Chemistry Reference
In-Depth Information
while '
convection
' is defined as the vertical transport of matter, and indirectly, a coupled
thermal/density/gravitational effect. We will restrict ourselves to only using the word
'advection' in this chapter because of the confusion of convection with thermally driven
processes that are coupled to gravity. However, we will discuss 3-D advective processes!
A little reflection can explain why the advective term is nonlinear in velocity: if
we follow a differential volume
PdV
in space as it is advected by flow, the total change in
the velocity vector of fixed scalar magnitude due to transport around an obstacle is the
product of the magnitude of the scalar value multiplied by the magnitude of the spatial
derivative of
.In the simple case of pure circular motion with a fixed radius of curvature
R
we find that the magnitude of the advective derivative is given by
, which
is nothing more than the centripetal acceleration
of a point particle travelling
in a circle of radius
R
with speed
, as every freshman physics student knows. Of course,
is in fact non-linear in
v
as promised. Thus, these formidable vector calculus
equations actually have an intuitive meaning if one is able to see through the mathematics.
The advective acceleration term
, which is the density times the spatial
acceleration of velocity field, will later be called (confusingly!) the inertial force, rather
like confusingly calling the centripetal acceleration the “centrifugal force”. The inertial
force is no more a force than the centrifugal force is, but the term is hopelessly embedded
in the literature and fighting it makes one look like a pedant.
Since Equation 3.2 is a nonlinear partial differential equation, there is no obvious
closed form solution to this equation, and hence even “dry water”, to use Feynman's
evocative phrase, can consume a lifetime of study easily, and indeed has for some people!
However, in spite of its complexities it does have one thing going for it: since there is no
viscosity there is no loss of kinetic energy and one can write down a potential function
whose gradient is the velocity. But this is of little help in the real world since we have to
add viscosity to have any hope of entering the nanofluidic world where shear forces
become extremely important. Viscous drag originates from transverse shear of the velocity
and the subsequent transfer of particle momentum between layers. When viscosity is added
to the Equation 3.2 the Navier-Stokes (N-S) equation is obtained:
(3.3)
Since viscous drag is dissipative it results in the “sucking” of kinetic energy from a volume
element in shear and results in non-energy conserving flows. Thus, the curl of the velocity
r
×∇ is not necessarily zero in the presence of viscosity flow and one can no longer write
a potential function from which all the velocities can be computed at subsequent time, as
you can for dry water. Since the Navier-Stokes equation has the double misfortune of
being both nonlinear in
r
and also not energy conserving it is an extraordinarily difficult
physics problem for arbitrary initial values of
r
,
ρ and
to calculate subsequent values of,
r
quite hopeless.
Fortunately (perhaps) some simplifications in the Navier-Stokes equation are
possible. The spatial derivatives in the Navier-Stokes Equation are numerically of the
form,
(3.4)
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