Geology Reference
In-Depth Information
Fig. 13.6 Parabolic
streamline pattern,
resulting from
superposition of a linear
flow and a source S .Itis
clearly visible a stagnation
point close to S .This
pattern is often used as a
model of the interaction
between a mantle plume
and the large-scale
asthenospheric flow (Sleep
1987 , 1990 )
(Fig. 13.6 ). The kinematic pattern at any given
time t may include stagnation points ,thatis,
points where v D 0, either in the fluid interior
or along its boundary. Two or more streamlines
may meet at a stagnation point, as illustrated in
Fig. 13.6 . In general, streamlines can be closed
lines, extend to infinity, or terminate at stagnation
points. It should be noted that the path r D r ( t )
of a fluid parcel does not coincide, in general,
with any streamline unless the motion is steady. A
flowfieldissaidtobe two - dimensional when the
velocity v ( r , t ) is at any point normal to some fixed
direction. In this instance, it is always possible to
choose a Cartesian reference frame such that v ( r )
has components ( u ( r ),0, ( r )) at any point r .Two-
dimensional velocity fields in fluid materials have
some interesting properties that greatly simplify
the solution of the Navier-Stokes equations when
the fluid can be considered incompressible.
For a two-dimensional incompressible fluid,
the mass conservation Eq. ( 13.8 ) reduces to:
Fig. 13.7 Areal flux through a path between two points
A and B
The function § is termed the stream func-
tion and represents a valuable tool for solving a
number of problems in geodynamics. By ( 13.63 ),
we see that the stream function associated with
any two-dimensional flow is not unique, because
an arbitrary constant may be added to § that
generates the same velocity field ( u , ). Now let
us consider an arbitrary path in the ( x , z )plane
between two points A and B (Fig. 13.7 ). At any
point along this curve, the versor normal to the
tangent vector d r D ( dx , dy ) is given by:
dy
dr ;
@ u
@x C
@
@ z D 0
dx
dr
(13.62)
n D
(13.64)
This condition is clearly satisfied if u and
are
The areal flux across the path is defined as
the line integral of the normal component of the
velocity along the path. By ( 13.63 )and( 13.64 ),
this quantity is given by:
derivatives of a scalar field § D §( x , z , t ):
@ z I D
@x
u D
(13.63)
 
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