Jökull - 01.12.1983, Side 60

(6) Since the front is moving, the front condition
is inherently non-linear regardless of the physical
relations under (5) above.
(7) In general, lavas move on an uneven ground
and the shape of the ground surface will thus enter
into consideration.
(8) The productivity of the eruptive sources
feeding lava flows is generally highly variable and
unpredictable.
(9) Few detailed field data are available on lava
flows in motion. On the other hand, a great number ’
of frozen or fossil lava flows are available for observ-
ation in various regions of the world. While the fossil
lava data are useful, it is to be emphasized that the
form and dimensions of such formations represent
lavas in the final solidified state and may therefore
not directly apply to the case of advancing flows
that are still liquid.
There nevertheless is a justification for an
attempt at using the mechanism ofa thin-sheet flow
of a homogeneous Newtonean fluid as a semi-
quantitative model in the study of the more maflc
lava flows. It is quite possible that some conclusions
as to the behavior of real lavas can be reached on the
basis of such a theory. Attempts at furnishing the
initial steps in this direction will be presented
below.
THE THIN NEWTONEAN SHEET
Consider a thin layer of a homogeneous incom-
pressible Newtonean liquid flowing over an almost
horizontal ground as indicated in Fig. 1. The vari-
ations in the altitude of the ground and all slopes are
assumed to be small.
n
X-
cLava<£Jl h S?<5íwií?bW*^<qround ?surfaceV4R , °
w%mmr
Fig 1. Model ofa Iava flow.
Mynd 1. Líkan af hraunrennsli.
We place a coordinate system with the origin in a
horizontal plane surface 2 below ground level and
with the z-axis vertically up. Let S = (x,y) be a point
on 2, f(S) be the ground altitude and h(S) be the
altitude of the fluid surface over the point S. More-
over, let Q be the density of the fluid, p and v be
the absolute and kinematic viscosities, respectively,
p(P,t) be the pressure and ú(P,t) = (u,v,w) be the
velocity in the fluid at the field point P = (S,z) and
at time t.
Since the fluid is assumed to be Newtonean, the
motion is governed by the Navier-Stokes equation
dö= _Vp +pv2a-g6, (1)
v Dt
where the material derivative
— = 6t + uðx + vðv + wðz
Dt
and g is the acceleration of gravity. The condition
of incompressibility is
V-ú = q
(3)
where q(P,t) is the volume source density. The
pressure at the free surface Í1 is constant and can be
assumed zero. The free surface boundary condition
is therefore
p on
n
(4)
The condition at the bottom of the fluid layer is that
the flow velocity be parallel to the ground surface
r. In the other words
ú'ú = 0
p on r
(5)
when n is the outward normal to T. Finally, an
initial condition may have to be adjoined to the
above conditions.
The mathematical problem of solving equation
(1) taking (2) to (5) into account is immense and
beyond present capabilities. For our purpose we
will make the following quite drastic but, never-
theless, plausible simplifications (a) to (d) which
will reduce the above problem to a more managable
form.
(a) In view of the very slow flow, we can assume
creep conditions where the mass forces can be
neglected as compared with the viscous and gravit-
ational forces. The term on the left ofequation (1)
will thus be neglected.
(b) Since the vertical velocity w in a creeping thin
almost horizontal layer is small as compared with
the horizontal components, we will assume that
(6)
and neglect w everywhere except in the conserv-
ation of volume condition (3).
(c) Moreover, the thin layer creep also implies
that
58 JÖKULL 33. ÁR