Flow and Fluid Behaviour of theMantle - Quantitative Plate Tectonics - page 336

d W D T .n/ v dS D v i ij n j dS
D ij v j n i dS D .£ v / dS

@ v j
@x i C k r

2 T C ¡ H

¡ P u D £ ij

(13.31)

(13.27)

The first term at the right-hand side of ( 13.31 )
can be expressed in terms of viscous dissipation
function and pressure power input. In fact, by the
symmetry of the stress tensor we have that:

where we have applied Cauchy's theorem ( 7.3 )
and the symmetry condition ( 7.2 ) for the stress
tensor. Therefore, using Gauss' theorem (see Ap-
pendix I) we see that the power done by surface
forces on S ( R ) is given by:

p• ij C £ 0 ij @ v i
@x j

@ v j
@x i D £ ji

@ v j
@x i D £ ij

@ v i
@x j D

£ ij

I Z

W D

.£ v / dS D r .£ v /dV (13.28)

D p @ v k

@x k C ˆ

(13.32)

S. R / R

Therefore, the conservation law ( 13.31 ) can be
rewritten as follows:

The total mechanical power on R is then
calculated as follows:

Z

2 T C ¡H (13.33)

¡ P u D p r v C ˆ C k r

W D

Œ r .£ v / C ¡g v dV

The internal energy density u in ( 13.33 )is
not independent from the temperature T and the
pressure p , thereby it is useful to find a form of the
energy conservation law such that u is substituted
by an expression of T and p . To this purpose,
we can use the first law of thermodynamics,
which states that in absence of mass exchange
the infinitesimal variation of internal energy, du ,
of a region is the sum of the energy per unit
mass absorbed in the from of heat, • q ,andthe
infinitesimal work done by the surrounding on the
system, • w :

R

@£ ij

@x i C ¡g j v j dV

Z

@x i v j C £ ij @ v j

D

R

Z @ ij

@x i C ¡g j v j C £ ij @ v j

dV

D

@x i

R

@£ jk

@x k C ¡g j v j C £ ij

dV

Z

@ v j
@x i

D

R

Z dV

P v j v j ¡ C £ ij @ v j
@x i

D

(13.29)

R

d u D •q C • w (13.34)

where we have used the Lagrangian version of
the equations of motion ( 13.10 ). Finally, the heat
power associated with the heat flow q into R
and with the radiogenic heat H can be obtained
easily generalizing ( 12.10 ) through inclusion of
the contribution of H :

Although irreversible processes occur in
a fluid, for example viscous dissipation, a
simple expression for du can be obtained only
assuming that the transformation is quasi-static
(i.e., reversible), thereby frictional heating and
other irreversible processes are considered of
secondary importance. In this hypothesis, the
first law of thermodynamics can be rewritten as
follows:

Z

k r

2 T C ¡H dV

Q D

(13.30)

R

The energy balance and conservation law can
be formulated by substituting the expressions
( 13.24 ), ( 13.26 ), ( 13.29 ), and ( 13.30 )into( 13.22 )
and taking into account that the equation holds for
any arbitrary region R . Therefore:

d u D •q pd

(13.35)

where D 1/¡ is the specific volume (volume per
unit mass). The heat per unit mass adsorbed by
the system can be expressed as follows: