J. metamorphic Geol., 1992, 10, 311-319
Hydrodynamic modelling of some metamorphic
processes
L. L . PERCHUK, Y. YU. PODLADCHIKOV A N D A . N . P O L Y A K O V
Institute of Experimental Mineralogy, Russian Academy of Sciences, Chernogolovka, Moscow District,
742432, Russia
A B S T R A C T The P-T paths for metamorphic complexes from the Precambrian shields and fold belts of different ages
may result from advection, i.e. one-cycle convective processes in the lithosphere. This conclusion has
been exemplified by the metamorphic evolution of several well-known complexes, for which an advective
model can be successfully applied. Numerical simulations of the above processes in terms of Newtonian
rheology by using a two-dimensional finite element program have been conducted.
Two representative models for intracontinental gravitational ordering initiated presumably by mantle
activity are considered: (i) a thermally activated multi-layered rhythmic sequence and (ii) huge rising
diapiars causing circulation, in which crustal lithologies underwent high-P metamorphism (above
10-15 kbar) and subsequent ascent toward the Earth's surface.
Key words: cratonization; gravitational ordering; metamorphic modelling.
INTRODUCTION
Direct genetic relationships between granulites and other
metamorphic rocks of greenstone belts have not yet been
found. Early Archaean greenstone belts are composed
mainly of basic volcanics with kamatiites at the lower part
of the stratigraphic column (Kuznetsov, 1985; Di Marco &
Lowe, 1989). In contrast, Kulikov et al. (1987) showed that
the lower part of the Karelian greenstone belt consists of
acid and andesitic volcanics, while its upper part is
composed of basalts and komatiites. This sequence reflects
an inverse volcanic (from acid to basic) activity in the early
stages of evolution of greenstone belts, that produces a
potentially unstable system in the gravity field. Such an
inverse sequence is also characteristic of modern marginal
sea floors (Perchuk, 1987).
The stratigraphic column for the Karelian belt, and for
some other belts (Kuznetsov, 1985; Lobach-Zhuchenko,
1988; Perchuk, 1989) shows several magmatic cycles. A
multi-layered rhythmic sequence is formed as a result of
such magmatic activity. Thus, any stratigraphic column
resulting from multi-cycle magmatic activity is potentially
unstable in the gravity field, regardless of a directjinverse
volcanic sequence.
The above sequence might be moved to high P-T
conditions in the course of several geological/tectonic
events (Es, 1972; Artushkov, 1983; Schreyer, 1988). An
increase in temperature leads to a decrease of viscosity and
initiates gravitational redistribution of the rocks according
to their densities. As a result of this process, the
greenstone belt may be transformed into a granulite facies
complex (Petrova & Levitskiy, 1986; Perchuk, 1989).
On the basis of geothermobarometry, Perchuk (1985,
1990) deduced P-T evolutionary paths for metamorphic
complexes of different geological settings. For granulite
facies rocks, P-T paths reflect only the retrograde stages
of a metamorphic event because the prograde stage is
erased at high temperature. Perchuk & Gerya (1990)
approximated data on the paths for granulite facies rocks
using the following equation:
P (kbar) = 0.02(f 3.7 x 10'~) x T(OC)- 6,8(f 2.5). (1)
This equation can be used for calculating the depth
corresponding to given temperatures of metamorphism for
granulite facies rocks. Equation (1) reflects a low
geothermal gradient (at high-T conditions). It is believed
to be a uniform characteristic for uprising portions of the
Precambrian lower crust. For Archaean greenstone belts
intruded by granites, West & Mareschal (1979) and Bickle
(1986) assumed a high dT/dP geotherm, but in this case,
the gradient reflects the lateral plutonic-metamorphic
conditions.
Thus, this study concentrates on a model for a
multi-layered rhythmic system of long-term development
of gravitational instability and the development of uniform
low dT/dP (at high temperature) gradients for granulite
facies complexes (Eq. 1).
The geodynamic histories of metamorphic complexes
formed in fold belts are a widely discussed problem. The
P-T paths for fold belts differ from those for the
granulite-amphibolite regional metamorphic terranes.
According to the evolution of thermodynamic parameters,
312 L. L. PERCHUK, Y. YU. PODLADCHIKOV & A. N. POLYAKOV
Perchuk (1977, 1989) divided fold belts into two groups: (i)
low-T, high-P, mainly eclogite-glaucophane schist formations and (ii) high-P, moderate-T complexes. A new type,
i.e. (iii), an ultra high-P coesite-bearing metamorphic
complex, was discovered by Chopin (1984) in the Dora
Maira massif of the Western Alps. On the basis of
paragenetic analyses of the Dora Maira rocks, he deduced
a P-T retrograde evolutionary path for this complex, with
a metamorphic peak at c. 37 kbar and c. 800" C.
Several localities of coesite-bearing eclogites are
described by Zhang et al. (1990) from the Jiangsu
Province, China. The rocks occur as pod swarms,
fragments or blocks in Proterozoic gneisses and range in
size from centimetres to several hundred metres. Three
(typical) mineral parageneses recognized in the coesitebearing eclogites are: (i) garnet omphacite phengite
quartz rutile,
(ii)
garnet omphacite kyanite
paragonite quartz epidote amphibole rutile,
(iii)
garnet omphacite epidote quartz Al-rich titanite
rutile. Quartz-coesite aggregates have been confirmed in
only one garnet grain from the Mengzhong eclogite
locality, although quartz pseudomorphs after coesite are
common. The rocks underwent intensive retrogression,
and retrograde minerals (amphiboles, paragonite around
kyanite, Si-poor phengite around Si-rich phengite, various
symplectites, etc.) were developed. The peak of
metamorphism is suggested by Zhang et al. (1990) to be at
730-840°C and 30kbar. The localities (Menzhong,
Qinglongshan, Jiagchang and Chizhuang) extend in the
fold belt for more than 1000 km along the border of the
northern China and Yangtz Cratons. Coesite-bearing rocks
are associated with granites and syenites (R. Zhang, pers.
comm.). Zhang et al. (1990) suggest that formation ifthe
belt involved '. . . the collision of two continental crusts
and resultant crustal thickening7 (p. 924). However, no
evidence for this model, or for the ascent of rocks to the
surface, is given by them.
Another example of a P-T path at ultra high pressure
was described by Sobolev et al. (1986) and Shatsky et al.
(1989) from the Kokchetav massif, northern Kazakhstan.
The Zerendinskiy granitoid batholith is there associated
with a metamorphic complex composed of eclogites, white
schists, kyanite-biotite-garnet gneisses, talc-kyanitegarnet- and kyanite-zoisite-quartz-bearing rocks (Fig. 1).
Isotopic ages of both granites and metamorphic rocks are
c. 530 Ma, while the age of the gneissic protolith is about
2000 Ma (Jagoutz et a[., 1989). Metapelites contain garnet
and zircon with diamond, which commonly shows
intergrowths with other very high-P minerals, such as
titanite containing 11 wt% AI,03, K-rich (up to 1.44 wt%)
clinopyroxene, Ti0,-rich (up to 3.7 wt%) phengite and
Al-rich rutile. Very recently coesite has also been found in
these rocks (N. V. Sobolev, pers. comm., 1991).
Co-genetic granite intrusions and the absence of thrust
structures are characteristics of both the areas discussed
above (Zaitsev, 1984; Shatsky et al., 1989; Zhang et al.,
1990). Those features lead us to model the geodynamic
evolution for ultra high-P complexes in terms of convective
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processes connected with magmatic activity; this is the
second aim of this paper.
I
CEO DY NAM l C FO R M U L A T I O N 0 F T H E
PROBLEM
There are many models for thermal convection in the deep
mantle and in the lithosphere (e.g. Christensen, 1984).
Multi-scale thermal convection in the Archaean crust was
proposed by Talbot (1968, 1971). However, all those
models raise serious problems concerning the redistribution of material within the continental crust, Mechanisms
for extensive redistribution must overcome the constraints
resulting from high viscosity and negligible effects of
thermal expansion because ( a p l aT), loq3.
While some models are based on thermal convection,
others suggest that the primary chemical inhomogeneity
might develop during cratonization (Ramberg, 1981;
Perchuk, 1989). Ramberg (1981) considered many models
for an early stage of gravitational instability. Schmeling
(1988) studied numerical models of Rayleigh-Taylor
instabilities (two-layered system) superimposed upon
convection. Weber (1986) considered the role of crustal
rheology for the tectonic development of the continental
crust during prograde metamorphism. He suggested the
formation of large-scale folds (composed of granulite facies
rocks of the lower crust), which 'pierce the overlying crust
during increasing amplification. Further crustal shortening
may produce granulite facies nappes from strongly
amplified large-scale fold structures leading to inverse
metamorphism which is generally accompanied by
granulite facies nappe complexes' (Weber, 1986, p. 95).
\
-
~ ~ l ~ multi-rhyt,,mic
i - l ~ ~ sequence
~ ~ ~ d
As shown above, volcanism at the marginal sea floor as
well as in some Precambrian ensialic greenstone belts
involves a regular variety of magmatic rocks including a
rhythmic sequence of rhyolites, basalts and komatiites.
These volcanics are often intercalated with sediments. For
modelling, such complex structures might be approximated
by simple multi-rhythmic sequences. Each rhythm consists
of three layers, i.e. of silicic, mafic and ultramafic rocks.
Consider such a simple multi-rhythmic sequence: at a
shallow level, this sequence is relatively stable in the
gravity field due to (i) its high viscosity and (ii) the small
thickness of a rhythm, S.. It can be shown that a period of
gravitational redistribution grows inversely with S,. and
directly with viscosity. Therefore, under the above
conditions this period is greater than is geologically
realistic. Moreover, the characteristic stresses might be
lower than the yield strength, and the sequence under
consideration is absolutely stable. Under high-grade
conditions, both viscosity and yield strength drop with
temperature, and gravitational redistribution of material
can occur in geologically realistic time spans.
Gravitational ordering may lead to the complete
transformation of a greenstone belt or any initial
multi-rhythmic sequence into a granulite facies complex.
I
i
I
HYDRODYNAMIC MODELLING OF METAMORPHIC PROCESSES 313
Fig. 1. Simplified geological map of Northern Kazakhstan (after Kushev &
Vinogradov, 1978). Scale 1:200,000. 1-4 =Early Palaeozoic metamorphic
suites having local names; 5 = Upper Proterozoic and Early Palaeozoic rocks,
which have not been divided into separate metamorphic units; 6 = Quaternary
sediments; 7 = Proterozoic intrusive rocks; 8 = Palaeozoic granites; 9 =
Zerendinskiy granitoid batholith (mid Cambrian); 10 = faults; 11 = axis of
anticlinal folds; 12 = axis of synclinal folds; 13= eclogites and diamondbearing mica schists (Shatsky et al., 1989).
Evidence for this process is reported in several papers
(Ramberg, 1981; Petrova & Levitskiy, 1986; Barbey &
Martin, 1987). Perchuk (1989) described a typical dome
structure (see Fig. 2) for granulite facies rocks from the
Sharyzhalgay complex (Lake Baikal, eastern Siberia),
whose geological structure has been studied in detail
(Hopgood & Bowes , 1989). This complex presumably
formed after the Archaean greenstone belt in the course of
a high-grade transformation (Petrova & Levitskiy, 1986).
However, a model for such transformations is not
available. We have studied this problem on the basis of
thermo-mechanical modelling of the gravitational ordering
of a multi-rhythmic sequence under high P-T conditions.
Ultra high-pressure complexes
There are a few geodynamic models for the origin of ultra
high-P rocks within the Earth's crust. For example, using
petrological data, Schreyer (1988) proposed a model for
the origin of the Dora Maira complex as a result of
subduction of continental crust to mantle depths.
However, the plate tectonic events related to the
generation of ultra high-P metamorphic complexes are
uncertain. The best example is the finding of
diarnond/coesite-bearing rocks from the Kokchetav massif
(see Fig. 1). Several tectonic models can be used 'to draw
down' rocks to 100 krn, or even more. The problem is to
create a model to account for the uplift of the rocks to the
Earth's surface without any tectonic denudation. An
uplift-erosion model is also inappropriate for ultra high-P
units, which are similar to the Kokchetav complex (see
Fig. 1).
The geological setting of these complexes can be also
explained in terms of a gravitational ordering model. The
P-T loops for high-P, rnoderate/high-T complexes from
fold belts of diffefent age may result from adbection, i.e.
one-cycle convective processes in the thick lithosphere
caused by ascent of a magmatic mass. Such complexes are
314 L. L. PERCHUK, Y .
YU. PODLADCHIKOV & A. N. POLYAKOV
Fig. 2. Schematic relationships between rocks within mantled dome in the Sharyzhalgay complex, which crops out along the shore of
Lake Baikal. 1 = biotite-garnet gneiss; 2 = migmatized metabasites; 3 = migmatized biotite-hypersthene gneiss; 4 = enderbite gneiss;
5 = enderbite with slightly developed gneissosity; 6 = boudins and xenoliths composed of crystalline schists and amphibolites;
7 =fractures; 8 = flow direction, i.e. direction along which material of different densities moved (after A. I. Melnikov, pers. comm.).
commonly associated with magmatic bodies of similar
ages. For example, the ascent of large masses of granite
magma within thick continental crust may initiate sinking
and compression (up to 4 g cm-" of relatively small masses
of the volcanic-sedimentary rocks at a depth of about
100krn and subsequent uplift of these ultra high-P
complexes toward the Earth's surface. This process may
lead to the formation of the coesite- and diamond-bearing
assemblages from gneissic and white schist complexes.
Dora Maira, Monte Rosa and Grand Paradiso in the
Western Alps are the Late Alpine internal crystalline
massifs in the fold belt surrounding the northern Italy
sedimentary basin of similar age. The relationships
between those complexes and Alpine granites are not
known. However, geophysical data suggest that the
basement of the above basin is composed of basaltic
material. Similar ages of high-P complexes and adjacent
basins may reflect a common advective process of
ascending basaltic magma, which initiates movement of a
metamorphic complex within a created convective cell.
Thus, the above geological and petrological data
highlight a problem that can be addressed by computer
simulation of the downward movement and subsequent
ascent of crustal lithologies due to convective circulation
driven by a rising magmatic body.
PHYSICAL M O D E L S A N D M E T H O D S
USED
The following equations were used for the numerical and
analytical simulation of the above geodynamic processes:
(a) the force equation:
where i = 1,2 and j = 1,2;
(b) Newtonian rheology equation:
(c) the mass conservation equation:
div V = 0;
(d) the heat transfer equation:
(4)
HYDRODYNAMIC MODELLING OF METAMORPHIC P R O C E S S E S 315
where: o;i = stress tensor: V = velocity (cm s-I);
p(X, Y, 2 )= density (g ~ r n - ~ )T;( X , Y, 2 )= temperature
("(2); g = 9.8 m s - ~ ; ,u, . exp(- yT) = viscosity (poise);
a = thermal diffusivity (cm2 s-I); Q = radioactive heat
- ~ aA/ar and aA/ar are the partial
production (J ~ r n s-I);
derivatives with respect to time and distance, respectively,
and A is any parameter from the density or viscosity field.
The boundary conditions are V ,= 0 and on,= 0.
Bolshoi & Podladchikov (1991) wrote a finite element
program in P- V (natural) variables and Cartesian
coordinates for numerical modelling of the advective
processes (see Appendix) using the above equations.
The modelling on the basis of Newtonian rheology
should use effective viscosities for rocks of real rheology
taking into account the following statements (see, for
example, Ranalli, 1987):
(1) the accuracy of numerous data on effective viscosity
for rocks of different compositions is very poor;
(2) the effect of phase transitions in minerals on
decreasing the effective viscosity in the course of granulite
facies metamorphism is not estimated quantitatively;
(3) the real strain rate values, which are extremely
important for estimates of effective viscosity (for example,
in terms of the power law rheology), are unknown;
(4) at high temperatures, the effective viscosity increases
greatly with melting temperature for rocks of different
compositions.
Therefore, selection of precise values of effective viscosity
o r the use of non-Newtonian rheology is inappropriate.
However, our results (time, z, and velocity, V) of
numerical modelling at given p,,,,,, H (total thickness) and
Fig. 3. Typical scenario for the evolution of
the 12-layer, four-rhythm succession with
discrete increases of density and viscosity
upwards in each rhythmic unit. Less dense
and less viscous material is shown with
lighter shadings. Four stages (from bottom,
upwards) correspond to the following
geological times: 10.5, 11.5, 19.2, and
34 Ma. FEM grid is 81 x 41, number of
markers are (81 x 41) x 9 = 29,889. H =
3 km, pmax
= lo2' poise and Ap =
r~rnaxl~lrn=
i n lo2.
Ap can be re-calculated for others (p:,,, H and Ap') using
the following rules:
Calculations show a weak (not more than one order of
magnitude) dependence of these values on the viscosity
variations. Therefore, the variations are not essential
parameters to solve the problem under consideration.
No chemical interactions between rocks of different
compositions can be considered at this moment.
RESULTS
Multi-layered multi-rhythmicsequence
A three-layered, four-rhythmic sequence has been chosen
for numerical simulation.
According to Eq. (I), the temperature difference within
a multi-layer sequence 1-3 km in total thickness is about
15-50" C. Therefore, variations of rock properties as a
function of temperature can be considered negligible.
Hence, we may conduct numerical simulation of
gravitational ordering in terms of isatherrnal
approximation.
As mentioned earlier, the period of gravitational
redistribution increases inversely with the thickness of a
rhythm, and even under granulite facies conditions a
layered non-rhythmic sequence some 1-3 km in thickness
316 L. L. PERCHUK, Y. YU. PODLADCHIKOV & A . N. POLYAKOV
may be relatively stable in the gravity field for a
considerable geological time. To overcome this difficulty,
Perchuk (1991) suggested that the gravitational redistribution of material could be accelerated within a layered
rhythmic system: the interaction between rhythms allows a
rapid, large-scale flow over the entire sequence (this is a
type of chain reaction mechanism).
Figure 3 shows the results of numerical simulation of the
four-rhythmic, three-layered sequence. Each rhythm of the
sequence consists of three layers each 250m thick
(H = 3 km) ; density/viscosity increase from p / p =
2.6 g cm-'/lo" poise (rhyolites) through p / p = 3.0 g cm-'/
lo1' poise
(basalts)
to
p i p = 3.3 g cm-"1O2" poise
(komatiites). These properties have been derived by
applying values taken from the P-T path and applying in
Eq. (11, taking into account the problem of effective
viscosity discussed above. Figure 3 shows the large-scale
movement that occurs because of the interaction between
rhythmic units during time spans of 10.5-34 Ma.
At present, the heat transfer Eq. (5) has not been used.
Consider a conventional multi-layered, multi-rhythmic
sequence 30 krn in total thickness as a model for
heterogeneous crust before gravitational ordering; its
temperature regime during metamorphism and periods of
deformation follows Eq. (1). This gradient is lower than
that for the theoretical conductive steady-state geotherm:
numerical calculations:
(1) the dimensionless strain heating value, A = uij. eij, is
close to constant, -lK3;
(2) the second term in the left-hand side of Eq. (10) is
negligible;
(3) the last term in the right-hand side of Eq. (10) is
negligible.
Taking into account these three statements, Eq. (10) can
be simplified as follows:
Equation (9) is then used as the initial condition for Eq.
(11). Equation (11) is well studied in the theory of flame
propagation as the chain reaction mechanism (Zeldovich &
Barenblat, '6959; Zeldovich et al., 1980). According to this
theory, the initial geotherm Eq. (9) is unstable for the two
following conditions for lower rhythms (as a part of t h e
sequence taken into consideration) where H = 10 km:
(the subscript dl denotes that Q is dimensionless;
Z =depth, krn) derived from Eq. (5) at the boundary
conditions: T = 0" C at P = 1 bar and T = 800" C at
P = 10 kbar, and radioactive heat production (see, for
example, Richter, 1985). It is clearly seen for granulite
facies rocks at shallow depth: temperatures of 400-415" C
at pressures of 1.2-1.5 kbar, while according to the above
conductive model, T = 100-120" C at the same pressures.
We suggest that this difference results from strain heating
during gravitational ordering of the heterogeneous crust
being taken into consideration. Numerical modelling of
this process in valuing an extremely heterogeneous
medium is limited by the computational power of the SUN
workstation used. However, with a one-dimensional model
it is possible to estimate the contribution of strain heating
using our results on numerical simulation for a
four-rhythmic sequence.
Consider Eq. (5) in a one-dimensional dimensionless
form using H as the distance scale, H y a as the time scale,
AT =8OO"C as the temperature scale and Ap as the
density scale.
where g = acceleration within the gravity field; Ap =
density difference between layers, exp(yT) = thermal term
of viscosity, p. The following statements result from our
(=a
.H - I
loss velocity.
-
9x
cm yr-I) is the conductive heat
Kydcharacterizes the hydrodynamic velocity:
The Dis term in Eq. (13) is a dissipation number:
A p - g .H
s 1.67.
C;p.AT
Thus, conditions (12) and (13) are satisfied. Instability of
the geotherm (9) results in a non-stationary heat front at a
high rate and, as a result, possible deviation of the
geotherm up to the high-T regime such as the geopath
from Eq. (1). This is the short-term transient regime in the
hydro- and thermodynamic evolution of the chosen
sequence, but during this period the metamorphic
reactions are recording a P-T path.
In Eq. (14), A is only one unknown parameter. For
is
higher values of the configurational parameter A, Vhyd
higher. Parameter A reflects an interaction between layers
of different density and their configuration. For noninteracted rhythms, A must be divided by the square of the
number of rhythms because the characteristic size, H, of
the convective cells is reduced to that of the one rhythm.
Differentiation of the Earth's crust into basaltic and
granitic layers could be explained by the proposed
mechanism and the process completed within a realistic
geological time period. In Fig. 3, strong accumulation of
large quantities of granitic material (shown by the lightest
shading) from different levels of the crust is depicted. Such
Dis =
HYDRODYNAMIC MODELLING OF METAMORPHIC PROCESSES 317
a large amount of granitic material cannot be obtained
from the mechanism of partial melting.
Advective model for ultra high-pressure
complexes
was
)
In this model the high-density material ( p = 4 g ~ m - ~
placed beneath the low-density material (e.g. granite
magma). Figure 4 shows a typical result of the numerical
simulation for ascent of high-density material as a result of
a convective circulation driven by a rising magmatic body
within thick continental crust. We did not study this
process systematically (for conditions of different rheology,
size of buoyant mass, etc.) because of lack of initial data as
discussed above.
To move the high-density rocks to very shallow depth an
additional granite intrusion is needed.
Fig. 4. Typical scenario for uplift of
relatively small fragments of high-density
material, of p = 4 g emm3(black) due to
influence of a large magmatic mass (lighter
grey) ascending toward the Earth's surface.
FEM grid is 41 x 21. Arrows indicate
relative rate of material moving within
advective cells. H = 100 km,p,,,,, = 10'
poise and Ap = pmax/pmin
= lo2. P-T
parameters for retrograde stage are as in
papers by Chopin (1984) and Shatsky et al.
(1989).
318 1. L . PERCHUK, Y . YU. PODLADCHIKOV & A . N. POLYAKOV
CONCLUSIONS
Numerical simulation has shown that rhythmically layered
sequences dramatically accelerate gravitational redistribution of material within the entire sequence. Using this
mechanism, a scenario for the cratonization of ensialic
greenstone belts situated in the middle of a continental
plate has been modelled.
The nature of uniform low dT/dP (at high temperature)
gradients for granulite facies complexes might result
from propagation of a heat front. This front is caused by
strain heating during gravitational ordering of multilayered and multi-rhythmic (heterogeneous) continental
crust. This phenomenon is described in terms of a
mathematical model for flame propagation as the chain
reaction mechanism (Zeldovich & Barenblat, 1959;
Zeldovich et al., 1980).
The P-T loops for high-P, moderate-T complexes
situated in fold belts were used for numerical modelling of
convective processes in thick continental crust. Ascent of
large masses of granite magma within thick crust may
initiate sinking and compression (up to 4 g cm-" of
relatively small masses of the volcanic-sedimentary rocks
at a depth of about 100 km, and subsequent uplift of these
ultra high-P complexes t o the Earth's crust. The possibility
that such advective process are exemplified by Kazakhstan
(Sobolev et al., 1986; Shatsky et al., 1989) are discussed.
ACKNOWLEDGEMENTS
We thank H. Schmeling for fruitful discussion o n the
problems of numerical simulation as well as for a
constructive review of the manuscript. W e are also grateful
to E. Grew for his comments o n the preliminary version of
this paper.
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APPENDIX: COMPUTER PROGRAM
We have chosen triangular elements with continuous quadratic
basis functions for velocity and discontinuous linear basis
functions for pressure. Discontinuous functions were used because
continuous functions were found to produce a 'chessboard' effect
(the incompressibility condition was satisfied for the domain as a
whole, but not for the individual elements).
Solution procedure
Problems are solved with a time-stepping procedure. At each time
step the velocity field is obtained by solving steady-state Stokes
Eqs (2)-(4) for a given geometry. The velocity field is then used
to move the material according to the continuity Eq. (6).
Velocity calculations
At each step our program first calculates the velocity field based
on the current density and distribution. The Galerkin method was
used to reduce Stokes equations to a symmetric, non-positive
definite system of linear equations. This system was assembled
and solved using the frontal method (Hood, 1976) with double
precision (the relative accuracy of the solution was approximately
Updating material properties
Moving a density or viscosity field with sharp discontinuities
through a discrete mesh is difficult due to problems with numerical
diffusion (Hirt & Nichols, 1981). These diffusion problems can be
overcome by using a method of characteristics based o n marker
points. Schmeling (Weinberg & Schmeling, 1991) developed a
marker technique which is very effective for multi-phase flow
where each phase has different rheological properties. Marker
points, containing material property information, are distributed
throughout the numerical mesh. The markers are moved through
the &sh according to the velocity field.
In Schmeling's method, each marker contains information on all
the material phases present. This allows the effective material
properties at the nodal points to be estimated correctly. Also,
uncertainties which arise when more than two phases are present
are avoided.
At each time step, new positions for the markers are calculated
from the current velocity field using the Runge-Kutta method,
The markers are then used to update the material properties of
the FEM mesh. Density and viscosity at each element integration
point is interpolated from the markers in the vicinity of that point.
A high-order, 13-point integration formula was used in order to
conserve detailed information from the marker field in the coarse
FEM mesh. A large number of markers is necessary in order to
reduce numerical diffusion.