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

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

2 Overview 1.1. Definition of the Continuous Medium Concept of Continuum Continuous Medium or Continuum 1.. Equations of Motion 1..1 Configurations of the Continuous Medium 1... Material and Spatial Coordinates Equation of Motion and Inverse Equation of Motion Mathematical Restrictions Example 1.3. Descriptions of Motion Material or Lagrangian Description Spatial or Eulerian Description Example Lecture 1 Lecture Lecture 3 Lecture 4 Lecture 5

3 Overview (cont d) 1.4. Time Derivatives Material and Local Derivatives Convective Rate of Change Example 1.5. Velocity and Acceleration Velocity Acceleration Example 1.6. Stationarity and Uniformity Stationary Properties Uniform Properties Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 3

4 Overview (cont d) 1.7. Trajectory or Pathline Equation of the Trajectories Example 1.8. Streamlines Equation of the Streamlines Trajectories and Streamlines Example Streamtubes 1.9. Control and Material Surfaces Control Surface Material Surface Control Volume Material Volume Lecture 11 Lecture 1 Summary 4

5 1.1 Definition of the Continuous Medium Ch.1. Description of Motion 5

6 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

7 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

8 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

9 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

10 1. Equations of Motion Ch.1. Description of Motion 10

11 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 Ω t of a continuous medium at a given time ( is the locus of the positions occupied by the material points of the continuous medium at the given time. 11

12 Configurations of the Continuous Medium Ω 0 : non-deformed (or reference) configuration, at reference time t 0. Γ 0 : non-deformed boundary. X : Position vector of a particle at reference time. t ϕ ( X, Γ 0 = 0 reference time t [ 0, T] Γ 0 Ω or Ω t : deformed (or presen configuration, at present time t. Γ or Γ t : deformed boundary. x : Position vector of the same particle at present time. current time Initial, reference or undeformed configuration Ω 0 X Ω x Present or deformed configuration 1

13 Material and Spatial Coordinates The position vector of a given particle can be expressed in: Non-deformed or Reference Configuration [ X] X1 X = X = Y material coordinates (capital letter) X 3 Z Deformed or Present Configuration [ x] x1 x = x = y spatial coordinates (small letter) x 3 z 13

14 Equations of Motion The motion of a given particle is described by the evolution of its spatial coordinates (or its position vector) over time. x= ϕ ( particle label, = x( particle label, x = ϕ ( particle label, i { 1,,3} The Canonical Form of the Equations of Motion is obtained when the particle label is taken as its material coordinates not T x = ϕ ( X, = x( X, particle label { X1 X X3} X x = ϕ ( X, X, X, i { 1,, 3} i i φ (particle label, is the motion that takes the body from a reference configuration to the current one. i i

15 Inverse Equations of Motion The inverse equations of motion give the material coordinates as a function of the spatial ones. Γ 0 ϕ ( X, Γ Ω Ω 0 X 1 ϕ ( x, x not 1 X = ϕ x = X x Xi = ϕi x1 x x3 t i (, (, 1 (,,, ) { 1,, 3} 15

16 Mathematical restrictions for φ and φ -1 defining a physical motion Consistency condition ( ) ϕ X,0 = X, as X is the position vector for t=0 Continuity condition 1 ϕ C Biunivocity condition, φ is continuous with continuous derivatives 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: ϕ ( X, ϕ i J = = det 0 X Xj The Jacobian of the equations of motion should be strictly positive ϕ ( X, ϕ i J = = det > 0 X Xj density is always positive (to be proven) 16

17 Example The spatial description of the motion of a continuous medium is given by: (, ) x X Find the inverse equations of motion. t t x1 = X1e x = Xe t t t x = X e y = Ye x3 5X1t X 3e = + z = 5Xt + Ze t t 17

18 Example - Solution t x1 = Xe 1 t x( X, x = Xe x3 = 5Xt 1 + Xe 3 t Check the mathematical restrictions: Consistency Condition ϕ X,0 = X? 1 Continuity Condition ϕ C? Biunivocity Condition? Density positive? ( ) 0 Xe 1 X1 0 x( X, t= 0) = Xe = X = X 0 5X1 0+ Xe 3 X 3 t i t t t t t J = = = 0 e 0 = e e e = e 0 t j x x x X X X x x x x X X X X 1 3 t x x x X X X e 5t e ( X, ϕ J = > 0 X J t = e > 0 18

19 Example - Solution t x1 = Xe 1 t x( X, x = Xe x3 = 5Xt 1 + Xe 3 t Calculate the inverse equations: x x = Xe X = = xe e t 1 t t 1 x x = Xe X = = xe e t t t ( ( ) ) x 5Xt x = 5Xt+ X e X = = x 5 xe t e = xe 5txe e t 3 1 t t t 4 t t X X = xe t t ( x, X xe t 4t X 3 = x3e 5tx1e ϕ = = 19

20 1.3 Descriptions of Motion Ch.1. Description of Motion 0

21 Descriptions of Motion The mathematical description of the particle properties can be done in two ways: Material (Lagrangian) Description Spatial (Eulerian) Description 1

22 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.

23 Spatial or Eulerian Description 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. 3

24 Example The equation of motion of a continuous medium is: x = X Yt x= x( X, y = Xt + Y z = Xt + Z Find the spatial description of the property whose material description is: ρ X + Y + Z 1+ t ( X,Y,Z,t ) = 4

25 Example - Solution x = X Yt x= x( X, y = Xt + Y z = Xt + Z Check the mathematical restrictions: Consistency Condition φ X,0 =? 1 Continuity Condition φ C? Biunivocity Condition? ( ) X X Y 0 X x( X, t = 0) = X 0+ Y = Y = X X 0 + Z Z x x x X Y Z 1 t 0 xi y y y J = = = t 1 0= ( t = 1+ t 0 t X j X Y Z t 0 1 z z z X Y Z φ( X, Any diff. Vol. must be positive J = > 0? X J = + > 1 t 0 5

26 Example - Solution x = X Yt x= x( X, y = Xt + Y z = Xt + Z Calculate the inverse equations: x = X Yt X = x + Yt y Y y xt y Y x + Yt = Yt + Y = y xt Y = y = Xt + Y X = t 1+ t t y xt x + xt + yt xt x + yt X = x + Yt = x + t = = 1+ t 1+ t 1+ t x + yt z + zt + xt + yt z = Xt + Z Z = z + Xt = z + t = 1+ t 1+ t x + yt X = 1+ t 1 y xt ϕ X ( x, = Y = 1+ t z + zt + xt + yt Z = 1+ t 6

27 Example - Solution Calculate the property in its spatial description: x + yt X = 1+ t 1 y xt ϕ X ( x, = Y = 1+ t z + zt + xt + yt Z = 1+ t ρ ( X,Y,Z,t ) ρ x + yt y xt z + zt + xt + yt + + X + Y + Z 1+ t 1+ t 1+ t x + y + yt + yt + z + zt = = = + t + t + t ( X,Y,Z,t ) ( 1 ) ( 1 ) ( t ) ( ) X + Y + Z x+ y + t+ t + z + t = = ρ 1+ t 1+ ( x, y,z,t ) 7

28 1.4 Time Derivatives Ch.1. Description of Motion 8

29 Material and Local Derivatives The time derivative of a given property can be defined based on the: Material Description Γ(X, TOTAL or MATERIAL DERIVATIVE Variation of the property w.r.t. time following a specific particle in the continuous medium. Γ X,t material derivative t Spatial Description γ(x, LOCAL or SPATIAL DERIVATIVE Variation of the property w.r.t. time in a fixed spot of space. local derivative γ t ( x, ( ) partial time derivative of the material description of the propery partial time derivative of the spatial description of the propery 9

30 Convective Derivative 30 Remember: x=x(x,, therefore, γ(x,=γ(x(x,,=γ(x, The material derivative can be computed in terms of spatial descriptions: not d not D Γ( X,) t material derivative = γ ( x, = γ ( x, = = dt Dt t d γ ( x, γ x γ ( x, t i ) γ x = γ ( x( X,, = + = + = dt t x i t t x t γ γ (,) t (,) t x x vx i t i vx (, γ ( x, Generalising for any property: material derivative ( x, ) χ( x, ) dχ t t = + v x dt t (, χ ( x, local convective rate of derivative change (convective derivative) REMARK The spatial Nabla operator is defined as: ê i x i

31 Convective Derivative Convective rate of change or convective derivative is implicitly defined as: v ( ) 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 d ( ) = 0 dt = t 31

32 Example Given the following equation of motion: x = X + Yt + Zt x(x, y = Y + Zt z = Z + 3Xt And the spatial description of a property ρ( x, = 3x + y + 3t, Calculate its material derivative. Option #1: Computing the material derivative from material descriptions Option #: Computing the material derivative from spatial descriptions 3

33 Example - Solution x = X + Yt + Zt x(x, y = Y + Zt z = Z + 3Xt ρ x, t = 3x + y + 3 ( ) t 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, : ( x ) ( xx ) ( X ) ( ) ( ) ρ, t = ρ (,, t = ρ, t = 3 X + Yt + Zt + Y + Zt + 3t = 3X + 3Yt + Y + 7Zt + 3t Calculate its material derivative as the partial derivative of the material description: ( x ) ( ) dρ, t ρ X, t = = ( 3X + 3Yt+ Y + 7Zt+ 3 = 3Y + 7Z + 3 dt t t x= xx (, ( x ) ( X ) dρ, t ρ, t = = 3+ 3Y + 7Z dt t x= xx (, 33

34 Example - Solution x = X + Yt + Zt x(x, y = Y + Zt z = Z + 3Xt ρ x, t = 3x + y + 3 ( ) t 34 Option #: Computing the material derivative from spatial descriptions dρ( x, ρ( x, = + v( x, ρ ( x, dt Applying this on ρ(x, : ( x, t ρ = ( 3x+ y+ 3 = 3 t t T Y + Z x T v( x, ( ) ( X Yt Z, ( Y Z, ( Z 3 X [ Y Z, Z, 3X ] Z = = = + = x= xx, t t t t t 3X T T ρ( x, ρ( x, ρ( x, ρ ( x, =,, = ( 3x+ y+ 3, ( 3x+ y+ 3, ( 3x+ y+ 3 = x y z x y z 3 T = [ 3,, 0 ] = 0

35 Example - Solution x = X + Yt + Zt x(x, y = Y + Zt z = Z + 3Xt ρ x, t = 3x + y + 3 ( ) t Option # The material derivative is obtained: dρ ( x, dt (, t ) x= xx 3 = 3 + [ Y + Z, Z,3 X] = 3+ 3Y + 3Z + 4Z v T 0 ρ v ( ρ) dρ ( x, dt = (, x xx = 3+ 3Y + 7Z 35

36 1.5 Velocity and Acceleration Ch.1. Description of Motion 36

37 Velocity Time derivative of the equations of motion. Material description of the velocity: Time derivative of the equations of motion x( X, V( X, = t xi ( X, Vi ( X, = i 1,,3 t REMARK Remember the equations of motion are of the form: (, (, x= ϕ X = x X Spatial description of the velocity: Velocity is expressed in terms of x using the inverse equations of motion: V( X( x,, v( x, not 37

38 Acceleration 38 Material time derivative of the velocity field. Material description of acceleration: Derivative of the material description of velocity: V( X, A( X, = t Vi ( X, Ai ( X, = i 1,,3 t Spatial description of acceleration: A(X, is expressed in terms of x using the inverse equations of motion: A X( x,, t a x, t ( ) ( ) Or a(x, is obtained directly through the material derivative of v(x,: dv( x, v( x, a( x, = = + v( x, v( x, dt t dv i( x, v i( x, vi ai( x, = = + v k ( x, ( x, i 1,, 3 dt t xk

39 Example Consider a solid that rotates at a constant angular velocity ω and has the following equation of motion: x ( ) ( ) x = R sin ωt+φ ( R, φ, y = R cos ωt +φ label of particle (non - canonical equations of motion) Find the velocity and acceleration of the movement described in both, material and spatial forms. 39

40 Example - Solution x = R y = R sin cos ( ωt + φ) ( ωt + φ) Using the expressions ( ) ( ) sin a± b = sin a cosb± cos a sin b cos a± b = cos a cosb sin a sin b The equation of motion can be rewritten as: ( ) ( ) ( ) ( ) ( ) ( ) x= R sin ω t+φ = R sin ωt cos φ+ R cos ωt sin φ y = R cos ω t +φ = R cos ω t cos φ R sin ω t sin φ For t=0, the equation of motion becomes: X = R sin φ Y = R cos φ Therefore, the equation of motion in terms of the material coordinates is: = Y = X ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x= R sin ω t+φ = R sin ωt cos φ+ R cos ωt sin φ= X cos ω t + Ysin ωt y = R cos ω t+φ = R cos ωt cos φ R sin ωt sin φ= X sin ω t + Ycos ωt = Y = X 40

41 Example - Solution ( ) sin ( ) ( ) cos( ) x = X cos ω t + Y ωt y = Xsin ω t + Y ωt The inverse equation of motion is easily obtained ( ) sin ( ) x= X cos ω t + Y ωt ( ) cos( ) y = X sin ω t + Y ωt X X ( ω ) ( ω cos( ) ( ω x Ysin t = cos y+ Y ωt = sin ( ) ( ω cos( ) ( ω x Ysin ωt y+ Y ωt = cos sin 41 X ( ( ) ( )) ( ) x ( ) ( ) ( ω cos( ω sin ( ) = cos( ω ( ω ) ( ω ) = ( ω ) + ( ω ) xsin t Ysin t ycos t Ycos t ( ) ( ω ) + ( ω ) = ( ω ) + ( ω ) xsin t ycos t Y cos t sin t = 1 ( ) ( ) ( ) ( ) ( ) x xsin ω t + ycos ωt sin ωt xsin ωt ycos ωt sin ωt = = = cos ωt cos ωt cos ωt cos ωt 1 sin = x y ωt

42 Example - Solution So, the equation of motion and its inverse in terms of the material coordinates are: (, x X (, X x ( ) sin ( ) ( ) cos( ) x= X cos ω t + Y ωt y = X sin ω t + Y ω t ( ) sin ( ) ( ) cos( ) X = xcos ωt y ωt Y = xsin ω t + y ω t canonical equations of motion inverse equations of motion 4

43 Example - Solution (, x X ( ) sin ( ) ( ) cos( ) x= X cos ω t + Y ωt y = X sin ω t + Y ω t Velocity in material description is obtained from V( X, = (, x X t (, V X x = ω + ω x( X, t t = = t y = ω + ω t t ( X cos( Ysin ( ) ( X sin ( Ycos( ) (, V X ( ) cos( ) ( ) sin ( ) V x X ω sin ω t + Y ω ωt = = V y X ω cos ωt Y ω ωt 43

44 Example - Solution ( ) sin ( ) ( ) cos( ) ( ) sin ( ) ( ) cos( ) X = xcos ωt y ωt Y = xsin ω t + y ω t x= X cos ω t + Y ωt y = X sin ω t + Y ω t Velocity in spatial description is obtained introducing X(x, into V(X,: y cos( ) sin ( ) v x X sin ( Y cos( v( x, V( X( x,, sin ( ) cos( ) v y X cos( Y sin ( x Alternative procedure (longer): ωxcos( ω sin ( ω +ωysin ( ω +ωxsin( ω cos( ω +ωycos ( ω v( x, ω xcos ( ω +ωysin ( ω cos ( ω ωxsin ( ω ωycos ( ω sin ( ω ω x( sin ( ω cos ( ω cos ( ω sin ( ω ) +ωy sin ( ω + cos ( ω x= X ω t + Y ωt y = X ω t + Y ω t ω + ω ω ωy = = = = ω ω ω ωx = = ( ) = 0 = 1 = ωx( sin ( ω + cos ( ω ) +ωy( sin ( ω cos( ω cos( ω sin ( ω ) = 1 = 0 44 (, v x v x ω y = = v y ω x

45 Example - Solution (, V X ( ) cos( ) ( ) sin ( ) X ω sin ω t + Y ω ωt = X ω cos ωt Y ω ωt Acceleration in material description is obtained applying: A( X, = (, V X t (, A X Vx = ω cos ω ω ω V( X, t = = t Vy = X ω sin ωt Y ω ωt t ( ) sin ( ) X t Y t ( ) cos( ) (, A X ( ω ) + ( ω ) ( ) cos( ) A x X cos t Ysin t = = ω A y X sin ω t + Y ωt 45

46 Example - Solution ( ) sin ( ) ( ) cos( ) X = xcos ωt y ωt Y = xsin ω t + y ω t (OPTION #1) Acceleration in spatial description is obtained by replacing the inverse equation of motion into A(X,: ( ) A X( x ) ( ) ( ( ) ( )) ( ) ( ( ) ( )) ( ) xcos ωt ysin ωt cos ω t + xsin ω t + ycos ωt sin ωt ( xcos( ω ysin ( ω ) sin ( ω + ( xsin ( ω + ycos( ω ) cos( ω a x, t =, t, t = (, a x (, a x = ω ( cos ( ω ) + sin ( ω )) + sin ( ω ) cos( ω ) + cos( ω ) sin ( ω ) = 1 ( ) x t t y t t t t = 0 = ω ( ) ( cos( ω ) sin ( ω ) + sin ( ω ) cos( ω )) + sin ( ω ) + cos ( ω ) x t t t t y t t = 0 = 1 a x ω x = = a y ω y 46

47 Example - Solution (, v x v x ω y = = v y ω x (OPTION #): Acceleration in spatial description is obtained by directly calculating the material derivative of the velocity in spatial description: (, ) v( x, ) dv x t t a( x, = = + v( x, v( x, dt t ωy x a( x, = + [ ω y, ω x] [ ω y, ω x] = t ωx y ( ωy) ( ωx) 0 x x 0 ω ω x = + [ ωy, ωx] = [ ωy, ωx] = 0 0 ω ( ωy) ( ωx) ω y y y ω x a( x, = ω y 47

48 1.6 Stationarity and Uniformity Ch.1. Description of Motion 48

49 Stationary properties A property is stationary when its spatial description is not dependent on time. χ( x, = χ( x) The local derivative of a stationary property is zero. χ ( x, χ( x, = χ( x) = 0 t The time-independence in the spatial description (stationarity) does not imply time-independence in the material description: (, = ( ) (, = ( ) χ x χ x χ X χ X REMARK This is easily understood if we consider, for example, a stationary velocity: ( ) ( ) ( ) REMARK In certain fields, the term steady-state is more commonly used. ( ) ( ) v x, t = v x = v x X, t = V X, t 49

50 Example Consider a solid that rotates at a constant angular velocity ω and has the following equation of motion: x= R sin ( ωt+ ϕ) y = R cos( ωt + ϕ) We have obtained: Velocity in spatial description (, v x Velocity in material description (, V X v x ω y = = v y ω x stationary ( ) cos( ) ( ) sin ( ) V x X ω sin ω t + Y ω ωt = = V y X ω cos ωt Y ω ωt 50

51 Uniform properties A property is uniform when its spatial description is not dependent on the spatial coordinates. χ ( x, = χ( If its spatial description does not depend on the coordinates (uniform character of the property), neither does its material one. χ x, t = χ t χ X, t = χ t ( ) ( ) ( ) ( ) 51

52 1.7 Trajectory (path-line) Ch.1. Description of Motion 5

53 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

54 Equation of the trajectories 54 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*. ( ) ϕ( X ) ( ) ϕ ( X ) x t =, t = φ() t X= X* xi t =, t = φ() i * i t i X= X Also, from the velocity field in spatial description, v(x,: A family of curves is obtained from: dx( = v( x(, x= φ ( C1, C, C3, [1] dt Particularizing for a given particle by imposing the consistency condition in the reference configuration: ( ) ( ) ( ) x t = X X = φ C1, C, C3, 0 C 0 i = χi X [] t= Replacing [] in [1], the equation of the trajectories in canonical form ( C1( ), C( ), C3( ), ϕ(, x= φ X X X = X

55 Example Consider the following velocity field: (, v x Obtain the equation of the trajectories. ω y = ω x 55

56 Example - Solution We integrate the velocity field: ( ) dx( = v x ( x, = ω = v( x, dy ( = v ( x, = ω dx t dt dt dt This is a crossed-variable system of differential equations. We derive the nd eq. and replace it in the 1 st one, y y x d y t dt ( ) dx( = ω = ω dt ( ) y t y +ω y = 0 56

57 Example - Solution The characteristic equation: r +ω =0 Has the characteristic solutions: r =± i ω j {1, } And the solution of the problem is: j iwt iwt { 1 } 1 ( ) ( ) y( = Real Part Z e + Z e = C cos ω t + C sin ωt And, using dy x, we obtain dt = ω 1 dy 1 x = = C ω sin ( ω + C ω cos( ω ω dt ω ( 1 ) So, the general solution is: ( 1,, ) = 1 sin ( ω ) cos( ω ) (,, ) = cos ( ω ) + sin ( ω ) x C C t C t C t y C1 C t C1 t C t 57

58 Example - Solution ( 1,, ) = 1 sin ( ω ) cos( ω ) (,, ) = cos ( ω ) + sin ( ω ) x C C t C t C t y C1 C t C1 t C t The canonical form is obtained from the initial conditions: x (,,0) C C = 1 This results in: X = 0 = 1 X = x( C1, C,0 ) = C1 sin ( ω 0 ) C cos( ω 0 ) = C Y = y( C1, C,0 ) = C1 cos( ω 0 ) + C sin ( ω 0) = C 1 = 1 = 0 xx (,) t ( ) ( ) ( ) ( ) x= Y sin ω t + X cos ωt y = Y cos ω t X sin ω t 58

59 1.8 Streamline Ch.1. Description of Motion 59

60 Streamline The streamlines are a family of curves which, for every instant in time, are the velocity field envelopes. Y time t 0 X Y time t 1 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. X REMARK Two streamlines can never cut each other. Is it true? 60

61 Equation of the Streamlines The equation of the streamlines is of the type: d x d y dz ( ) d x = = = dλ = ds = v vx vy vz dλ Also, from the velocity field in spatial description, v(x,t*) at a given time instant t*: A family of curves is obtained from: dx( λ ) = v( x( λ), t* ) x= φ( C 1, C, C 3, λ, t* ) dλ Where each group ( C 1, C, C 3) 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. v x v z v y 61

62 Trajectories and Streamlines For a stationary velocity field, the trajectories and the streamlines coincide PROOF: 1. If v(x,=v(x): Eq. trajectories: dx( = v x(, t x= φ C1, C, C3, t dt Eq. streamlines: ( ) ( ) ( ) dx λ ( ( λ), t* ) φ( C1, C, C3, λ, t* ) dλ = v x x = 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. 6

63 Trajectories and Streamlines For a stationary velocity field, the trajectories and the streamlines coincide PROOF:. If v(x,=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

64 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: at at dx( 0 d ( t ) 0 = = dt dt x 0 0 The eq. streamlines are: at at dx( λ ) 0 d ( λ ) 0 = = dλ dλ x 0 0 x at ( = 0 v 0 ( ) a t + C 1 t = 0 0 x atλ + C 1 t = 0 0 ( ) 64

65 Example Consider the following velocity field: xi vi = i 1,,3 1+ t { } Obtain the equation of the trajectories and the streamlines associated to this vector field. Do they coincide? Why? 65

66 Example - Solution Eq. trajectories: ( dx dt dxi dt ( = v x ( (, ( x( ) ) = v t, t i Introducing the velocity field and rearranging: i dx dt i xi = i 1+ t dx x i i dt = i 1+ t Integrating both sides of the expression: 1 1 dxi = dt x 1+ t ln xi = ln ( 1+ + lnci = lnci( 1+ i i 66 The solution: i i ( 1 ) x = C + t i

67 Example - Solution Eq. streamlines: ( λ ) dx dλ dxi dλ ( = v x ( ( λ ), t *) * ( x( λ ) t ) = v, i Introducing the velocity field and rearranging: i i dxi xi i dλ = 1+ t Integrating both sides of the expression: 1 1 dxi = dλ x 1+ t ln x i dx λ = + K 1+ t x i i i dλ = i 1+ t i λ K λ 1+ t + i K 1+ t i xi = e = e e = C i 67 The solution: i i λ 1+ t x = Ce i

68 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

69 1.9 Control and Material Surfaces Ch.1. Description of Motion 81

70 Control Surface A control surface is a fixed surface in space which does not vary in time. { f( xyz) } Σ= : x,, = 0 Mass (particles) can flow across a control surface. 8

71 Material Surface 84 A material surface is a mobile surface in the space constituted always by the same particles. In the reference configuration, the surface Σ 0 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 { X F ( XYZ ) } Σ : =,, = 0 0 ( ) ( ) F X, Y, Z = F X(,), x t Y(,), x t Z(,) x t = f(,) x t = f(, x y, z,) t { x ( ) } Σ t : = f xyzt,,, = 0 The set of spatial points belonging to the the surface depends on time The material surface moves in space

72 Material Surface Necessary and sufficient condition for a mobile surface in space, implicitly defined by the function f( xyzt,,, ), 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: d F( X) f ( x, f ( xx (,, = F( X, 0 = f ( x, = = 0 dt t Sufficient: if the material derivative of f(x, is null: d F( X,) t f ( x, f ( xx (,), t = F( X, 0 = f ( x, = F( X,) t F( X) dt t { ( ) } { ( ) } The surface Σ t : = x f x, t = 0 = X F X = 0 contains always the same set the of particles (it is a material surface) 85

73 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. { ( ) 0} V : = x f x REMARK The function f(x) is defined so that f(x)<0 corresponds to the points inside V. Particles can enter and exit a control volume. 87

74 Material Volume A material volume is a (mobile) volume enclosed inside a material boundary or surface. In the reference configuration, the volume V 0 will be defined in terms of the material coordinates: { X F( X) } V0 : = 0 The particles X in the volume are the same at all times In spatial description, the volume V t will depend on time. { x ( x ) } V : = f, t 0 t The set of spatial points belonging to the the volume depends on time The material volume moves in space along time 89

75 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

76 Summary Ch.1. Description of Motion 9

77 Summary 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: not (, (, (,,, ) { 1,, 3} x= ϕ X = x X x = ϕ X X X t i i i 1 3 Reference or non-deformed not 1 X= ϕ x = X x i i (, (, 1 (,,, ) { 1,, 3} X = ϕ x x x t i 1 3 Mathematical Restrictions: φ( X,0) = X Consistency Condition 1 Continuity Condition φ C φ Biunivocity Condition ( X, J = 0 X Any differential volume must be positive J = ( X, φ X > 0 Actual or deformed 93

78 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

79 Summary (cont d) Time derivatives Velocity Material description: Spatial description: Acceleration Material description: Spatial description: ( x, ) χ( x, ) dχ t t = + v x dt t material derivative x( X, (, = t ( ) = V X( x ) V X ( ) v x, t, t, t V( X, A( X, = t (, = A X( x,, t a x local derivative ( ) (, χ ( x, convective rate of change (, ) v( x, ) dv x t t a( x, = = + v x, t v x, t dt t ( ) ( ) A property is stationary when its spatial description is independent of time. χ ( x, χ( x, = χ( x) = 0 t A property is uniform when its spatial description is independent of the spatial coordinates. χ( x, = χ( 95

80 Summary (cont d) Trajectory or pathline: locus of the positions occupied by a given particle in space throughout time x( = ϕ ( x, dx( X= X* = v x(, t x= φ C, C, C, t dt ( ) ( 1 3 ) Streamlines: family of curves which, for every instant in time, are the velocity field envelopes. dx dy dz dx( λ ) = = v x( λ), t* x φ C, C, C, λ, t* u v w ( ) ( 1 3 ) dλ = = Streamtube: surface composed of streamlines which pass through the points of a closed contour fixed in space. ( C1, C, C3( C1, C,, λ ( C1, C,, ( C1, C, x= f = h In a stationary problem, the trajectories and streamlines coincide. 96

81 Summary (cont d) Material surface or boundary: mobile surface in the space constituted always by the same particles. In reference configuration: Σ 0 : = { X F( X, Y, Z ) = 0} In spatial configuration: Σ : = { x f ( x, y, z, = 0} 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. Σ= : { X f( xyz,, ) = 0} Material volume: volume enclosed inside a material boundary or surface. It is always constituted by the same particles. In reference configuration: V : = X F X 0 0 In spatial configuration: : = x f x, t 0 V t t ( x, ) { ( ) } { ( ) } df t f = + v f = 0 x Σt t dt t Control volume: group of fixed points in space situated in the interior of a closed control surface, which does not vary in time. { ( ) 0} V : = x f x 97