Flow and Fluid Behaviour of theMantle - Quantitative Plate Tectonics - page 332

This equation shows that the density may
locally change, even in the case of incompressible
fluids. The hypothesis of incompressibility has
been widely used in the modelling of mantle flow.
It implies that the volume dV of each mantle
parcel remains invariant, thereby it is assumed
that changes in pressure do not determine neither
adiabatic compressional heating nor adiabatic ex-
tensional cooling during mantle convection.

where ǜ is the viscosity and p is the thermody-
namic pressure. As the fluid is incompressible,
by ( 13.8 ) we have that the trace of the strain
rate tensor is zero. Therefore, taking the trace of
( 13.11 ) we obtain:

1
3 £ kk

p D (13.12)

This is an interesting relation that links the
thermodynamic pressure of incompressible flu-
ids, which must satisfy an equation of state that
involves temperature and density, to an invariant
of the stress tensor. In particular ( 13.12 ), estab-
lishes the equivalence between thermodynamic
pressure and mean mechanical pressure in the
case of incompressible fluids.
Finally, combining ( 13.11 )and( 13.10 )and
taking into account of ( 13.8 ) gives the equations
of motion for an incompressible Newtonian fluid
with uniform viscosity:

13.2 Navier-Stokes Equations

Now we are going to formulate the equations of
motion for a fluid in the framework of the Eule-
rian representation. To this purpose, we can start
from the Cauchy momentum Eq. ( 7.50 ) , taking
into account that the acceleration of a volume
element must be calculated as the material deriva-
tive of the velocity, a D d v / dt , and that gravity is
the unique relevant body force in the context of
mantle dynamics. Therefore, if £ D £ ( r , t )isthe
stress field and f D ¡ g represents the gravitational
body force density, applying ( 13.2 )wehavethat
in index notation the Eulerian version of the
equations of motion assumes the form:
¡ @ v i

¡ @ v i

@x j v j

@x i C ǜ @ 2 v i

@ v i @p

@t C

D

@x j C ¡g i
(13.13)

@x j v j

Equations ( 13.13 ) are known as Navier - Stokes
equations for an incompressible fluid. The first
two terms at the right-hand side of these equa-
tions represent surface forces (per unit volume)
exerted on a fluid particle. They are, respectively,
the pressure force ,andthe viscous force .More
general equations of motion for Newtonian fluids
can be written releasing the incompressibility
constraint. The most general form of the constitu-
tive equation describing a Newtonian fluid reads
(e.g., Schubert et al. 2004 ):

@£ ij
@x j C ¡ g i

@ v i

@t C

D

(13.10)

These equations are completely general (ex-
cept for the assumption of a specific body force
field) and do not depend from a particular rheol-
ogy of the material. To be used in the solution of
geodynamical problems, they must be combined
with a constitutive rheological equation that spec-
ifies the relation existing between stress, kine-
matic variables, which now are represented by
strain rates, intrinsic parameters of the material,
and state variables (see Sect. 7.4 ) . In the case
of an incompressible Newtonian fluid (see Sect.
7.4 ) , it is possible to show that the constitutive
equation reads (e.g., Ranalli 1995 ):
£ ij D p• ij C 2ǜ P © ij D p• ij C ǜ @ v i

£ ij D p• ij C 2ǜ P © ij C œ P © kk • ij (13.14)

where œ is called second viscosity . Note that
£ 0 ij D 2ǜ P © ij C œ P © kk • ij represents the analog of
the deviatoric stress introduced in Chap. 7 (see
Eq. 7.26 ) . The average pressure associated with
the stress tensor is:

@ v j
@x i @x j C

(13.11)