STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 26

Unit 8 : Gauss’ Law; Equation of Continuity

Essential tools to study Fluid Mechanics / Electrodynamics, etc.

PCD_STiCM 1 Previous Unit: Unit 7

Physical examples of fields.

Potential energy function.

Gradient, Directional Derivative, ….

PCD_STiCM 2 d uˆ  Mathematical ds  r dr Connections….. uˆ lim s0  s ds Differentiation Potentials Fields

Integration / Constant of Integration Boundary Value problem

SCALAR VECTOR POINT FUNCTIONS POINT FUNCTIONS

PCD_STiCM 3 Johann Gauss’ Law; Carl Friedrich Equation of Continuity. Gauss Hydrodynamics & 1777 - 1855 Electrodynamics illustrations.

Learning goals: WhenWhat there can is no cause source and a nochange sink, the density in the of matter indensity a volume element of the can fluid change inside if and only the of kettle?matter flows in, or out, of that region across the surface that boundsAt what that volume rate regioncan .that density Thechange? divergence theorem : an exact mathematical PCD_STiCM 4 expression of a conservation principle. 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 http://www.gap-system.org/~history/PictDisplay/Gauss.html

 ˆ dV  A()() r   A r  dSn volume surface regionTerrestrial Magnetism enclosing that region PCD_STiCM 5 Consolidated expressions for the GRADIENT Cartesian Coordinate System d    uˆ  ds =eˆx  e ˆ y  e ˆ z x  y  z  r dr uˆ lim s0 Cylindrical Polar Coordinate System  s ds 1   sr =eˆ  e ˆ  e ˆ     z z

Spherical Polar Coordinate System 11   =eˆ  e ˆ  e ˆ r r r  r sin    PCD_STiCM 6 d uˆ  The ‘GRADIENT’ is a vector operator ds – 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

Ar( ) : CURL of a VECTOR POINT FUNCTION PCD_STiCM 7 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!

PCD_STiCM 8 Field strength: Field Lines 1 q VECTOR POINT E() r eˆ 2 r FUNCTION 4 0 r Field intensity fall like 1/r2

PCD_STiCM 9 Vector Fields: V V()(,,) r V x y z ‘Point’ V V(,) r t V(,,)(,,) r  V   z function

PCD_STiCM 10 Vector Fields: V V()(,,) r V x y z ‘Point’ V V(,) r t function V(,,)(,,) r  V   z

In the ‘continuum model’, the velocity field V  V() r is a vector point function.

PCD_STiCM 11 Ar( ): A vector point function. Define: “Flux” of a vector point function Flux crossing a surface A ( r ) dSnˆ surface Flux: additive property - obtained by integrating the quantity what is the direction/orientation of nˆ ? UNIT NORMAL TO THE SURFACE AT A GIVEN POINT .... but which surface?

.. what is the directionPCD_STiCM of the unit normal?12 Ar( ) : A vector point function.

Flux crossing a surface A ( r ) dSnˆ surface

Elemental directed/oriented ‘Area’

PCD_STiCM 13 Flux crossing a surface The direction of the vector  A() r dSnˆ surface element must be defined surface in a manner that is consistent with the forward-movement of a right-hand screw. C traversed one way

right-hand-screw convention must be C traversed followed. PCD_STiCM the other way14 PCD_STiCM 15 Consider the sense/direction in … right which the rim of the hand net can be traversed screw 

What about some other point ?

… and what if you ‘pinched’ the net and pulled it ‘above’ the rim?

PCD_STiCM 16 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 PCD_STiCMlike the butterfly net? 17 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? PCD_STiCM 18 How is the earring?

It is MOBIUS !!!

PCD_STiCM 19 http://mathssquad.questacon.edu.au/mobius_strip.html 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,PCD_STiCM and do not have a twist. 20 Maurits Cornelius Escher 1898 - 1972

http://www.mcescher.com/ “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 takePCD_STiCM cognizance of them. ” 21 Flux of a vector field

Ar( ) : A vector nrˆ( ) : unit vector point function normal to the surface at the point r

Flux crossing a surface  A() r dSnˆ surface PCD_STiCM 22 Flux crossing a surface Is there any net accumulation of  A() r dSnˆ the flux in a volume element? surface Source Sources and Sinks may be present in the region !

Sink What happens when the size of the volume element shrinks,

becoming infinitesimallyPCD_STiCM small? 23 Consider a mass/charge density mc or crossing a certain cross-section of area at a certain rate.

m (rr )v( ) has the dimensions -3 1 -2 1 ML LT ML T

-3 1 -2 1 c (rr )v( ) has the dimensions  QL LT QL T Amount of charge crossing unit area in unit time Amount of mass/charge crossing unit area in unit time PCD_STiCM 24 Consider a point P ( x0 , y 0 , z 0 ) in a region of the vetor field Ar ( )

How shall we determine the flux crossing a surface;  A() r dSnˆ ? surface PCD_STiCM 25 Z Y

δx

(0,0,0) δz X

δy PCD_STiCM 26 Z Y eˆz δx Two faces parallel to the xy plane δz

(x0,y0,z0) The point P ( x0 , y 0 , z 0 ) is in the δy X region of the vectore field Ar ( ) eˆz NET Flux through the xy -face, (perpendicular to eˆz ), is zz    Az x 0, y 0,, z 0   A z  x 0, y 0 z 0    x y 22   

Note! Only the z-component Flux crossing a surface contributes to the total flux ˆ through the faces parallel to  A() r dSn the xy plane. PCD_STiCM surface 27 Flux crossing a surface Z Y  A() r dSnˆ surface δx NET Flux through the xy -face, (perpendicular to eˆz ), is zz    A x y,, z  A x y z   x y z 0, 0 0  z  0, 0 0  (x0,y0,z0) δz 22    δy X

A Az  z z  x  y  V  z z (,)x0, y 0 z 0 (,)x0, y 0 z 0 Flux crossing all the six surface elements Adding the flux through all faces, total flux that enclose the cell A Ax y Az  A() r dSnˆ A( r ) dSnˆ      dV surface x  y  z whole PCD_STiCM 28 cube Flux crossing a surface Adding the flux through Z Y  A() r dSnˆ all faces, total flux surface A δx Ax y Az A( r ) dSnˆ     dV   x  y  z closed (x ,y ,z ) surface 0 0 0 δz X The integrand of the volume integral is δy called the divergence of the vector. A Ax y Az div AA      x  y  z

 ˆ d   A()() r   A r  dSn volume surface region enclosing that region PCD_STiCM 29 Gauss’ Divergence Theorem Divergence of a polar vector is a scalar

Divergence of an axial-vector is a pseudo-scalar

PCD_STiCM 30 We shall take a break here…….

Questions ? Comments ? [email protected] http://www.physics.iitm.ac.in/~labs/amp/

Next: L27 Equation of Continuity

PCD_STiCM 31 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 27

Unit 8 : Gauss’ Law; Equation of Continuity

PCD_STiCM 32 Flux crossing a surface Adding the flux through  A() r dSnˆ all faces, total flux surface

 ˆ d   A()() r   A r  dSn volume surface region enclosing that region

Gauss’ Divergence Theorem

PCD_STiCM 33   ˆ Application:  d E(r)   Er( )  dSn volume surface electric intensity region enclosing field due to a that region point charge

q d Er() eˆˆ r2 sin d  d  e   2 rr  volume surface 4 0r region enclosing that region

q 1   ddEr()       d  volume0 0 volume volume 0 region region region   Er()  Differential (or ‘point’) form of PCD_STiCM 34 0 the Gauss’s law Flux crossing a surface Z Y  A() r dSnˆ surface A δx Ax y Az A( r ) dSnˆ      dV closed x  y  z surface (x0,y0,z0) δz X The integrand of the volume integral is δy called the divergence of the vector. dV  A()() r  A r  dSnˆ Gauss’   Divergence volume surface region enclosing Theorem that region A Ax y Az div AA      x  y  z Cartesian expression of ‘divergence of the vector’ not its definition! PCD_STiCM 35 Physical meaning of ‘divergence’; definition free from coordinate system

 ˆ Take the limit of dV  A()() r   A r  dSn volume surface the ratio of region enclosing that total flux over δs region to δV  A(r) ndSˆ enclosing flux per unit volume, surface divAA lim    at that point V 0 V 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 spacePCD_STiCM 36 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  ˆ dV  A()() r   A r  dSn volume surface region enclosing that region

since S is a closed surface, the unit normal nˆ of dS (elemental area) is the outward normal

Conversion of a surface integral to a volume integral.

PCD_STiCM 37  ˆ dV  A()() r   A r  dSn volume surface region 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.

PCD_STiCM 38 A Ax y Az How shall we express div AA      x  y  z ‘divergence’ in cylindrical polar  A  coordinate system? 1   ˆ ˆ ˆ ˆ ˆ ˆ e e   ez  e(,,)e(,,)e(,,) A   z   A    z  z A z   z     z vector algebra There are TWO OPERATIONS here! calculus take derivatives  A  ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e   z zz A (,,)   z   1  ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z    eˆ e ˆA (,,)e  z  ˆ A (,,)e   z  ˆ A (,,)   z z PCD_STiCM   z z 39 z A   ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e   z zz A (,,)   z   1  ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z    ˆ ˆ ˆ ˆ ez eA  (,,)e  z   A  (,,)e   z  z A z (,,)   z z

Note that the components A ,A ,Az each depends on ( , ,z)

.... but the unit vectors eˆˆ ,e also depends on   ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z   1  ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z    ˆ ˆPCD_STiCM ˆ ˆ 40 ez eA  (,,)e  z   A  (,,)e   z  z A z (,,)   z z  ˆ ˆ ˆ ˆ A  e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z   1  ˆ ˆ ˆ ˆ e e A  (,,)e  z   A  (,,)e(,,)   z zz A   z    ˆ ˆ ˆ ˆ ez eA  (,,)e  z   A  (,,)e   z  z A z (,,)   z z eeˆˆ 0,eˆ ,  eeˆˆ 0,  eˆ   1  A  A(,,)(,,)  z  A   z   1  A ( ,  , z ) A (  ,  , z ) PCD_STiCM  z z 41 Expression for ‘divergence’ in spherical polar coordinate system eˆ eˆ eˆ  r  0  0  0 r r r eˆ eˆr eˆ   eˆ eˆ  0   r  ˆ er ˆ  sineˆ eˆ e    coseˆ  coseeˆˆ  sin     r  A  11   ˆ ˆ ˆ ˆ ˆ ˆ er e  e   e(,,)e(,,)e(,,) rA r r    A  r     A  r   r r  r sin    

12 1  1  r Ar ( r , ,  ) A ( r ,  ,  )sin  A ( r ,  ,  ) r2  r rsin   r sin    PCD_STiCM 42 Examples for solenoidal and nonsolenoidal fields

ˆ ˆ A0 A=(x-y)ex  (x y ) ey

Influx balances the outflux A2 Solenoidal Example: B A Ax y Az div AA     x  y  z PCD_STiCM 43 Flux crossing a surface Is there any net accumulation of  A() r da surface the flux in a volume element? Source Sources and Sinks may be present in the region !

Sink

What happens when the size of the volume element shrinks,

becomingPCD_STiCM infinitesimally small? 44 Consider a mass/charge density mc or crossing a certain cross-section of area at a certain rate.

m (rr )v( ) has the dimensions -3 1 -2 1 ML LT ML T

-3 1 -2 1 c (rr )v( ) has the dimensions  QL LT QL T Amount of charge crossing unit area in unit time Amount of mass/charge crossing unit area in unit time PCD_STiCM 45 Consider a mass/charge density mc 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

Current Density Vector J( r )  ( r )v( r ) current crossing unit area

Mass/Charge Current Density Vector Remember! Sources/Sinks may be present in the region !

PCD_STiCM 46 Divergence theorem: Conservation principle Conservation of mass or charge (rt , ) represents mass/charge density J( r , t ): mass/charge current density What shall we get if we integrate the flux emanating from all the six enclosing surfaces? dV  J()() r  J r  dSnˆ J() r ndsˆ  I    volume surface S region enclosing that region q  i. e .  J ( r ). ndsˆ  total    dV    dV SVVt  t  t Negative sign: Net current oozing out of that region. Outward flux is at the expense of the charge inside! PCD_STiCM 47 Divergence theorem: Conservation principle

dV  J()() r   J r  dSnˆ volume surface region enclosing J() r ndSˆ  I that  region S

qtotal  i. e . J ( r ) ndSˆ   dV  - dV  tt  S t VV

Compare the integrands of the definite volume integrals Integral and   Jr()   Equation Differential forms of t of the equation of  Continuity  Jr( )   0 continuity: t conservation principle PCD_STiCM 48 Divergence theorem: Conservation principle Equation of Continuity dV  J()() r   J r  dSnˆ volume region   Jr()    t dV  J( r )   0  t  volume  Jr( )   0 region 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. PCD_STiCM 49 We shall take a break here…….

Questions ? Comments ? [email protected]

Next: L28 Equation of Fluid Motion

‘continuum limit’ Lagrangian / Euler description of fluid flow

http://www.physics.iitm.ac.in/~labs/amp/

PCD_STiCM 50 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 28

Unit 8 : Gauss’ Law; Equation of Fluid Motion

PCD_STiCM 51 Divergence theorem: Conservation principle

dV  J()() r   J r  dSnˆ J() r ndSˆ  I volume surface  region enclosing S that region

qtotal  i. e .  J ( r ) ndSˆ   dV   - dV S t ttVV Compare the integrands of the definite volume integrals Integral and   Jr()   Equation Differential forms of t of the equation of  Continuity  Jr( )   0 continuity: t conservation principle PCD_STiCM 52 FLUID MECHANICS

We consider an incompressible fluid.

Under the application of a force, a solid gets pushed/pulled/spun…. or deformed.

A fluid ‘flows’ Deformation of solids, fluid flow: “rheology” Non-Newtonian fluids --- example paints, foams, molten plastics…..

Ever walked on a fluid ? PCD_STiCM 53 http://www.youtube.com/watch?v=f2XQ97XHjVw

PCD_STiCM 54 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 withPCD_STiCM “IDEAL” liquids 55 m Density (r)  limit V 0 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

PCD_STiCM 56 S does not depend on the direction of uˆN . Pascal's law.

Stress at the point P is S . A P uˆN  A

The unit normal u ˆ N can take any orientation. Blaise Pascal 1623-1662 “Let no one say that I have said nothing new...

the arrangementPCD_STiCM of the subject is new. …………”57 http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Pascal.html S does not depend on the direction of uˆN . Pascal's law.

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.

PCD_STiCM 58 some definitions….. To understand the term ‘ideal’ fluid, we first define (i) ‘tension’, (ii) ‘compressions’ and (iii) ‘shear’. Consider the force F on a tiny elemental area  A passing through point P in the liquid.

Stress at the point P is S . A P uˆN  S uˆN  S S: Tension A

S uˆ  0 S: Shear N Compression S: The unit normal u ˆ N can take any S uˆN   S orientation.

An ideal fluid is one in which stress at any point is essentially one of PCD_STiCMCOMPRESSION. 59 S uˆN  S Tension

S uˆN  0 Shear

S uˆN   S Compression Ideal fluid

DIRECTION uˆ of C traversed one N way the ‘ORIENTATION’ OF THE SURFACE

ELEMENT C traversed PCD_STiCM the other way 60 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.

PCD_STiCM 61 At time t0 At time t > t0

In the Lagrangian viewpoint, one tracks the evolution in phase space of the entire continuous medium; has huge amount of detailed information.

- notPCD_STiCM so in the Eulerian description. 62 In the Eulerian description, one is interested in quantities such as the density  ( r ) or the velocity v ( r ) and pressure p ( r ) of an arbitrary fluid particle at r . PCD_STiCM 63 How do we track density  (r) & velocity v(r) at a point in a river?

Sindhu Indus Vitastha Jhelum Asikini Chenab Airavati Ravi Satadru Sutlej

PCD_STiCM 64 How do we track density  (r) & velocity v(r) at a point in a river?

Deoprayag ("Divine confluence") -Bhagirathi & Alakananda, Himalayan tributaries of GANGA.

http://www.google.co.in/imgres?imgurl=http://personal.carthage.edu/jlochtefeld/picturepages/pilgrimage/deoprayag05.jpg&im grefurl=http://personal.carthage.edu/jlochtefeld/picturepages/pilgdeoprayag.html&h=481&w=700&sz=87&tbnid=ZQt0BRpsi0EPCD_STiCM 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 How do we have TriweNi SANGAM at Prayag?

PCD_STiCM 66 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? Jamuna Saraswathi Ganga

Prayag Triweni SangamPCD_STiCM 67 ‘LAGRANGIAN’ TRACKING: The waters of Yamuna would mix with the waters of Saraswathi and bring them to Prayag, into the Ganga!

PCD_STiCM 68 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

Joseph-Louis Lagrange 1736 - 1813 Eulerian Approach:

Follow the velocity and density of fluid at a particular point; this must be done for all points in space PCD_STiCM Leonhard Euler69 (1707-1783) Quantities of interest: *velocity : v(r). G C * pressure: p(r,t). H D P F B *density : (r,t). E r A

Mass Current Density Vector Amount of mass of fluid crossing face EFGH in unit J(r,t)  (r,t)v(r,t) time = m V lim  lim Dimensions : ML2T 1 t0 t t0 t Measure of the amount of mass  x yz  lvim  x  yz crossing unit area in unit time. t0 t

 J x, EFGH yz

PCD_STiCM 70 Amount of mass of fluid crossing face EFGH in unit time = m V  lim  lim G C t0 t0 t t H D P xyz F B  lim  vxyz t0 E t r A xyz eˆx  lim  vxyz t0 t Net OUTWARD flux  J   x   J (r)  x  yz through the two faces  x     orthogonal to x-axis =   x  P  2  J x    xyz Amount of mass of fluid  x  crossing face ABCD in unit P time   J x  J x  x   V  J x (r)   yz      x  P   x  P  2 PCD_STiCM 71 Net OUTWARD flux through the two faces orthogonal to x-axis = G C D J x  H   xyz P  x  P F B E J x  r A    V  x  eˆx P This quantity Net OUTWARD flux through the whole must be parallelepiped, per unit volume = equal to J J x y J z      J  x  y  z  P t P P P  The choice of the term ‘DIVERGENCE’   J   is thus wellPCD_STiCM justified. 72t Equation of Continuity Conservation of Matter

(r,t)   J (r,t)   t J(r,t) uˆ  mass flux in the direction of uˆ (r,t)   J (r,t)   0 t

PCD_STiCM 73  

eˆy Face ‘1’ of the Face ‘2’ parallelepiped of the eˆ region parallelepiped x region eˆz Ideal fluid: stress at any point is essentially one of COMPRESSION.

PCD_STiCM 74 We can now develop the EQUATION of MOTION for a Newtonian Fluid

PCD_STiCM 75  p x ˆ  1 2  F(1)  p(r)    yz ex   x  P 2  eˆx  p x Ideal fluid: stress ˆ F(2)  p(r)    yz ex  at any point is  x  P 2  essentially one of COMPRESSION. p p ˆ ˆ F(1)  F(2)    xyz ex     V ex x  P x  P 6 “HYDROSTATIC force”  F(i)  p V There may be some additional i1 external force acting, such as gravity.

Net force per unit volume   p = negative gradient of PCD_STiCM pressure 76 Net HYDROSTATIC force acting on the parallelepiped per unit volume  p  

External force (such as gravity) Total {hydrostatic + external acting on the parallelepiped (gravity)} force acting on the per unit volume parallelepiped per unit F external volume  lim V 0 V m d v lim F external m V 0 V dt  lim V 0 m V d v (r)  p  g(r) F external dt  lim (r) V 0 m Mass x Acceleration “Cause-Effect”  g(r) Newton’s law: PCD_STiCM Equation of Motion 77 m d v Mass x Acceleration / “Cause-Effect” lim Newton’s law: Equation of Motion V 0 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

r Euler Fixed point in space, not a function of time d v Is the ACCELERATION of actual material/fluid particle/molecule, and not just the rate at which dt velocity of the fluidPCD_STiCM is changing at a fixed point in 78 space. Mass x Acceleration / “Cause-Effect” d v (r)  p  g(r) Newton’s law: Equation of Motion dt dv  d   d  v( r ( t ) , t) vx( t ), y ( t ),) z(, t t dt dt   dt  vdx  v dy  v dz  v     x dt  y dt  z dt  t

dv dx v dy  v dz  v  v     dt dt x dt  y dt  z  t   v   v t “CONVECTIVE DERIVATIVE OPERATOR” The term ‘convection’ d    is a reminder of the fact that in the i . e .  v    convection process, the transport  PCD_STiCM 79 dt   t  of a material particle is involved. Mass x Acceleration / “Cause-Effect” d v (r)  p  g(r) Newton’s law: Equation of Motion dt dv(r,t) p p   g   External force dt external field, which we (r) (r) considered to be gravity

   dv(r,t) p v  v(r,t)     F viscous  t  dt (r)

Hydrodynamic Viscous, frictional, term dissipative term. This terms makes “dry water wet” PCD_STiCM - Feynman 80 We shall take a break here…….

Questions ? Comments ? [email protected] http://www.physics.iitm.ac.in/~labs/amp/

Next: L29 Unit 9 – Fluid Flow / Bernoulli’s principle

….. but which Bernoulli ?

PCD_STiCM 81