STiCM. Select / Special Topics in Classical Mechanics. STiCM Lecture 26. Unit 8 : Gauss Law; Equation of Continuity

Transcription

1 STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics Indian Institute of Technology Madras Chennai STiCM Lecture 26 pcd@physics.iitm.ac.in Unit 8 : Gauss Law; Equation of Continuity Essential tools to study Fluid Mechanics / Electrodynamics, etc. 1

2 Previous Unit: Unit 7 Physical examples of fields. Potential energy function. Gradient, Directional Derivative,. 2

3 Mathematical Connections.. Potentials Differentiation d uˆ ds uˆ lim s0 r dr s ds Fields Integration / Constant of Integration Boundary Value problem SCALAR POINT FUNCTIONS VECTOR POINT FUNCTIONS 3

4 Gauss Law; Equation of Continuity. Hydrodynamics & Electrodynamics illustrations. Johann Carl Friedrich Gauss Learning goals: What can cause a change in the When there is no source and no sink, the density of matter in a volume element can change if and only of matter density of the fluid inside the kettle? flows in, or out, of that region across the surface that At what rate can that density bounds that volume region. change? The divergence theorem : an exact mathematical expression of a conservation principle. 4

5 The result is equally consequential with regard to fields just as well as for matter. We shall develop further handle on methods of vector calculus and apply the techniques to study fluid dynamics and electrodynamics. Johann Carl Friedrich Gauss and Wilhelm Weber volume region dv A( r) ( ) ˆ A r dsn surface Terrestrial Magnetism enclosing that region 5

6 Consolidated expressions for the GRADIENT Cartesian Coordinate System =eˆ eˆ eˆ x y z x y z Cylindrical Polar Coordinate System 1 =eˆ eˆ eˆ z z d uˆ ds r uˆ lim s0 s s r dr ds Spherical Polar Coordinate System 1 1 =eˆ eˆ eˆ r r r r sin 6

7 d uˆ ds The GRADIENT is a vector operator it is of course not a vector. The operator would operate on an operand and generate new entities as a result of the operation. Operand : SCALAR POINT FUNCTION. RESULT : Other operations using GRADIENT OPERATOR Ar ( ) : DIVERGENCE of a VECTOR POINT FUNCTION ( ) : CURL of a VECTOR POINT FUNCTION Ar 7

8 GRADIENT of SCALAR POINT FUNCTION. RESULT : Other operations using GRADIENT OPERATOR A( r) : DIVERGENCE of a VECTOR POINT FUNCTION This is NOT a scalar product of two vectors! Ar ( ) : CURL of a VECTOR POINT FUNCTION This is NOT a vector product of two vectors! 8

9 Field Lines Field strength: E( r) q r 2 eˆ r VECTOR POINT FUNCTION Field intensity fall like 1/r 2 9

10 Vector Fields: Point function V V ( r) V ( x, y, z) V ( r,, ) V (,, z) V V ( r, t) 10

11 Vector Fields: Point function V V ( r) V ( x, y, z) V ( r,, ) V (,, z) V V ( r, t) In the continuum model, the velocity field V V ( r) is a vector point function. 11

12 Ar ( ) : A vector point function. Define: Flux of a vector point function Flux crossing a surface A( r) surface dsnˆ Flux: additive property - obtained by integrating the quantity.. what is the direction of the unit normal? 12 what is the direction/orientation of nˆ? UNIT NORMAL TO THE SURFACE AT A GIVEN POINT... but which surface?

13 Ar ( ) : A vector point function. Flux crossing a surface A( r) surface dsnˆ Elemental directed/oriented Area 13

14 The direction of the vector surface element must be defined in a manner that is consistent with the forward-movement of a right-hand screw. Flux crossing a surface surface A() r dsnˆ C traversed one way right-hand-screw convention must be followed. C traversed 14 the other way

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16 Consider the sense/direction in which the rim of the net can be traversed right hand screw What about some other point? and what if you pinched the net and pulled it above the rim? 16

17 Are there non-orientable surfaces? The surface under consideration, however, better be a well-behaved surface! A cylinder open at both ends is not a well-behaved surface! A cylinder open at only one end is well-behaved ; isn t it already like the butterfly net? 17

18 Consider a rectangular strip of paper, spread flat at first, and given two colors on opposite sides. Now, flip it and paste the short edges on each other as shown. Is the resulting object three-dimensional? How many edges does it have? How many sides does it have? 18

19 How is the earring? It is MOBIUS!!! 19

20 The Möbius strip used to be common in belt drives (eg. car fan belt). Modern belts are made from several layers of different materials, with a definite inside and outside, and do not have a twist. 20

21 Maurits Cornelius Escher The laws of mathematics are not merely human inventions or creations. They simply 'are'; they exist quite independently of the human intellect. The most that any(one)... can do is to find that they are there, and to take cognizance of them. 21

22 Flux of a vector field nr ˆ( ) : unit vector normal to the surface at the point r Ar ( ) : A vector point function Flux crossing a surface surface A() r dsnˆ 22

23 Flux crossing a surface surface A() r dsnˆ Source Is there any net accumulation of the flux in a volume element? Sources and Sinks may be present in the region! Sink What happens when the size of the volume element shrinks, becoming infinitesimally small? 23

24 Consider a mass/charge density or crossing a certain cross-section of area at a certain rate. m c ( )v( ) has the dimensions m r r ML LT ML T ( r )v( r ) has the dimensions c QL LT QL T Amount of charge crossing unit area in unit time Amount of mass/charge crossing unit area in unit time 24

25 Consider a point P( x, y, z ) in a region of the vetor field Ar ( ) How shall we determine the flux crossing a surface; surface A() r dsnˆ? 25

26 Z Y δx (0,0,0) δz X δy 26

27 Two faces parallel to the xy plane The point P( x, y, z ) is in the region of the vectore field Ar ( ) δy NET Flux through the xy-face, (perpendicular to eˆ z ), is z z Az x0, y0, z0 Az x0, y0, z0 xy 2 2 Z δx eˆz (x 0,y 0,z 0 ) Y eˆz δz X Note! Only the z-component contributes to the total flux through the faces parallel to the xy plane. Flux crossing a surface surface A() r dsnˆ 27

28 Flux crossing a surface surface A() r dsnˆ NET Flux through the xy-face, (perpendicular to eˆ z ), is Z δx Y z z Az x0, y0, z0 Az x0, y0, z0 xy 2 2 δy (x 0,y 0,z 0 ) δz X Flux crossing all surface A the six surface elements that enclose the cell A() r dsnˆ z z x y A z z ( x0, y0, z0 ) x0, y0 z0 z (, ) V Adding the flux through all faces, total flux whole cube A A x y A z A( r) dsnˆ dv x y z 28

29 Flux crossing a surface surface closed surface A() r dsnˆ Adding the flux through all faces, total flux A A x y A z A( r) dsnˆ dv x y z The integrand of the volume integral is called the divergence of the vector. δy Z δx (x 0,y 0,z 0 ) Y δz X div A A A x y Az x y z A volume region d A( r) A( r) dsnˆ surface enclosing that region Gauss Divergence Theorem 29

30 Divergence of a polar vector is a scalar Divergence of an axial-vector is a pseudo-scalar 30

31 We shall take a break here. Questions? Comments? pcd@physics.iitm.ac.in Next: L27 Equation of Continuity 31

32 STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics Indian Institute of Technology Madras Chennai STiCM Lecture 27 pcd@physics.iitm.ac.in Unit 8 : Gauss Law; Equation of Continuity 32

33 Flux crossing a surface surface A() r dsnˆ Adding the flux through all faces, total flux volume region d A( r) A( r) dsnˆ surface enclosing that region Gauss Divergence Theorem 33

34 volume region d E( r) Er ( ) dsnˆ surface enclosing that region Application: electric intensity field due to a point charge volume region q d Er ˆ 2 ( ) e 2 r r sindder 4 surface 0r enclosing that region ˆ volume 0 0 volume volume 0 region region region q 1 d Er ( ) d d Er ( ) 0 Differential (or point ) form of the Gauss s law 34

35 Flux crossing a surface surface closed surface A() r volume region dsnˆ A A x y A z A( r) dsnˆ dv x y z The integrand of the volume integral is called the divergence of the vector. dv A( r) A( r) dsnˆ surface enclosing that region δy Z δx (x 0,y 0,z 0 ) Gauss A A x y Az div A A x y z Cartesian expression of divergence of the vector not its definition! Y δz Divergence Theorem 35 X

36 Physical meaning of divergence ; definition free from coordinate system volume region dv A( r) A( r) dsnˆ surface enclosing that region Take the limit of the ratio of total flux over δs to δv diva lim enclosing surface V 0 V A(r) nds ˆ A flux per unit volume, at that point remember: flux is defined through a surface, whereas divergence is defined at a point Flux is a scalar quantity. It is not a scalar field; it is not a local quantity it is not a point function. Divergence is a scalar field; it is a scalar point function, it is defined at each point of space 36

37 Gauss s Divergence Theorem If a volume V is bounded by a surface S, then, for vector A, The surface integral of the normal component of a vector A taken over a closed surface is equal to the integral of the divergence of A taken over the volume enclosed by the surface volume region dv A( r) A( r) dsnˆ since S is a closed surface, the unit normal (elemental area) is the outward normal surface enclosing that region ˆn of ds Conversion of a surface integral to a volume integral. 37

38 volume region dv A( r) A( r) dsnˆ surface enclosing that region Physical Meaning: Integration of the faucets (source of vector field) over a volume is equal to the flux flowing out through the surface enclosing the volume. 38

39 div A A x y Az A A x y z A 1 eˆ eˆ eˆ e ˆ (,, ) e ˆ (,, ) e ˆ z A z A z z Az (,, z) z vector algebra There are TWO OPERATIONS here! A How shall we express divergence in cylindrical polar coordinate system? calculus take derivatives eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) 1 eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) ˆ e e ˆ (,, ) e ˆ (,, ) e ˆ z A z A z z Az (,, z) z 39

40 A A eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) 1 eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) ˆ e e ˆ (,, ) e ˆ (,, ) e ˆ z A z A z z Az (,, z) z Note that the components A,A,A z each depends on (,,z)... but the unit vectors e ˆ,e ˆ also depends on eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) 1 eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) ˆ e e ˆ (,, ) e ˆ (,, ) e ˆ z A z A z z Az (,, z) z 40

41 e e A (,, z) e A (,, z) e zaz(,, z) A ˆ ˆ ˆ ˆ 1 eˆ e ˆ (,, ) e ˆ (,, ) e ˆ A z A z zaz(,, z) ˆ e e ˆ (,, ) e ˆ (,, ) e ˆ z A z A z z Az (,, z) z eˆ ˆ e 0, eˆ, eˆ eˆ 0, eˆ 1 A A(,, z) A(,, z) 1 A (,, z) Az (,, z) 41 z

42 Expression for divergence in spherical polar coordinate system eˆ r 0 r eˆ r eˆ eˆ r sineˆ eˆ 0 r eˆ eˆ r eˆ coseˆ A 1 1 eˆ eˆ eˆ e ˆ (,, ) e ˆ (,, ) e ˆ r r Ar r A r A ( r,, ) r r r sin (,, ) (,, )sin (,, ) 2 r A 2 r r A r A r r r r sin r sin eˆ 0 r eˆ 0 eˆ coseˆ sin eˆ r 42

43 Examples for solenoidal and nonsolenoidal fields A0 Influx balances the outflux Solenoidal Example: B ˆ A=(x-y)e x ( x y) e y A2 A A x y A z div A A x y z 43 ˆ

44 Flux crossing a surface surface A() r da Source Is there any net accumulation of the flux in a volume element? Sources and Sinks may be present in the region! Sink What happens when the size of the volume element shrinks, becoming infinitesimally small? 44

45 Consider a mass/charge density or crossing a certain cross-section of area at a certain rate. m c ( )v( ) has the dimensions m r r ML LT ML T ( r )v( r ) has the dimensions c QL LT QL T Amount of charge crossing unit area in unit time Amount of mass/charge crossing unit area in unit time 45

46 Consider a mass/charge density or crossing a certain cross-section of area at a certain rate. Amount of mass/charge crossing unit area in unit time: Physical quantity of interest: Density x Velocity m c J( r) ( r)v( r) Current Density Vector current crossing unit area Mass/Charge Current Density Vector Remember! Sources/Sinks may be present in the region! 46

47 Divergence theorem: Conservation principle ( rt, ) represents mass/charge density Conservation of mass or charge J( r, t) : mass/charge current density What shall we get if we integrate the flux emanating from all the six enclosing surfaces? S J ( r) nds ˆ I volume region dv J ( r ) J ( r) dsnˆ surface enclosing that region qtotal i. e. J ( r). nds ˆ dv dv t t t S V V Net current oozing out of that region. Negative sign: Outward flux is at the expense of the charge inside! 47

48 Divergence theorem: Conservation principle dv J ( r) J ( r) dsnˆ volume region surface enclosing that region S J ( r) nds ˆ I qtotal i. e. J ( r) nds ˆ t S dv - t t V V dv Compare the integrands of the definite volume integrals Integral and Differential forms of the equation of continuity: conservation principle Jr ( ) t Jr ( ) 0 t Equation of Continuity 48

49 Divergence theorem: Conservation principle volume region dv J( r) J( r) dsnˆ volume region dv J ( r) 0 t Equation of Continuity Jr ( ) t Jr ( ) 0 t Divergence theorem: In the absence of the creation or destruction of matter (no source or sink ), the density within a region of space can change only by having matter flow into or out of the region through the surface that bounds it. 49

50 We shall take a break here. Questions? Comments? pcd@physics.iitm.ac.in Next: L28 Equation of Fluid Motion continuum limit Lagrangian / Euler description of fluid flow 50

51 STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics Indian Institute of Technology Madras Chennai STiCM Lecture 28 pcd@physics.iitm.ac.in Unit 8 : Gauss Law; Equation of Fluid Motion 51

52 Divergence theorem: Conservation principle volume region qtotal i. e. J ( r) nds ˆ t S dv J ( r) J ( r) dsnˆ surface enclosing that region dv - t t V Jr ( ) t Jr ( ) 0 t S J ( r) nds ˆ I Compare the integrands of the definite volume integrals Integral and Differential forms of the equation of continuity: conservation principle V dv Equation of Continuity 52

53 We consider an incompressible fluid. Under the application of a force, a solid gets pushed/pulled/spun. or deformed. A fluid flows FLUID MECHANICS Deformation of solids, fluid flow: rheology Non-Newtonian fluids --- example paints, foams, molten plastics.. Ever walked on a fluid? 53

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55 FLUID MECHANICS Non-Newtonian fluids --- example paints, foams, molten plastics.. Matter behaves like a solid when pressure is exerted on it, and like a liquid when only little or no pressure is exerted on it. The fluid's viscosity depends not on shear but on the rate of change of shear --- complex systems; but we shall work with IDEAL liquids 55

56 Density ( r) limit V 0 m V What is the meaning of the limit V 0? Classical fluid: continuum mechanics, continuously divisible matter. V dimensions of the fluid medium V dimensions of the molecules of fluid 56

57 S does not depend on the direction of uˆ N. Pascal ' s law. Stress at the point P is S. P ˆ u N A A Blaise Pascal The unit normal uˆn can take any orientation. Let no one say that I have said nothing new... the arrangement of the subject is new. 57

58 S does not depend on the direction of Pascal ' s law. uˆ N. The pressure is the same in every direction. The shape of the container does not matter. The pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the container. 58

59 To understand the term ideal fluid, we first define (i) tension, (ii) compressions and (iii) shear. F A Consider the force on a tiny elemental area passing through point P in the liquid. some definitions.. S S S Stress at the point P is S. uˆ uˆ uˆ N N N S 0 S S: Tension S: S: Shear Compression P The unit normal orientation. ˆ u N A A û N can take any An ideal fluid is one in which stress at any point is 59 essentially one of COMPRESSION.

60 S uˆ N S Tension S uˆ N 0 Shear S uˆ N Ideal fluid S Compression DIRECTION uˆn of the ORIENTATION C traversed one way OF THE SURFACE ELEMENT C traversed the other way 60

61 The state of a fluid is completely determined by FIVE quantities: (1,2,3) : Three components of the velocity at each point : v( r). (4) : The pressure p( r, t). (5) : The density ( r, t). Above, we consider Eulerian position vector of a point with reference to a chosen frame of reference. It is not the position vector of any particular fluid molecule/particle. 61

62 At time t 0 At time t > t 0 In the Lagrangian viewpoint, one tracks the evolution in phase space of the entire continuous medium; has huge amount of detailed information. - not so in the Eulerian description. 62

63 In the Eulerian description, one is interested in quantities v( r) r such as the density or the velocity and p(r) (r) pressure of an arbitrary fluid particle at. 63

64 How do we track density (r) & velocity v(r) at a point in a river? Sindhu Vitastha Asikini Airavati Satadru Indus Jhelum Chenab Ravi Sutlej 64

65 How do we track density (r) & velocity v(r) at a point in a river? Deoprayag ("Divine confluence") -Bhagirathi & Alakananda, Himalayan tributaries of GANGA. grefurl= 65 J::&tbnh=96&tbnw=140&prev=/images%3Fq%3Dbhagirathi%2Briver%2Bpicture&hl=en&usg= 899jmh0E8OAKvcBWqaIJPd ON_k8=&sa=X&oi=image_result&resnum=1&ct=image&cd=1

66 How do we have TriweNi SANGAM at Prayag? 66

67 Saraswathi Jamuna Is merely the position vector of a particular point in space, or is it the position vector of a moving molecule of water Ganga, Yamuna or Saraswathi? Ganga Prayag Triweni Sangam 67

68 LAGRANGIAN TRACKING: The waters of Yamuna would mix with the waters of Saraswathi and bring them to Prayag, into the Ganga! 68

69 Equation of motion for fluids two basic approaches Lagrangian Approach: Follow the motion of some particle of the fluid; this must be done for all particles of the fluid Eulerian Approach: Joseph-Louis Lagrange Follow the velocity and density of fluid at a particular point; this must be done for all points in space Leonhard Euler ( ) 69

70 Quantities of interest: *velocity : v( r). * pressure: p( r, t). * density : ( r, t). H E G F r P D A C B Mass Current Density Vector J ( r, t) Dimensions ( r, t)v( r, t) : ML 2 T 1 Measure of the amount of mass crossing unit area in unit time. Amount of mass of fluid crossing face EFGH in unit time = lim t0 J x y z t y z x, EFGH lim m t lim t0 t0 v x V t yz 70

71 71 P r A B C D E F G H Amount of mass of fluid crossing face EFGH in unit time = t V t m t t 0 0 lim lim z y t z y x x t v lim 0 Amount of mass of fluid crossing face ABCD in unit time z y x x J r J P x x 2 ) ( V x J z y x x J P x P x Net OUTWARD flux through the two faces orthogonal to x-axis = ê x z y x x J r J P x x 2 ) ( z y t z y x x t v lim 0

72 Net OUTWARD flux through the two faces orthogonal to x-axis = J x x J x x P xyz P V H E ê x Net OUTWARD flux through the whole parallelepiped, per unit volume = J J x y J z J x P y z P P The choice of the term DIVERGENCE is thus well justified. G F r P P D A C B This must be equal to t J quantity 72 t

73 Equation of Continuity Conservation of Matter J ( r, t) ( r, t) t J ( r, t) uˆ mass flux in the direction J ( r, t) ( r, t) t 0 of uˆ 73

74 ê z ê y ê x Face 1 of the parallelepiped region Face 2 of the parallelepiped region Ideal fluid: stress at any point is essentially one of COMPRESSION. 74

75 We can now develop the EQUATION of MOTION for a Newtonian Fluid 75

76 p x F( 1) p( r) x P 2 p x F( 2) p( r) y z x P 2 p p F( 1) F(2) x x x 6 i1 F( i) p V Net force per unit volume p P yz eˆ x eˆ x yz eˆ x V eˆ x = negative gradient of pressure 1 2 ê x P 76 Ideal fluid: stress at any point is essentially one of COMPRESSION. HYDROSTATIC force There may be some additional external force acting, such as gravity.

77 Net HYDROSTATIC force acting on the parallelepiped per unit volume p External force (such as gravity) acting on the parallelepiped per unit volume lim V 0 lim V 0 lim V 0 F F F g( r) external V external m external m m V ( r) Total {hydrostatic + external (gravity)} force acting on the parallelepiped per unit volume m lim V 0 V ( r) d v dt d v dt p Mass x Acceleration Cause-Effect Newton s law: Equation of Motion g( r) 77

78 Mass x Acceleration / Cause-Effect Newton s law: Equation of Motion m lim V 0 V d v dt d v ( r) p g( r) dt r : LAGRANGIAN position vector of a moving/flowing fluid particle/molecule, not the EULERIAN position vector of a fixed point in space. r Lagrangian r(t) This is a function of time d v dt r Euler Fixed point in space, not a function of time Is the ACCELERATION of actual material/fluid particle/molecule, and not just the rate at which velocity of the fluid is changing at a fixed point in space. 78

79 d v dt Mass x Acceleration / Cause-Effect Newton s law: Equation of Motion ( r) p g( r) d v d d v( r( t), t) v x( t), y( t), z( t), t dt dt dt v dx v dy v dz v x dt y dt z dt t d v dx v dy v dz v v dt dt x dt y dt z t CONVECTIVE DERIVATIVE OPERATOR The term convection d is a reminder of the fact that in the i. e. convection process, the transport v 79 of a material particle is involved. dt v t v t

80 d v dt Mass x Acceleration / Cause-Effect Newton s law: Equation of Motion ( r) p g( r) d v( r, t) dt p ( r) g external p ( r) External force field, which we considered to be gravity v t v( r, t) d v( r, t) dt p ( r) F viscous Hydrodynamic term Viscous, frictional, dissipative term. This terms makes dry water wet - Feynman 80

81 We shall take a break here. Questions? Comments? pcd@physics.iitm.ac.in Next: L29 Unit 9 Fluid Flow / Bernoulli s principle.. but which Bernoulli? 81