spaces.

Assumption that the medium and is made of infinite

particles (of infinitesimal size) whose properties are describable by continuous functions with continuous derivatives.

7

Exceptions to the Continuous Medium

Exceptions will exist where the theory will not account for all the observed properties of matter. E.g.: fatigue cracks. In occasions, continuum theory can be used in combination

with empirical information or information derived from a physical theory based on the molecular nature of material.

The existence of areas in which the theory is not applicable does not destroy its usefulness in other areas.

8

Continuum Mechanics

Study of the mechanical behavior of a continuous medium when subjected to forces or displacements, and the subsequent effects of this medium on its environment.

It divides into: General Principles: assumptions and consequences

applicable to all continuous media. Constitutive Equations: define the mechanical behavior of

a particular idealized material.

9

Ch.1. Description of Motion

1.2 Equations of Motion

10

Material and Spatial points, Configuration

A continuous medium is formed by an infinite number of particles which occupy different positions in space during their movement over time.

MATERIAL POINTS: particles

SPATIAL POINTS: fixed spots in space

The CONFIGURATION Ωt of a continuous medium at a given time (t) is the locus of the positions occupied by the material points of the continuous medium at the given time.

11

Γ0

Ω0 X

Ω

Γ

x

( ), tϕ X

Configurations of the Continuous Medium

Ω or Ωt: deformed (or present) configuration, at present time t.

Γ or Γt : deformed boundary. x : Position vector of the same

particle at present time.

Ω0: non-deformed (or reference) configuration, at reference time t0.

Γ0 : non-deformed boundary. X : Position vector of a particle at

reference time.

Initial, reference or undeformed configuration

Present or deformed configuration

0 0t = → reference time [ ]0,t T∈ → current time

12

Material and Spatial Coordinates

The position vector of a given particle can be expressed in: Non-deformed or Reference Configuration

Deformed or Present Configuration

[ ]1

2

3

XX XX YX Z

= = ≡

material coordinates (capital letter)

[ ]1

2

3

xx xx yx z

= = ≡

spatial coordinates (small letter)

13

The motion of a given particle is described by the evolution of its spatial coordinates (or its position vector) over time.

The Canonical Form of the Equations of Motion is obtained when the “particle label” is taken as its material coordinates

Equations of Motion

( ) ( )( )

, ,

, 1, 2,3i i

t t

x t i

ϕ

ϕ

= =

= ∈

x xparticle label particle label

particle label

φ (particle label, t) is the motion that takes the body from a reference configuration to the current one.

1 2 3TX X X≡ ≡ Xparticle label

( ) ( )( ) 1 2 3

, ,

, , , 1,2,3

not

i i

t t

x X X X t i

x X x Xϕ

ϕ

= =

= ∈

14

Γ0

Ω0 X

Ω

Γ

x

( ), tϕ X

( )1 ,x tϕ−

Inverse Equations of Motion

The inverse equations of motion give the material coordinates as a function of the spatial ones.

( ) ( )( )

1

11 2 3

, ,

, , , 1,2,3

not

i i

t t

X x x x t i

X x X xϕ

ϕ

−

−

= =

= ∈

15

Mathematical restrictions for φ and φ-1 defining a “physical” motion

Consistency condition , as X is the position vector for t=0

Continuity condition , φ is continuous with continuous derivatives

Biunivocity condition φ is biunivocal to guarantee that two particles do not occupy

simultaneously the same spot in space and that a particle does not occupy simultaneously more than one spot in space. Mathematically: the “Jacobian” of the motion’s equations should be different from zero:

The “Jacobian” of the equations of motion should be

“strictly positive”

( ),0ϕ =X X

1Cϕ∈

( ),det 0i

j

tJ

ϕ ϕ ∂ ∂= = >

∂ ∂

XX X

( ),det 0i

j

tJ

ϕ ϕ ∂ ∂= = ≠

∂ ∂

XX X

16

density is always positive (to be proven)

Example

The spatial description of the motion of a continuous medium is given by:

Find the inverse equations of motion.

( )

2 21 1

2 22 2

2 23 1 3

,5 5

t t

t t

t t

x X e x Xet x X e y Ye

x X t X e z Xt Ze

− −

= = ≡ = ≡ = = + = +

x X

17

Example - Solution

Check the mathematical restrictions:

Consistency Condition ?

Continuity Condition ?

Biunivocity Condition ?

Density positive ?

( ),0ϕ =X X

1Cϕ∈

( ),0

tJ

ϕ∂= >

∂XX

( )

21 1

22 2

23 1 3

,5

t

t

t

x X et x X e

x X t X e

−

=

≡ = = +

x X

( )

2 01 1

2 02 2

2 01 3 3

, 05 0

X e Xt X e X

X X e X

⋅

− ⋅

⋅

= = = = ⋅ +

x X X

1 1 1

21 2 3

2 2 2 2 22 2 2

1 2 3 2

3 3 3

1 2 3

0 00 0 05 0

t

t t t t ti

j t

x x xX X X

ex x x xJ e e e e e tX X X X

t ex x xX X X

− −

∂ ∂ ∂∂ ∂ ∂

∂ ∂ ∂ ∂= = = = ⋅ ⋅ = ≠ ∀∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂

2 0tJ e= >

18

Example - Solution

Calculate the inverse equations:

( )

21 1

22 2

23 1 3

,5

t

t

t

x X et x X e

x X t X e

−

=

≡ = = +

x X

2 211 1 1 12

t tt

xx X e X x ee

−= ⇒ = =

2 222 2 2 22

t tt

xx X e X x ee

−−= ⇒ = =

( )( )2 2 2 2 43 13 1 3 3 3 1 3 12

55 5 5t t t t tt

x X tx X t X e X x x e t e x e tx ee

− − − −−= + ⇒ = = − = −

( )

21 1

1 22 2

2 43 3 1

,5

t

t

t t

X x et X x e

X x e tx eϕ

−

−

− −

=

≡ = = = −

X x

19

Ch.1. Description of Motion

1.3 Descriptions of Motion

20

Descriptions of Motion

The mathematical description of the particle properties can be done in two ways: Material (Lagrangian) Description

Spatial (Eulerian) Description

21

Material or Lagrangian Description

The physical properties are described in terms of the material coordinates and time.

It focuses on what is occurring at a fixed material point (a particle, labeled by its material coordinates) as time progresses.

Normally used in solid mechanics.

22

The physical properties are described in terms of the spatial coordinates and time.

It focuses on what is occurring at a fixed point in space (a spatial point labeled by its spatial

coordinates) as time progresses.

Normally used in fluid mechanics.

Spatial or Eulerian Description

23

Example

The equation of motion of a continuous medium is:

Find the spatial description of the property whose material description is:

( ), x X Yt

t y Xt Yz Xt Z

= −= ≡ = + = − +

x x X

( ) 21X Y ZX,Y,Z,t

tρ + +

=+

24

Example - Solution

Check the mathematical restrictions:

Consistency Condition ?

Continuity Condition ?

Biunivocity Condition ?

Any diff. Vol. must be positive ?

( ) XX =0,φ

1C∈φ

( ) 0,>

∂∂

=XX tJ φ

( )0

, 0 00

X Y Xt X Y Y

X Z Z

− ⋅ = = ⋅ + = = ⋅ +

x X X

( ) 2

1 01 0 1 1 1 1 1 00 1

i

j

x x xX Y Z t

x y y yJ t t t t tX X Y Z

tz z zX Y Z

∂ ∂ ∂∂ ∂ ∂ −

∂ ∂ ∂ ∂= = = = ⋅ ⋅ − ⋅ − ⋅ = + ≠ ∀∂ ∂ ∂ ∂

−∂ ∂ ∂∂ ∂ ∂

21 0J t= + >

( ), x X Yt

t y Xt Yz Xt Z

= −= ≡ = + = − +

x x X

25

Example - Solution

Calculate the inverse equations:

( ), x X Yt

t y Xt Yz Xt Z

= −= ≡ = + = − +

x x X

221

x X Yt X x Yty Y y xtx Yt Yt Y y xt Yy Y t ty Xt Y X

t

= − ⇒ = + − −⇒ + = ⇒ + = − ⇒ =− += + ⇒ =

2 2

2 2 21 1 1y xt x xt yt xt x ytX x Yt x t

t t t− + + − + = + = + = = + + +

2 2

2 21 1x yt z zt xt ytz Xt Z Z z Xt z t

t t+ + + + = − + ⇒ = + = + = + +

( )

2

12

2 2

2

1

,1

1

x ytXt

y xtt Yt

z zt xt ytZt

ϕ−

+ = +−≡ = = +

+ + += +

X x

26

Example - Solution

Calculate the property in its spatial description:

( )( )

2 2

2 2 2 2 2

22 2 2

1 1 11 1 1

x yt y xt z zt xt ytt t tX Y Z x y yt yt z ztX,Y,Z,t

t t tρ

+ − + + + + + + + ++ + + + + + + = = =+ + +

( )

2

12

2 2

2

1

,1

1

x ytXt

y xtt Yt

z zt xt ytZt

ϕ−

+ = +−≡ = = +

+ + += +

X x

( ) ( ) ( )( )

( )2 2

22 2

1 1

1 1

x y t t z tX Y ZX,Y,Z,t x, y,z,tt t

ρ ρ+ + + + ++ +

= ⇒ =+ +

27

Ch.1. Description of Motion

1.4 Time Derivatives

28

Material and Local Derivatives

The time derivative of a given property can be defined based on the: Material Description Γ(X,t) TOTAL or MATERIAL DERIVATIVE

Variation of the property w.r.t. time following a specific particle in the continuous medium.

Spatial Description γ(x,t) LOCAL or SPATIAL DERIVATIVE Variation of the property w.r.t. time in a fixed spot of space.

( ), tt

∂Γ ≡ → ∂

X partial time derivative of the material derivative

of the properymaterial description

( ), tt

γ∂ ≡ → ∂

x partial time derivative of the local derivative

of the properyspatial description

29

Convective Derivative

Remember: x=x(X,t), therefore, γ(x,t)=γ(x(X,t),t)=Γ(X,t) The material derivative can be computed in terms of

spatial descriptions:

Generalising for any property:

( ) ( )

( )( ) ( )

( ) ( , ) ( , )

( , ) ( , )

( , ), ,

, ,, ,

not not

i

i

i t it t

t t

d D tt tdt Dt tt txd t t

dt t x t t tγγ

γ

γ γ

γ γγ γγ

∂∂ ⋅

∂Γ→ = = = =

∂∂ ∂∂∂ ∂ ∂

= = + ⋅ = + ⋅ =∂ ∂ ∂ ∂ ∂ ∂

x x v xv x x

Xx x

x x xx Xx

material derivative

∇∇∇

( ) ( ) ( ) ( ), ,, ,

d t tt t

dt tχ χ

χ∂

= + ⋅∇∂

x xv x x

REMARK The spatial Nabla operator is defined as:

eiix

∂∇ ≡

∂material

derivative local

derivative convective rate of

change (convective derivative) 30

Convective Derivative

Convective rate of change or convective derivative is implicitly defined as:

The term convection is generally applied to motion related phenomena. If there is no convection (v=0) there is no convective rate of change

and the material and local derivatives coincide.

( )⋅∇ •v

( ) ( ) ( )0d

dt t• ∂ •

⋅∇ • = =∂

v

31

Example

Given the following equation of motion:

And the spatial description of a property ,

Calculate its material derivative.

Option #1: Computing the material derivative from material descriptions

Option #2: Computing the material derivative from spatial descriptions

+=+=

++=≡

XtZzZtYy

ZtYtXxt

32),(Xx

( ) tyxt 323, ++=ρ x

32

Example - Solution

Option #1: Computing the material derivative from material descriptions

Obtain ρ as a function of X by replacing the Eqns. of motion into ρ(x,t) : Calculate its material derivative as the partial derivative of the material description:

+=+=

++=≡

XtZzZtYy

ZtYtXxt

32),(Xx

( ) tyxt 323, ++=ρ x

( ) ( ) ( ) ( ) ( ), ( , ), , 3 2 2 33 3 2 7 3

t t t t X Yt Zt Y Zt tX Yt Y Zt t

ρ ρ ρ= = = + + + + +

= + + + +

x x X X

( )( )

( ),

, ,3 3 7

x x X

x X

t

d t tY Z

dt tρ

=

ρ ∂= = + +

∂

( )( )

( ) ( ),

, ,3 3 2 7 3 3 7 3

x x X

x X

t

d t tX Yt Y Zt t Y Z

dt t tρ

=

ρ ∂ ∂= = + + + + = + +

∂ ∂

33

Example - Solution

Option #2: Computing the material derivative from spatial descriptions

Applying this on ρ(x,t) :

( ) ( ) ( ) ( ), ,, ,

d t tt t

dt tρ ρ

ρ∂

= + ⋅∇∂

x xv x x

( ) ( )

( ) ( ) ( ) ( ) ( ) [ ]

( ) ( ) ( ) ( ) ( ) ( ) ( )

[ ]

,

,3 2 3 3

, , 2 , 3 , 2 , 3 23

, , ,, , , 3 2 3 , 3 2 3 , 3 2 3

33, 2, 0 2

0

x x X

x

xv x

x x xx

TT

T T

T

t

tx y t

t tY Z

t X Yt Zt Y Zt Z Xt Y Z Z X Zt t t t

X

t t tt x y t x y t x y t

x y z x y z

=

∂ρ ∂= + + =

∂ ∂+

∂ ∂ ∂ ∂ = = + + + + = + = ∂ ∂ ∂ ∂

∂ρ ∂ρ ∂ρ ∂ ∂ ∂∇ρ = = + + + + + + = ∂ ∂ ∂ ∂ ∂ ∂

= =

+=+=

++=≡

XtZzZtYy

ZtYtXxt

32),(Xx

( ) tyxt 323, ++=ρ x

34

Example - Solution

Option #2 The material derivative is obtained:

+=+=

++=≡

XtZzZtYy

ZtYtXxt

32),(Xx

( ) tyxt 323, ++=ρ x

( )( )

,

3,

3 [ , 2 , 3 ] 2 3 3 3 40

( )

x x X

x

v

v Tt

d tY Z Z X Y Z Z

dt

ρ

ρ

=

ρ = + + = + + +

⋅

∇

∇

( )( ),

,3 3 7

x x X

x

t

d tY Z

dt =

ρ= + +

35

Ch.1. Description of Motion

1.5 Velocity and Acceleration

36

Velocity

Time derivative of the equations of motion. Material description of the velocity:

Time derivative of the equations of motion

Spatial description of the velocity:

Velocity is expressed in terms of x using the inverse equations of motion:

( )( ) ( ), , ,t t tV X x v x

( ) ( )

( ) ( )

,,

,, 1, 2,3i

i

tt

tx t

V t it

∂= ∂

∂ = ∈ ∂

x XV X

XX

REMARK Remember the equations of motion are of the form:

( ) ( ), ,x X x Xnot

t tϕ= =

37

Material time derivative of the velocity field. Material description of acceleration:

Derivative of the material description of velocity:

Spatial description of acceleration:

A(X,t) is expressed in terms of x using the inverse equations of motion: Or a(x,t) is obtained directly through the material derivative of v(x,t):

Acceleration

( ) ( )

( ) ( )

,,

,, 1, 2,3i

i

tt

tV t

A t it

∂= ∂

∂ = ∈ ∂

V XA X

XX

( )( ) ( ), , ,t t tA X x a x

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, ,, , ,

v , v , v, v , , 1, 2,3i i ii k

k

d t tt t t

dt td t t

a t t t idt t x

∂= = + ⋅∇ ∂

∂ ∂ = = + ⋅ ∈ ∂ ∂

v x v xa x v x v x

x xx x x

38

Example

Consider a solid that rotates at a constant angular velocity ω and has the following equation of motion:

Find the velocity and acceleration of the movement described in both, material and spatial forms.

( )( )

sin( , , )

cos

x R tR t

y R t

ω

ω

= + φφ → = + φ

→

xlabel of particle

(non - canonical equations of motion)

39

Example - Solution

Using the expressions

The equation of motion can be rewritten as:

For t=0, the equation of motion becomes:

Therefore, the equation of motion in terms of the material coordinates is:

( )( )

sin sin cos cos sin

cos cos cos sin sin

a b a b a b

a b a b a b

± = ⋅ ± ⋅

± = ⋅ ⋅ ∓

( ) ( ) ( )( ) ( ) ( )

sin sin cos cos sin

cos cos cos sin sin

x R t R t R t

y R t R t R t

= ω + φ = ω φ+ ω φ

= ω + φ = ω φ− ω φ

sin cos

X RY R

= φ = φ

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

sin sin cos cos sin cos sin

cos cos cos sin sin sin cos

x R t R t R t X t Y t

y R t R t R t X t Y t

= ω + φ = ω φ+ ω φ = ω + ω

= ω + φ = ω φ− ω φ = − ω + ω

X=

X=Y=

Y=

( )( )

φ+ω=φ+ω=

tRytsinRx

cos

40

Example - Solution

The inverse equation of motion is easily obtained

( ) ( )

( ) ( )

cos sin

sin cos

x X t Y t

y X t Y t

= ω + ω

= − ω + ω

( )( )

( )( )

sincos

cossin

x Y tX

t

y Y tX

t

− ω=

ω

− + ω=

ω

( )( )

( )( )

sin coscos sin

x Y t y Y tt t

− ω − + ω=

ω ω

( ) ( ) ( ) ( )2 2sin sin cos cosx t Y t y t Y tω − ω = − ω + ω

( ) ( ) ( ) ( )( )2 2

1

sin cos cos sinx t y t Y t t

=

ω + ω = ω + ω

( ) ( )( ) ( )( ) ( )

( )( )

( ) ( )( )

( )( )( )

( )

2

2

cos

sin cos sin sin cos sincos cos cos cos

1 sinsin

cos

t

x x t y t t x t y t txXt t t t

tx y t

t= ω

− ω + ω ω ω ω ω= = − − =

ω ω ω ω

− ω= − ω

ω

41

( ) ( )( ) ( )

cos sin

sin cos

x X t Y t

y X t Y t

= ω + ω

= − ω + ω

Example - Solution

So, the equation of motion and its inverse in terms of the material coordinates are:

( )( ) ( )( ) ( )

cos sin,

sin cos

x X t Y tt

y X t Y t

= ω + ω→ →= − ω + ω

x X canonical equations of motion

( )( ) ( )( ) ( )

cos sin,

sin cos

X x t y tt

Y x t y t

= ω − ω→ →= ω + ω

X x inverse equations of motion

42

Example - Solution

Velocity in material description is obtained from ( ) ( ),,

x XV X

tt

t∂

=∂

( ) ( ) ( ) ( )( )

( ) ( )( )

cos sin,,

sin cos

x X t Y tt t ttyt X t Y tt t

∂ ∂ = ω + ω∂ ∂ ∂= = ∂ ∂∂ = − ω + ω ∂ ∂

x XV X

( )( ) ( )( ) ( )

cos sin,

sin cos

x X t Y tt

y X t Y t

= ω + ω→ = − ω + ω

x X

( )( ) ( )( ) ( )

sin cos,

cos sinx

y

X t Y tVt

V X t Y t

− ω ω + ω ω = = − ω ω − ω ω

V X

43

Example - Solution

Velocity in spatial description is obtained introducing X(x,t) into V(X,t):

Alternative procedure (longer):

( ) ( )( ) ( ) ( )( ) ( )

v sin cos, , ,

v cos sinx

y

yX t Y tt t t

xX t Y t

y

x

v x V X x ω− ω + ω ω = = = = − ω− ω − ω ω

−

( ) ( )( ) ( )

cos sin

sin cos

X x t y t

Y x t y t

= ω − ω

= ω + ω

( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( )( )

2 2

2 2

2 2

2 2

0 1

1

cos sin sin sin cos cos,

cos sin cos sin cos sin

sin cos cos sin sin cos

sin cos

x t t y t x t t y tt

x t y t t x t y t t

x t t t t y t t

x t t

ω

ω

ω

ω= =

=

− ω ω +ω ω +ω ω ω +ω ω = = − ω +ω ω ω −ω ω −ω ω ω

ω ω − ω ω +ω ω + ω

=− ω + ω

v x

( ) ( ) ( ) ( )( )

0

sin cos cos siny t t t t=

+ω ω ω − ω ω

( )v

,v

x

y

yt

x ω = = −ω

v x

( ) ( )( ) ( )

cos sin

sin cos

x X t Y t

y X t Y t

= ω + ω →= − ω + ω

44

( ) ( )( ) ( )

cos sin

sin cos

x X t Y t

y X t Y t

= ω + ω

= − ω + ω

Example - Solution

Acceleration in material description is obtained applying: ( ) ( ),,

V XA X

tt

t∂

=∂

( )( ) ( )( ) ( )

sin cos,

cos sin

X t Y tt

X t Y t

− ω ω + ω ω = − ω ω − ω ω

V X

( ) ( ) ( ) ( )

( ) ( )

2 2

2 2

cos sin,,

sin cos

x

y

V X t Y tt ttVt

X t Y tt

∂ = − ω ω − ω ω∂ ∂= = ∂∂ = ω ω − ω ω ∂

V XA X

( )( ) ( )( ) ( )

2cos sin

,sin cos

x

y

X t Y tAt

A X t Y t

ω + ω = = −ω − ω + ω

A X

45

Example - Solution

(OPTION #1) Acceleration in spatial description is obtained by replacing the inverse equation of motion into A(X,t):

( ) ( )( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )

2

, , ,

cos sin cos sin cos sin

cos sin sin sin cos cos

t t t

x t y t t x t y t t

x t y t t x t y t tω

= =

ω − ω ω + ω + ω ω = − − ω − ω ω + ω + ω ω

a x A X x

( ) ( )( ) ( )

cos sin

sin cos

X x t y t

Y x t y t

= ω − ω

= ω + ω

( )

( ) ( )( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( )( )

2 2

2

2 2

01

0 1

cos sin sin cos cos sin

,cos sin sin cos sin cos

x t t y t t t t

tx t t t t y t t

==

= =

ω + ω + − ω ω + ω ω = −ω

− ω ω + ω ω + ω + ω

a x

( )2

2, x

y

a xt

a y

−ω = = −ω

a x

46

Example - Solution

(OPTION #2): Acceleration in spatial description is obtained by directly calculating the material derivative of the velocity in spatial description:

( ) ( ) ( ) ( ) ( ), ,, , ,

v x v xa x v x v x

d t tt t t

dt t∂

= = + ⋅∇∂

( ) [ ] [ ]

[ ]( ) ( )

( ) ( )[ ]

2

2

, , ,

0 0, ,

0 0

y xt y x y xxt

y

y xxx xy x y xyy x

y y

= =

= = =

∂ ω ∂ ∂ + ω −ω ω −ω ∂−ω∂ ∂

∂ ∂ ω −ω −ω −ω ∂ ∂ + ω −ω ω −ω ∂ ∂ ω −ω ω −ω ∂ ∂

a x

( )2

2,

xt

y

−ω = −ω

a x

( )v

,v

x

y

yt

x ω = = −ω

v x

47

Ch.1. Description of Motion

1.6 Stationarity and Uniformity

48

Stationary properties

A property is stationary when its spatial description is not dependent on time.

The local derivative of a stationary property is zero.

The time-independence in the spatial description (stationarity) does

not imply time-independence in the material description:

( ) ( ), tχ χ=x x

( ) ( ) ( ),, 0

tt

tχ

χ χ∂

= =∂x

x x

( ) ( ) ( ) ( ), ,t tχ χ χ χ= =x x X X REMARK This is easily understood if we consider, for example, a stationary velocity: ( ) ( ) ( )( ) ( ), , ,t t t= = =v x v x v x X V X

49

REMARK In certain fields, the term steady-state is more commonly used.

Example

Consider a solid that rotates at a constant angular velocity ω and has the following equation of motion:

We have obtained:

Velocity in spatial description

stationary

Velocity in material description

( )( )

sin

cos

x R t

y R t

ω ϕ

ω ϕ

= +

= +

( )( ) ( )( ) ( )

sin cos,

cos sinx

y

X t Y tVt

V X t Y t

− ω ω + ω ω = = − ω ω − ω ω

V X

( )v

,v

x

y

yt

xωω

= = − v x

50

Uniform properties

A property is uniform when its spatial description is not dependent on the spatial coordinates.

If its spatial description does not depend on the coordinates (uniform

character of the property), neither does its material one.

( ) ( ),x t tχ χ=

( ) ( ) ( ) ( ), ,t t t tχ χ χ χ= =x X

51

Ch.1. Description of Motion

1.7 Trajectory (path-line)

52

Trajectory or pathline

A trajectory or pathline is the locus of the positions occupied by a given particle in space throughout time.

REMARK A trajectory can also be defined as the path that a particle follows through space as a function of time.

53

Equation of the trajectories

The equation of a given particle’s trajectory is obtained particularizing the equation of motion for that particle, which is identified by it material coordinates X*.

Also, from the velocity field in spatial description, v(x,t):

A family of curves is obtained from:

Particularizing for a given particle by imposing the consistency condition in the reference configuration:

Replacing [2] in [1], the equation of the trajectories in canonical form

( ) ( )( ) ( )

*

*

, ( )

, ( )ii i

t t t

x t t t iϕ φ=

=

= =

= = ∈

X X

X X

x X

X

ϕ φ

1,2,3

( ) ( )( ) ( )1 2 3, , , ], [1d t

t t C C C tdt

= =x

v x x φ

( ) ( ) ( )1 2 30, , , [2]0 i it

t C C C C χ== = =x X X Xφ

( ) ( ) ( )( ) ( )1 2 3, , , ,x X X X XC C C t tφ ϕ= =

54

Example

Consider the following velocity field:

Obtain the equation of the trajectories.

( )

, y

tx

ω = −ω

v x

55

Example - Solution

We integrate the velocity field:

This is a crossed-variable system of differential equations. We derive the 2nd eq. and replace it in the 1st one,

( ) ( )

( ) ( )

( ) ( )

v ,,

v ,

x

y

dx tt yd t dtt

dt dy tt x

dt

= = ω=

= = −ω

xxv x

x

( ) ( ) ( )2

22

d y t dx ty t

dt dt= −ω = −ω 2 0y y′′ + ω =

56

Example - Solution

The characteristic equation:

Has the characteristic solutions:

And the solution of the problem is:

And, using , we obtain

So, the general solution is:

2 2 0r +ω = 1,2jr i j= ± ω ∈

( ) ( )1 2 1 2( ) cos siniwt iwty t Real Part Z e Z e C t C t−= + = ω + ω

dy xdt

= −ω

( ) ( )( )1 21 1 sin cosdyx C t C t

dt= − = − − ω ω + ω ω

ω ω

( ) ( ) ( )( ) ( ) ( )

1 2 1 2

1 2 1 2

, , sin cos

, , cos sin

x C C t C t C t

y C C t C t C t

= ω − ω

= ω + ω

57

Example - Solution

The canonical form is obtained from the initial conditions:

This results in:

( )1 2, ,0C C =x X( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 2

1 2 1 2 1

0 1

1 0

, ,0 sin 0 cos 0

, ,0 cos 0 sin 0

X x C C C C C

Y y C C C C C

= =

= =

= = ω⋅ − ω⋅ = = = ω⋅ + ω⋅ =

( ) ( )( ) ( )

sin cos( , )

cos in

x Y t X tt

y Y t X s t

= ω + ω→ = ω − ω

x X

58

( ) ( ) ( )( ) ( ) ( )

1 2 1 2

1 2 1 2

, , sin cos

, , cos sin

x C C t C t C t

y C C t C t C t

= ω − ω

= ω + ω

Ch.1. Description of Motion

1.8 Streamline

59

Streamline

The streamlines are a family of curves which, for every instant in time, are the velocity field envelopes.

Streamlines are defined for any given time instant and change with

the velocity field.

REMARK The envelopes of vector field are the curves whose tangent vector at each point coincides (in direction and sense but not necessarily in magnitude) with the corresponding vector of the vector field.

time – t0

X

Y

X

Y time – t1

REMARK Two streamlines can never cut each other. Is it true?

60

Equation of the Streamlines

The equation of the streamlines is of the type:

Also, from the velocity field in spatial description, v(x,t*) at a given time instant t*: A family of curves is obtained from:

Where each group identifies a streamline x(λ) whose

points are obtained assigning values to the parameter λ. For each time instant t* a new family of curves is obtained.

( ) ( )( ) ( )1 2 3, * , , , , *d

t C C C tdλ

λ φ λλ

′ ′ ′= =x

v x x

( )v v vx y z

d x d y dz dd dsd

λλ

= = = = =x v

61

( )1 2 3, ,C C C′ ′ ′

vx

v y

vz

Trajectories and Streamlines

For a stationary velocity field, the trajectories and the streamlines coincide – PROOF: 1. If v(x,t)=v(x):

Eq. trajectories:

Eq. streamlines:

The differential equations only differ in the denomination of the integration parameter (t or λ), so the solution to both systems MUST be the same.

( ) ( )( ) ( )1 2 3, , , ,d t

t t C C C tdt

φ= =x

v x x

( ) ( )( ) ( )1 2 3, * , , , , *d

t C C C tdλ

λ φ λλ

= =x

v x x

62

Trajectories and Streamlines

For a stationary velocity field, the trajectories and the streamlines coincide – PROOF: 2. If v(x,t)=v(x) the envelopes (i.e., the streamline) of the field do not

vary throughout time.

A particle’s trajectory is always tangent to the velocity field it

encounters at every time instant. If a trajectory starts at a certain point in a streamline and the

streamline does not vary with time and neither does the velocity field, the trajectory and streamline MUST coincide.

63

Trajectories and Streamlines

The inverse is not necessarily true: if the trajectories and the streamlines coincide, the velocity field is not necessarily stationary – COUNTER-EXAMPLE:

Given the (non-stationary) velocity field:

The eq. trajectory are:

The eq. streamlines are:

( )t 00

at =

v

( )

212

t 00

a t C + =

x

( )1

t 00

at Cλ ′+ =

x

( ) ( )t0 t 00 0

at atd

d dtdt

= =

xx

( ) ( )0 00 0

at atd

d ddλ

λ λλ

= =

xx

64

Example

Consider the following velocity field:

Obtain the equation of the trajectories and the streamlines associated to this vector field.

Do they coincide? Why?

v 1,2,31

ii

x it

= ∈+

65

Example - Solution

Eq. trajectories:

Introducing the velocity field and rearranging:

Integrating both sides of the expression:

The solution:

( ) ( )( )( ) ( )( )

,

v ,ii

d tt t

dtdx t

t t idt

=

= ∈

xv x

x 1,2,3

1i idx x i

dt t= ∈

+1,2,3

1i

i

dx dt ix t

= ∈+

1,2,3

1 11i

i

dx dtx t

=+∫ ∫ ( ) ( )ln ln 1 ln ln 1i i ix t C C t i= + + = + ∈1,2,3

( )1i ix C t i= + ∈1,2,366

Example - Solution

Eq. streamlines:

Introducing the velocity field and rearranging:

Integrating both sides of the expression:

The solution:

( ) ( )( )( ) ( )( )*

, *

v ,ii

dt

ddx t

t id

λλ

λ

λλ

=

= ∈

xv x

x 1,2,3

1i idx x i

d tλ= ∈

+1,2,3

1i

i

dx d ix t

λ= ∈

+1,2,3

1 11i

i

dx dx t

λ=+∫ ∫ ln

1i ix Kt

λ= +

+

1 1i

ii

Kt tKx e e ei

λ λ

++ + = = ∈ 1,2,3

1i i

tx C e iλ

+= ∈1,2,3

iC=

67

Streamtube

A streamtube is a surface composed of streamlines which pass through the points of a closed contour fixed in space. In stationary cases, the tube will remain fixed in space

throughout time. In non-stationary cases, it will vary (although the closed contour line is fixed).

69

Ch.1. Description of Motion

1.9 Control and Material Surfaces

81

Control Surface

A control surface is a fixed surface in space which does not vary in time.

Mass (particles) can flow across a control surface.

( ) : x , , 0f x y zΣ = =

82

Material Surface

A material surface is a mobile surface in the space constituted always by the same particles. In the reference configuration, the surface Σ0 will be defined in terms

of the material coordinates:

The set of particles (material points) belonging the surface are the same at all times

In spatial description

The set of spatial points belonging to the the surface depends on time The material surface moves in space

( ) 0 : , , 0F X Y ZΣ = =X

( ) : , , , 0t f x y z tΣ = =x

( ) ( ), , ( , ), ( , ), ( , ) ( , ) ( , , , )F X Y Z F X t Y t Z t f t f x y z t= = =x x x x

84

Necessary and sufficient condition for a mobile surface in space, implicitly defined by the function , to be a material surface is that the material derivative of the function is zero: Necessary: if it is a material surface, its material description does not depend

on time:

Sufficient: if the material derivative of f(x,t) is null:

The surface contains always the same set the of particles (it is a material surface)

( ) ( ) ( ) ( ) ( ), ( , ), , 0 , 0Xx x X X xd Ff t f t t F t f td t t

∂→ = = = =

∂

( ) ( ) : , 0 0t f t FΣ = = = =x X Xx

Material Surface

( ), , ,f x y z t

( ) ( ) ( ) ( ) ( , ), ( , ), , 0 , ( , ) ( )Xx x X X x X Xd F tf t f t t F t f t F t Fd t t

∂→ = = = ≡

∂

85

Control Volume

A control volume is a group of fixed points in space situated in the interior of a closed control surface, which does not vary in time. Particles can enter and exit a control volume.

( ) 0|: ≤= xx fV

REMARK The function f(x) is defined so that f(x)<0 corresponds to the points inside V.

87

Material Volume

A material volume is a (mobile) volume enclosed inside a material boundary or surface. In the reference configuration, the volume V0 will be defined in terms

of the material coordinates:

The particles in the volume are the same at all times

In spatial description, the volume Vt will depend on time.

The set of spatial points belonging to the the volume depends on time The material volume moves in space along time

( ) 0 : | 0V F= ≤X X

( ) : | , 0tV f t= ≤x x

X

89

Material Volume

A material volume is always constituted by the same particles. This is proved by reductio ad absurdum:

If a particle is added into the volume, it would have to cross its

material boundary.

Material boundaries are constituted always by the same particles, so, no particles can cross.

Thus, a material volume is always constituted by the same particles (a material volume is a pack of particles).

90

Ch.1. Description of Motion

Summary

92

CONTINUOS MEDIUM: Matter studied at a macroscopic scale: it completely fills the space & medium and its properties are describable by continuous functions with continuous derivatives.

Equations of Motion and Inverse Equations of Motion:

Mathematical Restrictions: Consistency Condition Continuity Condition Biunivocity Condition Any differential volume must be positive

Summary

Reference or non-deformed

Actual or deformed

( ) ( )( ) 1 2 3

, ,

, , , 1, 2,3

x X x Xnot

i i

t t

x X X X t i

ϕ

ϕ

= =

= ∈

( ) ( )( )

1

11 2 3

, ,

, , , 1, 2,3

X x X xnot

i i

t t

X x x x t i

ϕ

ϕ

−

−

= =

= ∈

( ) XX =0,φ1C∈φ

( ) 0,>

∂∂

=XX tJ φ

( ) 0,≠

∂∂

=XX tJ φ

93

Summary (cont’d)

Descriptions of Motion Material or Lagrangian Description: physical properties are described in

terms of the material coordinates and time.

Spatial or Eulerian Description: physical properties are described in terms of the spatial coordinates and time.

94

Summary (cont’d)

Time derivatives

Velocity Material description: Spatial description:

Acceleration Material description: Spatial description:

A property is stationary when its spatial description is independent of time.

A property is uniform when its spatial description is independent of the

spatial coordinates.

( ) ( )( ), , ,v x V X xt t t=

( ) ( ),,

x XV X

tt

t∂

=∂

( ) ( ),,

V XA X

tt

t∂

=∂

( ) ( ) ( ) ( ) ( ), ,, , ,

v x v xa x v x v x

d t tt t t

dt t∂

= = + ⋅∇∂

( ) ( )( ), , ,a x A X xt t t=

( ) ( ) ( ),, 0

tt

tχ

χ χ∂

= =∂x

x x

( ) ( ),x t tχ χ=

( ) ( ) ( ) ( ), ,, ,

d t tt t

dt tχ χ

χ∂

= + ⋅∇∂

x xv x x

material derivative

local derivative

convective rate of change

95

Summary (cont’d)

Trajectory or pathline: locus of the positions occupied by a given particle in space throughout time

Streamlines: family of curves which, for every instant in time, are the velocity field envelopes.

Streamtube: surface composed of streamlines which pass through the points of a closed contour fixed in space.

In a stationary problem, the trajectories and streamlines coincide.

( ) ( )*

,X X

x xt tϕ=

= ( ) ( )( ) ( )1 2 3, , , ,d t

t t C C C tdt

φ= =x

v x x

d d d= =

x y zu v w

( ) ( )( ) ( )1 2 3, * , , , , *d

t C C C tdλ

λ φ λλ

= =x

v x x

( ) ( )( ) ( )1 2 3 1 2 1 2 1 2, , , , , , , , , ,C C C C C t C C t t C C tλ= =x f h

96

Summary (cont’d)

Material surface or boundary: mobile surface in the space constituted always by the same particles. In reference configuration: In spatial configuration: The necessary and sufficient condition for a mobile surface in space to be material:

Control surface: fixed surface in space which does not vary in time.

Material volume: volume enclosed inside a material boundary or surface. It is always constituted by the same particles. In reference configuration: In spatial configuration:

Control volume: group of fixed points in space situated in the interior of a closed control surface, which does not vary in time.

( ) 0,, :0 ==Σ ZYXFX

( ) 0,,, : ==Σ tzyxft x

( ),0 t

df t f f tdt t

∂= + ⋅∇ = ∀ ∈Σ ∀∂

xv x

( ) : , , 0f x y zΣ = =X

( ) 0 : | 0V F= ≤X X

( ) 0,|: ≤= tfVt xx

( ) 0|: ≤= xx fV

97