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4. THE EQUATION OF MOTION

Newton’s law for a fluid element is

!dV

dt= F!!!, (4.1)

where ! is the mass density (mass per unit volume), V is the velocity of the fluid element, and F is the force per unit volume acting on the element. The latter is composed of two types: volumetric forces and surface forces.

Volumetric forces act throughout the volume of the fluid element. They can be thought of as acting at the centroid. Examples are:

1. Gravity: Fg = !g , where g is the gravitational acceleration. If the force is a central force, then Fg = !"# , where ! is the gravitational potential.

2. Electromagnetic forces: Since the fluid can conduct electricity, it can have a current density J = n!q!V!!" (with the sum being over all species of ions and electrons, and n

!, q

!, and V

! the number density, electric charge, and

velocity of species ! ), and, in principle, a net electric charge per unit volume, !q . The electromagnetic force (per unit volume) are then the electric force, Fq = !

qE (E is the electric field), and the Lorentz force,

FL= J ! B (B is the magnetic field).

Surface forces are more complicated. Consider the forces acting on a surface S . We assume the convention that the material in front of S exerts a force on the material behind S that is given by

F = S !P!!!, (4.2)

or Fi = SiPij !!!. (4.3)

We consider three orientations for S , along each of the three coordinate directions. If S = e

1,

F = P11e1+ P

12e2+ P

13e3!!!, (4.4)

which is a vector. Similarly, if S = e2, then

F = P21e1+ P

22e2+ P

23e3!!!, (4.5)

and if S = e3, then

F = P31e1+ P

32e2+ P

33e3!!!. (4.6)

It therefore takes 9 numbers to define the force on the surface S . These are the components of the stress tensor, Pij . (We will not prove that P is a tensor, but it is!)

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The total surface force acting on a fluid element is the sum of the forces on its faces. We want the total force acting on the volume. Since F has been defined as the as the force exerted by the material in front of S acting on the material behind S , all of the material within the element is behind the faces, and the net force is the negative of the surface forces, i.e.,

F = ! dS "S

!# !P!!!, (4.7)

or, by Gauss’ theorem,

F = ! " #PdVV

$ !!!. (4.8)

As V ! 0 , we obtain the net force per unit volume as f = !" #P!!!. (4.9)

This is the volumetric equivalent of the surface forces. The equation of motion considering only surface forces is then

!dV

dt= "# $P!!!. (4.10)

We will now prove that the stress tensor is symmetric, i.e., Pij = Pji . Consider the angular momentum per unit volume of fluid,

L = r ! "V!!!. (4.11)

Using Equation (4.10), the total (i.e., Lagrangian) time rate of change of L for a fluid with fixed volume V

0 is

!L = !r "dV

dtdV

V0

# = $ r " % &PdVV0

# !!!, (4.12)

or, in Cartesian tensor notation

!Li = ! "ijkrj #lPlk( )dVV0

$ !!!. (4.13)

Since !lrj = " lj , we can write

rj !lPlk( ) = !l rjPlk( ) " Pjk !!!, (4.14)

so that

!Li = ! "ijk #l rjPlk( ) ! Pjk$%

&'dV

V0

( !!!,

!!!!= ! #i "ijkrjPlk( )dVV0

( + "ijkPlkdVV0

( !!!,

!!!!= ! dSl"ijkrjPlkS0

!# + "ijkPlkdVV0

# !!!, (4.15)

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where we have used Gauss’ theorem, and S0 is the surface bounding V

0. We recognize

the first term on the right hand side of Equation (4.15) as the total external torque applied to the surface of the volume. The remaining term is the rate of change of internal angular momentum of the fluid. In the absence of applied torque, we require

!Li= 0 , or

!ijkPlkdVV0

" = 0!!!, (4.16)

which can be written as 1

2!ijkPjk + !ikjPkj( )dV

V0

" =1

2!ijk Pjk # Pkj( )dV

V0

" = 0!!!. (4.17)

(The first expression comes from interchanging the dummy indices j and k ; the second follows from the properties of !ijk .) Since Equation (4.15) must hold for an arbitrary volume, we have Pjk = Pkj , which is the desired result.

The symmetry of the stress tensor is a very general result; it can be considered a general principle of physics. It is independent of the properties of the medium, which can be solid, liquid, or gas (or even plasma). It is required to prevent the internal angular momentum of the system from increasing without bound.

The stress tensor (indeed, any tensor) can always be decomposed as P = pI! "!!!, (4.18)

or Pij= p!

ij" #

ij!!!. (4.19)

The first term on the right hand side is called the scalar pressure. The second term is called the viscous stress tensor.

Including all the volumetric and equivalent volumetric forces, Equation (4.1) becomes

!dV

dt= !qE + J " B # $p +$ %&!!!, (4.20)

which is the Lagrangian form of the equation of motion. The Eulerian form is

!"V"t

+ V #$V%&'

()*= !

qE + J + B , $p +$ #-!!!. (4.21)

As always, they are equivalent. We now compute W

V= V !F

V, the work done on a volume element by the viscous

force FV= ! "# :

WV= V

i!j"

ji= !

jVi"

ji( ) # " ji!jVi!!!,

or W

V= ! " # "V( ) $ # :!V!!!. (4.22)

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Then

WVdV

V0! = " # $ #V( )dV

V0

! % $ :"VdVV0

! !!!,

!!!!!!!!!!!!!!!= dS ! " !V( )S0

!# $ " :%VdVV0

# !!!. (4.23)

The first term on the right hand side is the work done on the surface S ; the second term is the work done throughout the volume. In the absence of the surface term, kinetic energy is lost from the fluid if ! :"V > 0 ; it must show up as internal (thermal) energy. We therefore identify the volumetric viscous heating rate as

QV= ! :"V!!!. (4.24)

We remark that the equation of motion has introduced 6 new dependent variables; i.e., the 6 independent components of the stress tensor..This is a further example of the closure problem first mentioned in Section 3.