Flow and Fluid Behaviour of theMantle - Quantitative Plate Tectonics - page 335

of the overlying plate. This is a consequence of
a fundamental empirical law of fluid dynamics
that is known as no - slip boundary condition (e.g.,
Pozrikidis 2009 ), which states that in a fluid the
tangential component of the velocity field is con-
tinuous across a solid boundary. Accordingly, the
average strain rate through a cross-section of the
fluid sheet in Fig. 13.1 will be: P © D v =h.If£ is
the shear stress that the overlying plate exerts on
the fluid, then the total force is F D £ A , thereby
the power dissipated by shearing is: W D F v D
£Ah P © D £V P ©. In general, the power dissipated
in a fluid by viscous deformation is referred to
as viscous dissipation and is associated with an
irreversible conversion of mechanical energy into
temperature increases.
If w is the work per unit volume associated
with the viscous flow, than the viscous dissipation
function, ˆ, has the expression:

The rate of change of the kinetic energy can be
obtained taking the total derivative of the volume
integral ( 2.19 ) . If § is an arbitrary scalar field,
then the material derivative of the volume integral
of § over a region R is given by:

Z Z d

dt C §.r;t/ r v dV

d
dt

§.r;t/dV D

R R

Z @§

@t C @x i .§ v i / dV

@

D

(13.23)

R

Therefore, using the continuity Eq. ( 13.7 ), for
the kinetic energy we obtain:

Z Z

d
dt

1
2

K D

¡ v 2 dV D ¡ v k P v k dV (13.24)

R R

Let us consider now the internal energy U ,
which can be written as follows:

@ v i
@x j

ˆ D w D £ 0 ij (13.20)

Z

U.t/ D ¡.r;t/ u .r;t/dV (13.25)

where £ 0 is the deviatoric stress tensor. Substi-
tuting the deviatoric part of expression ( 13.14 )
gives:

R

where u is the internal energy per unit mass.
Using ( 13.23 ) we obtain for the rate of change
of the internal energy:

ǜ @ v i

@x k • ij @ v i

@ v j
@x i

C œ @ v k

ˆ D

@x j C @x j

(13.21)

d

dt .¡ u / C ¡ u r v dV

Z Z

d
dt

U D

¡ u dV D

R R

Therefore, both dynamic and second viscosities
lead to dissipation in a fluid. Now we are ready
to formulate the law of energy conservation for a
fluid parcel. Differently from the heat diffusion
Eq. ( 12.15 ) , the energy balance will take into
account of the advection of material, thereby the
partial derivative of the temperature will be sub-
stituted by a material derivative. In general, the
conservation of energy of a mass of fluid requires
that the rate of change of the kinetic energy, K,
plus the rate of change of the internal energy, U ,
be equal to the mechanical power input, W ,plus
the rate of heat produced within or entering the
body, Q:

Z Z

D Œ u P ¡ C ¡ P u C ¡ u r v dV D ¡ P u dV

R R

(13.26)

The total energy content of the region of fluid
R may change as a consequence of flow through
its surface S ( R ), heat conduction, work done by
the gravity field or by surface tractions, and ra-
dioactive heat production. The mechanical power
input arises from surface forces applied along
S ( R ) and from the gravity force exerted on each
volume element in R . Let us consider first the
surface force exerted on a surface element dS
with orientation n .If T ( n ) is the traction, then the
power input on dS is given by:

K C U D W C Q

(13.22)