CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
NON-NEWTONIAN VISCOSITY EFFECTS AT SHOCKED
FLUID INTERFACES *
STEVEN M. VALONE
Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract. The regime under consideration here is one in which a polymeric material is shocked
strongly enough to behave as a fluid, but not as a simple fluid. The present work takes a case-study
approach in which several models of shear-rate-dependent viscosities are applied to a RichtmyerMeshkov shock instability at high-viscosity polymeric interface. The early instability growth regime is
modeled with the viscosity dependence of Mikaelian and the late growth regime is modeled with an
aerodynamic viscous-drag model. Shear-rate dependencies of the viscosities can change the shape of
the early growth, while altering the decay in growth rate at later times. Both of these effects should be
observable in experiments. The results suggest that the shear-rate relaxation times will need to be
known to better than an order of magnitude for prediction purposes.
INTRODUCTION
Inviscid,
compressible
shock passage
The subject of hydrodynamic instabilities in
shocked fluid interfaces began very soon after
shock physics became a field of inquiry. The
fluids studied in early investigations were primarily
gases (1,2), treated as incompressible, inviscid
materials. Several later efforts (3,4) introduced
viscous effects into the evolution of the instability
fronts. It was only in the early to middle 1990's
that compressibility effects (5) and drag effects
(6,7) were introduced. However, a general picture
of the flow characteristics associated with the
Richtmyer-Meshkov instability (8,9) emerged. The
evolution is broken down into three stages (Fig. 1):
An initial, inviscid compressible stage associated
with passage of the shock wave through the
material interface; a front growth stage which is
linear in the perturbation amplitude, if treated
inviscidly, and exponentially in time, if treated
with Newtonian viscosity (10); and an
incompressible flow stage characterized by
hydrodynamic drag. The second stage will be
growth rate
Early, viscous
incompressible
\
Late, viscous
incompressible
asymptote
Figure 1. Growth stages of a material interface instability as a
shock wave passes from the light fluid to the heavy one.
referred to as the early stage and the third stage
will be referred as the late stage. Presently,
instability growth in viscous, compressible
materials have not been treated at the level of (e.g)
ZhangandSohn(ll).
The evolution equations for the instability
growth rates are valid for non-Newtonian fluids. It
will always be assumed that the evolution begins at
the early stage. We will study the results of intro-
* Work supported by the US Department of Energy under contract W-7405-ENG-36 to the University of California.
295
proposed and lead to different asymptotic behavior
in the instability growth rate of the peak amplitude.
In these models, it is necessary to separate the
evolution of the spikes (heavy fluid) and the
bubbles (light fluid). All of the models take the
form
ducing non-Newtonian viscosity into existing
interface evolution models. The purpose of this
inquiry is to gauge the sensitivity of the evolution
to the viscosity model and, specifically to shear
rate, and to discern whether or not non-Newtonian
effects should be experimentally observable.
h"=-Ch'lhVA,
MODELS
where A is some characteristic length-scale for the
perturbation and C is some function of r| and A.
The models differ in their assignments for A and C.
For A, Alon et al. (7) (AHOS model) set it to the
ratio of volume-to-surf ace area which is ~A,. The
asymptotic decay for the instability growth rate
goes as 1/t. Dimonte and Schneider (12) (DS
model) and Cheng et al. (13) (COS model) set A =
h. The asymptotic decay then ranges between 1
and 1/t.
For C, CAHOS = cd fAHOs(A)> where cd is a
dimensionless drag coefficient and fAHOs(A) = *
for bubbles and (1-A)/(1+A) for spikes. The DG
model takes CDS = cd with no dependence on A.
Finally, the CGS model (14) takes C = cd.fCGS(A),
where fCGS(A) = (l-A)/2 for bubbles and (l+A)/2
for spikes. The presence of A couples the spike
and bubble flows. The model used for cd is that for
simple laminar flow around a sphere, cd = 24/Re,
where Re = (A/2)2e'p/r| and s' = (h'spike
SDike +
h\bubble
If TI is constant in time, Eqs. (3-4) can be
integrated analytically. For arbitrary t|, Eqs. (3-4)
are simple to integrate numerically. To date, only
constant r| have been considered (7-13)
Viscosity Models: The inviscid, T| = 0, and
constant (zero-shear-rate) viscosity, r| = TJQ, models
have been mentioned in regards to earlier work.
We consider three shear-rate-dependent models
which have been used in various applications of
polymer viscosity modeling. No attempt is made
here to find optimal models for particular
applications or to be comprehensive in covering all
important types of models.
The two models are associated with Bird (14)
andGraessley (15)
Early-time Growth Models: The original RM
model takes the growth rate as constant after the
passage of the shock wave. Letting h be the peak
amplitude of a single-mode instability and h' its
time derivative, the growth-rate constant is
h' = Ah 0 kAu,
(1)
where A is the Atwood number, h0 is the initial
amplitude of the perturbation in the interface, k is
the wavenumber of the perturbation, and Au is the
velocity jump after passage of the shock. The
Atwood number is the ratio
= (p 2 -Pi)/(p 2 + P i ) >
(2)
where p j is the density of the material which is
shocked first and p2 is the density of the material
shocked second as the shock front traverses the
interface. The wavenumber k = 2n/K, where A, is
the wavelength of the perturbation.
Mikaelian (3, 10) derives a viscosity
dependence to the early-time growth by analyzing
the lowest-order moment expansion of the
momentum equation for incompressible flow:
h' = Ah0 k Au exp(-k2T|t),
(4)
(3)
where r| is the viscosity at time t. Mikaelian
considers Newtonian viscosity only. The growth
rate is further modified here with the decay
modeled by Zhang and Sohn (11).
This
modification entails dividing the RHS of Eq. (3) by
a polynomial function of A, k, and t.
The early-time models are considered to be
valid during the time required for the perturbation
to grow to ~ 0.1 A. Eqs. (1) and 3 are considered
valid after the compression time, estimated as the
shock-passage time, ho/D, where D is the shock
speed.
Late-time Drag Models: Several different
versions of the drag evolution equations have been
a
296
(5)
and
(6)
respectively, where TJ^ is the infinite shear-rate
viscosity, £' is the shear rate, T is a characteristic
relaxation time, and a is a constant. The function f
in Eq. (6) is a given in (15). The transcendental
equation must be solved for each value of T|G. Fig.
2 illustrates the shear-rate dependence of the three
models. The Eyring (16) and Graessley models are
fairly similar. The Bird model can be calibrated as
needed for work hardening or shear thinning.
0.00
0.00
0.05
0.10
0.15
0.20
0.25
t(ns)
Figure 3. Shear-rate dependence of the early-time growth. T is
a relaxation time.
0.1
E
'•5
0.01
- 3 - 2 - 1
0
1
2
3
Iog10i 5s/oi(dimensionless)
Figure 2. Shear-rate dependence of various viscosity models
mentioned in the text. I is a relaxation time.
o.oo
0.330
0.335
0.340 0.345
0.350
0.355
0.360
t(ns)
SIMULATIONS AND RESULTS
Figure 4. Shear-rate dependence of the DS growth rate. I is a
relaxation time.
Applying the Newtonian, Bird, Graessley
viscosity models to both the early and late stages
illustrates the differences and capabilities of the
models. Only the DS drag model will be used.
The early-time first-order ODE and the latetime second-order ODEs are integrated with a 5thorder Runga-Kutta algorithm. The same initial
amplitude, aQ = 0.1 Jim, change in shock speed
across the interface, Au = 1 km/s, and T|0 = 103
were used in all cases. See the results in Figs. 3
and 4.
At early times, the viscosity model
dramatically influences the shape of the growth
rate curve. High shear-rate effects cause even
high-viscosity fluids (in terms of TIO) to behave
inviscidly, making it possible for them to exhibit
interface instability. The duration of the growth is
strongly influenced by both T and r|0. During latestage growth, i and T|0 control the decay rate in the
growth profile. These results suggest that T and r|0
determined for one growth/viscosity model
combination are unlikely to be usable for another
model combination. Newtonian viscosity models,
as has always been suspected, seem entirely
inappropriate over any significant range of shock
strengths.
Order-of-magnitude changes in either T or r|0
297
8.
should be readily observable in experiments on
either growth stage. The ID-ODE approaches for
modeling growth may be valuable as a viscositymodel screening tool . The justification requires a
posteriori comparison with experiment.
9.
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