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AD-A173 423 
UNCLASSIFIED 


analysis of NONLINEAR PROBLEMS in mydrooynahics and 
REACTION-DIFFUSION(U) RENSSELAER POLYTECHNIC INST TROY 
NY D A DREM OCT 86 ARO-11498. 16-NA DRHC04-73-C-8828 

F/G 28/4 

















































^UNCLASSIFIED _ 


AD-A175 423 REPORT DOCUMENTATION PAGE 





security classification authority 


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ARO 11490.16-MA: 14063.23-MA; 16753.24-MA 


:>a NAME OF PERFORM.NG ORGANIZATION 

Rensselaer Polytechnic Inst, 



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(if applicable) 

1. S. Arm; Research Oft ice 


Tb ADDRESS (City. State and ZIP Code) 

P. 0. Box 13 2 1 1 
Research iri in c * e ■ vi k. .. 


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rsa NAME OF .NDING / SPONSORING 
ORGANIZATION 

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P. 0. Box 13311 


8b OFFICE SYMBOL 9 PROCUREMENT .NSTRuMENT IDENTIFICATION NUMBER 

(If applicable) DAHC04-73-C-0028; DAAC29-76-G-03 15 

DAAC29-79-C-0146; DAAC29-82-K-0185 


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PROGRAM 

PROJECT 

TASK 

Park, NC 37709-2211 

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’ ii title (Include Security Classification) 

Analysis of Nonlinear Problems in Hydrodynamics and Reaction-Diffusion 


i 12 PERSONAL AUTHOR(S) 


13b TIME COVE RE 0 14 DATE OF REPORT (Year. Month. Day) 

from 9/1/73 TO 6/30/86 October 1986 


15 PAGE COUNT 

11 


i TYPE OF REPORT 

Final Report 


SUPPLEMENTARY NOTATION 

The view, opinions and/or findings contained in this report are those 
t,the author(s).and should not ,be . const rued as an official Department of the Army position 

11 . N , r fu'i' 1 1 nr . un ' -.r* il.nci h •• • ■ • r h »■» r* if i.Sn __ _' 


COSATi CODES 


GROUP 


SUB-GROUP 


'0 Subject t ERMS ,Continue on revent if necessary and identify by block number) 

Biological dynamics. Morphogenesis, Hydrodynamics 
Fluid dynamics. Lubrication 



OO FORM 1473,84 MAR 


83 APR edition may be uted until e*nau»t#d 
All other editions are obsolete 


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UNCLASSIFIED 


































. 1 ; 


uriciassi1 leu 


SECURITY CLASSIFICATION of This P AGEfWTon om Bnfnd) 


20 . ABSTRACT CONTINUED 


A J e d 


^"^Reaction-diffusion equations occur in many natural and 
technological situations. i< -Two—have been "studied extensively.' 
under this contract-. First, reaction diffusion systems in 
biology include the the release, transport and action of 

neurotransmitters. The effects of other chemicals that enhance 
or block the actions of the neurotransmitter ions have been 
studied,/> -Second, the fluid dynamics of combustion processes have ncXu 
been extensively studied. The stability of flows to 

inhomogeoeities in fuel and temperature^have been described. 


Unclassified __ 

SECURITY CLASSIFICATION OF This PAGEO*T>»n D»r« Enfr»d) 


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ARO 11490.16-MA 
ARO 14063.23-MA 
ARO 16753.14-MA 
ARO 19501.15-MA 


Final Report 

ANALYSIS OF NONLINEAR PROBLEMS IN HYDRODYNAMICS 
AND REACTION'DIFFUSION 


Contract No. DAHC04-73-C-0028 

after 1 September 1976 

Contract No. DAAG 29-76-G-0315 

after September 1, 3979 as 
Contract No. DAAG 29-79-C 0146 

and after September 1, 1982 as 
Contract No. DAAG 29-82-K0185 

Rensselaer Polytechnic Institute 
Troy, New York 12180-3590 


Prepared by D. A. Drew 











The work supported by this contract is focussed on fluid 
dynamics and hydrodynamic stability and related issues. Other 
work on this contract included lubrication work, 
reaction-diffusion systems, dynamics of biochemical systems and 
multiphase flows. 

The problem on the stability and bifurcations of the flow 
between two rotating cylinders was studied for its simplicity, 
importance, and its richness in possible flow patterns. This 
flow situation is important because it is a model for the 
atmosphere (near the equator) and a model for a lubricated 
journal bearing. It is also a good physical situation for the 
study of turbulence. The flow is purely azimuthal for 
sufficiently slow flows, bifurcating to toroidal vortices (Taylor 
vortices) for sufficiently fast flows, bifurcating again to wavy 
vortices at still faster flows, eventually leading to turbulence. 
End conditions due to the finiteness of the apparatus cause the 
bifurcation to be "gradual," with weak vortices existing for very 
slow flows. 

The analysis of this situation centered around the 
quasi-linear stability analyses that start from the linear 
stability problem and assume that the nonlinear solution is an 
eigensolution of the linear problem, modulated by a slowly 
varying amplitude. The slowly varying amplitude (or amplitudes, 
when a second mode occurs, as in the wavy vortices) is governed 
by a nonlinear ordinary differential equation. The structure of 
the equations is quite rich, in that many different bifurcation 
possibilities exist. 

The work on lubrication studied the Reynolds equation for 
two-dimensional and unsteady flows. The Reynolds equation is an 
equation for the pressure in the lubrication area of a bearing, 
derived by assuming the fluid film is thin. In two dimensional 
slider bearings, it is important to describe the leakage out of 
the sides of the bearing. In order to do so, it is necessary to 
examine the dynamics near the edges of the bearing These 
equations were derived and studied. In addition, some results 
were obtained on squeeze films, which are unsteady lubrication 
flows. 

The work on biological dynamics focussed on the stability of 
motions of cells and chemicals from the point of view of 
morphogenesis, or the formation of patterns. It. was discovered 
that spatial pattern formation could be viewed as an instability 
in chemotaxis, the response of cells to secretion and subsequent 
decay of chemical attractors by the cells. Work stemming from 
this early recognition of the possibility has led to similar 
models for morphogenesis in embryos. 

Reaction-diffusion equations occur in many natural and 
technological situations. Two have been studied extensively 
under this contract. First, reaction diffusion systems in 
biology include the the release, transport and action of 














neurotransmitters. The effects of other chemicals that enhance 
or block the actions of the neurotransmitter ions have been 
studied. Second, the fluid dynamics of combustion processes have 
been expensively studied. The stability of flows to 

inhomogeneities in fuel and temperature have been described. 

Polymerization is the process by which long chain 
molecules are formed from monomers. The process is dominated by 
reaction and diffusion. A model for this process has been 
derived, and results giving the rate of formation of polymers 
have been obtained. 

Multiphase fluid dynamics is also important in many 
industrial and natural processes. Under the contract, the 
equations of motion, constitutive equations and predictions from 
these equations have been studied. Terms to make the equations 
well posed have been found. The effect of viscosity in shear 

flows has been quantified. A solution making use of an 

asymptotic analysis of small and large terms in the combustion of 
monopropellant particles has been found. 

People supported by this contract over its long life 

include: 

R. C. DiPrima, who was a pioneer in the Taylor vortex 

flow, but passed away before the end of the contract; 

L. A. Segel, who was one of the early workers in 

biomathematics and reaction-diffusion equations in biology; 

J. T. Stuart of Imperial College, London, a noted fluid 
stability researcher; 

A. R. Kapila, who did much to couple the fluid dyanmics 
to the combustion processes in deflagrations and detonations; 

D. A. Drew, who worked on multiphase flows; 

B. Ng, who did numerical work on the stability and 
bifurcation in Taylor cells; 

P. Hall, who did some basic work on the stability of 
curved flows; 

P. Eagles, who did some of the difficult numerical 
solutions of the Orr Summerfeld equation necessary to quantify 
the bifurcations; 

A. Fridor, who did some numerical work. 













The following students received degrees after some support 
under this contract. 


W. Steinmetz (Ph. D.) 

J. Schmitt (Ph. D. ) 

E. F. Pate (Ph. D. ) 

6. Ganser (Ph. D.) 

T. Jackson (Ph. D.) 

M. Bentrcia (Ph. D. ) 

Publications under this contract and related work’¬ 
ll. C. DiPrima and S. Kogelman, "Stability of Spatially Periodic 
Supercritical Flows in Hydrodynamics", Physics of Fluids 13. 

1-11 (1970). 

R. C. DiPrima and R. N. Grannick, "A Non linear Investigation of 
the Stability of Flow Between Counter-rotating Cylinders", 
Instability of Continuous Systems (IUTAM Symposium Uerrenalb 
1969) Springer-Verlag, Berlin, 1971, 55-60. 

R. C. DiPrima, W. Eckhaus and L. Segel, "Non-linear Wave-number 
Interaction in Near-critical Two-dimensional Flow", Journal of 
Fluid Mechanics, 49, 705-744 (1971). 

R. C. DiPrima and J. T. Stuart, "Flow Between Eccentric Rotating 
Cylinders", Journal of Lubrication Technology, 94, 266 274 
(1972). 

R. C. DiPrima, and J. T. Stuart, "Non-local Effects in the 
Stability of Flow Between Eccentric Rotating Cylinders", Journal 
of Fluid Mechanics, 54, 393-416 (1972). 

R C. DiPrima, "Asymptotic Methods for an Infinitely Long Step 
Slider Squeeze Bearing", Journal of Lubrication Technology, 95, 
208 216 (1973). 

R. C. DiPrima and R. Wollkind, "Effect of a Coriolis Force on 
the Stability of Plane Poiseuille Flow", The Physics of Fluids, 
16, 2045 2051 (1973). 

R. C. DiFrima and J. T. Stuart, "The Nonlinear Calculation of 
Taylor-Vortex Flow in a Lightly-Loaded Journal Bearing", Journal 
of Lubrication Technology, 96, 28-35 (1974). 

R. C. DiPrima and J. Schmitt, "Asymptotic Methods for an 
Infinite Slider Bearing with a Discontinuity in Film Slope", 
Journal of Lubrication Technology, 98, 446-452 (1976). 

R. C. DiPrima and N. Liron, "Effect of Secondary Flow on the 
Decaying Torsional Oscillations of a Sphere and a Plane", The 
Physics of Fluids, 19, 1450-1458 (1976). 














R. C. DiPrima and P. H. Eagles, "Amplification Rates and Torques 
for Taylor-Vortex Flows Between Rotating Cylinders", The Physics 
of Fluids, 20, 171 175 (1977). 

R. C. DiPrima, "Basic Research in Science", Mechanics, 6, 85 88 
(1977). 

R. C. DiPrima and J. A. Schmitt, “Asymptotic Methods for a 
General Finite Width Gas Slider Bearing", Journal of Lubrication 
Technology, 100, 254-260 (1978). 

R. C. DiPrima and J. Flaherty, "Effect of a Coriolis Force on 
the Stability of Tlane Poiseuille Flow", The Physics of Fluids, 
21, 718 726 (1978). 

R. C. DiPrima and J. T. Stuart, "The Eckhaus and Benjamin Feir 
Resonance Mechanisms", Proceedings of the Royal Society of 
London, A362 . 27-41 (1978). 

R C. DiPrima, P. M. Eagles and J. T. Stuart, "The Effects of 
Eccentricity on Torque and Load in Taylor-Vortex Flow", Journal 
of Fluid Mechanics, 87, 209-231 (1978). 

R. C. DiPrima and A. Pridor, "The Stability of Viscous Flow 
Between Rotating Concentric Cylinders with an Axial Flow", 
Proceedings of the Royal Society of London, A366 . 555 573 (1979). 

R. C. DiPrima, "Nonlinear Hydrodynamic Stability", 0. S. 

National Congress of Applied Mechanics, Eighth, Proceedings: 
University of California at Los Angeles, June 26-30, 1978; 
published by Western Periodicals Company, 39 60 (1979). 

R. C. DiPrima and J. T. Stuart, "On the Mathematics of 

Taylor Vortex Flows in Cylinders of Finite Length", Proceedings 

of the Royal Society of London, A372 . 357 365 (1980). 

R. C. DiPrima and H. L. Swinney, "Instabilities and Transition 
in Flow Between Concentric Rotating Cylinders", Tonics in 
Applied Physics (1981), 4.5, Hydrodynamic Instabilities and the 
Transition to Turbulence:, 139 180, Springer Verlag (ed. by H. L. 
Swinney and J. P. Gollub). 

R. C. DiPrima, "Transition in Flow Between Rotating Concentric 
Cylinders", Transition and Turbulence (1981), 1-24, Academic 
Press (Proceedings of a Symposium conducted by the Mathematics 
Research Center, University of Wisconsin-Madison, October 13-15, 
1980, ed. R. E. Meyer. 

J. J. Shepherd and R. C. DiPrima, "Asymptotic Analysis of a 
Finite Gas Slider Bearing of Narrow Geometry", Journal of 
Lubrication Technology, 105 . 491 495 (1983). 















R. C. DiPrima and J. Sijbrand, “Interactions of Axi symmetric and 
Non-axisymmetric Disturbances in the Flow between Concentric 
Rotating Cylinders: Bifurcations near Multiple Eigenvalues", 
Stability in the Mechanics of Continua, 383 386, 1982, 

Springer-Verlag, ed. F. H. Schroeder (Proceedings of a IUTAM 
Symposium, Numbrecht, Germany, August 31 - September 4, 1981). 

R. C. DiPrima and J. T. Stuart, "Hydrodynamic Stability", 

Journal of Applied Mechanics, j>0, 983-991 (1983). 

R. C. DiPrima and P. Hall, “Complex Eigenvalues for the 
Stability of Couette Flow", Proceedings of the Royal Society of 
London, £, 396, 75-94 (1984). 

V. K. Garg and R. C. DiPrima, "The Effect of Beating on the 
Stability of Couette Flow", Physics of Fluids, 27, 812 820 
(1984). 

R. C. Diprima, P. M. Eagles, and B. S. Ng, "The Effect of Radius 
Ratio on the Stability of Couette Flow and Taylor Vortex Flow", 
Physics of Fluids. 

D. J. Marsh and L. A. Segel, “Analysis of Countercurrent 
Diffusion Exchange in Blood Vessels of the Renal Medulla", 
American Journal of Physiology 221 . 817-828, (1971), 

L. A. Segel, "Simplification and Scaling", SIAM Review 14, 

547 571 (1972). 

L. A. Segel and B. Stoeckley, "Instability of a Layer of 
Cherootactic Cells, Attractant, and Degrading Enzyme", Journal of 
Theoretical Biology 37, 561-585 (1972). 

L. A. Segel and J. L. Jackson, "Dissipative Structure: an 
explanation and an ecological example". Journal of Theoretical 
Biology 37, 545=559 (1972). 

L. A. Segel and J. L. Jackson, "Theoretical Analysis of 
Chemotactic Movement in Bacteria", Journal of Mechanochemistry 
and Cell Motility 2, 25-34 (1973). 

J. W. Scanlon and L. A. Segel, "Some Effects of Suspended 
Particles on the Onset of Benard Convection", Physics of Fluids 
16, 1573-1578 (1973). 

T. Scribner, L. A. Segel and E. B. Rogers, "A Numerical Study of 
the Formation and Propagation of Traveling Bands of Chemotactic 
Bacteria", Journal of Theoretical Biology 46, 189-219 (1974). 

A. Levitzki, L. A. Segel and M. Steer, "Cooperative Response of 
Oligometric Protein Receptors Coupled to Non-cooperative Liqand 
Binding", Journal of Molecular Biology 91, 125-130 (1975). 

















L. A. Segel, "Incorporation of Receptor Kinetics into a Model 
for Bacterial Chemotaxis", Journal of Theoretical Biology 57, 
23-42 (1976). 


S. A. Levin and L. A. Segel, "Hypothesis for Origin of 
Planktonic Patchiness”, Mature 259, 659 (1976). 

L. A. Segel, “On Relation Between the Local Interaction of Cells 
and their Global Transformation", Proceedings of Fourth 
International Conference on Theoretical Physics and Biology 
(Versailles, 1973) A. Marois, ed., Amsterdam: North-Holland 
Press (1976). 


L. A. Segel and S. A. Levin, “Applications of nonlinear 
stability theory to the study of the effects of dispersion on 
predator prey interactions". Selected Topics in Statistical 
Mechanics and Biophysics (R. Piccirelli, ed.) American Institute 
of Physics Symposium 27, 123-152 (1976). 

S. Hardt, A. Naparstek, L. A. Segel, and S. R. Caplan, 
"Spatio-temporal structure formation and signal propagation in a 
homogeneous enzymatic membrane". Analysis and Control of 
Immobolized Enzyme Systems (D. Thomas and J. Kernevez, eds.) 
Amsterdam: North-Holland Publishing Co., 9-15 (1976). 

L. A. Segel, "An introduction to continuum theory." Proceedings 
of the SIAM-AMS Summer Seminar in Applied Mathematics Modern 
Modeling of Continuum Phenomena (Lectures in Applied Mathematics 
16, R. C. DiPrima, ed.) American Mathematical Society, 
Providence, RI, 1-60 (1977). 

L. A. Segel, "A theoretical study of receptor mechanisms in 
bacterial chemotaxis." SIAM Journal on Applied Mathematics 32, 
653-665 (1977). 

H. Parnas and L. A. Segel, "Computer evidence concerning 
chemotactic response in aggregating Dictyostelium discoideum." 
Journal of Cell Science 25, 191-204 (1977). 

L. A. Segel, I. Chet, and Y. Hennis, "A simple quantitative 
assay for bacterial motility." Journal of General Microbiology 
98, 329 337 (1977). 

A. Goldbeter and L. A. Segel, "Unified mechanism for relay and 
oscillation of cyclic AMP in Dictyostelium discoideum." 
Proceedings National Academy of Sciences (U.S.A.) 4, 1543-1547 
(1977). 

L. A. Segel, "Mathematical models for cellular behavior." 

Studies in Mathematical Biology (S. Levin, ed.) Mathematical 
Association of America, 156-190 (1978). 















H. Parnas and L. A. Segel, “A computer simulation of pulsatile 
aggregation in Dictyostelium discoideum." Journal of 
Theoretical Biology 71, 185-207 (1978). 

L. A. Segel, "A singular perturbation approach to diffusion 
reaction equations containing a point source, with application 
to the hemolytic plague assay." Journal of Mathematical Biology 
6, 75 85 (1978). 

J. Gressel and L. A. Segel, “The paucity of plants evolving 
genetic resistance to herbicides: possible reasons with 
implications." Journal of Theoretical Biology, 349 371 (1978). 

0. Kedem, I. Rubinstein, and L. A. Segel, "Reduction of 
polarization by ion-conduction spacers: theoretical evaluation 
of a model system." Desalination 27, 143-156 (1978). 

L. A. Segel, "On deducing the nature and effect of 
attractant-receptor binding from population movements of 
chemotactic bacteria." Physical Chemical Aspects of Cell 
Surface Events in Cellular Regulation (C. DeLisi and R. 
Blumenthal, eds.) New York: Elsevier North Holland Publishing 
Co., 293 302 (1979) . 

I. Rubinstein and L. A. Segel, "Breakdown of a stationary 
solution to the Nernst-Planck-Poisson equations." J. Chem. Soc. 
Faraday Transactions II, 75, 936-940 (1979). 

H. Parnas and L. A. Segel, " A theoretical study of calcium 
entry in nerve terminals, with application to neurotransmitter 
release." Journal of Theoretical Biology 84, 3-29 (1980). 

L. A. Segel, " A mathematical model relating to herbicide 
resistance, 1 17, in: Case studies in Mathematical Modelling (W 
Boyce, ed) Pitman Publishing Ltd, London 1981 

A. Goldbeter and L. A. Segel, "Control of developmental 
transitions in the cyclic AMP signalling system of Dictyostelium 
discoideum." Differentiation (in press). 

I. Rubinstein and L. A. Segel, "Sensitivity and instability in 
standing gradient flow." Proc. 28th Int. Congr. Physiological 
Sci. (1980). 

S. I. Rubinow, L. A. Segel and W. Ebel, "A mathematical 
framework for the study of morphogenetic development in the 
slime mold." (submitted for publication) 

M. S. Falkovitz and Lee A. Segel, "Polymerization and Diffusion 
in Unstirred Bulk", Submitted to SIAM Journal on Applied 
Mathematics. 

















B. Parnas and L. A. Segel, “A theoretical study of Calcium entry 
in nerve terminals with application to neurotransmitter 
release.” J. Theoretical Biology, 91, 125 (1981). 

H. Parnas and L. A. Segel, "Ways to discern the presynaptic 
effect of drugs on neurotransmitter release." J. Theoretical 
Biology (1982). 

M. S. Falkovitz and L. A. Segel, "Some Analytical Results 
Concerning the Accuracy of the Continuous Approximation in a 
Polymerization Problem", SIAM Journal on Applied Mathematics, 

42, 542-548 (1982). 

M. S. Falkovitz and L. A. Segel, "Spatially Inhomogeneous 
Polymerization in Unstirred Bulk", SIAM Journal of Applied 
Mathematics, 4J, 386-416 (1983). 

A. Novick Cohen and L. A. Segel, "Polymerization and diffusion 
in Unstirred Bulk", Submitted to Physica D: Journal of Nonlinear 
Analysis 

H. Parnas and L. A. Segel, "A Case Study of Linear versus 
Nonlinear Modelling", Journal of Theoretical Biology, 103 . 

549'580 (1983). 

B. Parnas and L. A. Segel, "Exhaustion of Calcium Does Not 
Terminate Evoked Neurotransmitter Release", Accepted for 
publication by the Journal of Theoretical Biology. 

A. K. Kapila, "Homogeneous branched-chain explosions: initiation 
to completion", J. Engineering Math.. 12 . 221 235 (1978). 

A K. Kapila and B. J. Matkowsky, "Reactive diffusive system 
with Arrhenius kinetics: Multiple solutions, ignition and 
extinction", SIAM J. AppI . Math. . 36, 373-389 (1979). 

A. K. Kapila, B. J. Matkowsky and J. Vega, “Reactive-diffusive 
system with Arrhenius kinetics: the Robin problem", SIAM J. 
AppI. Math. . 38, 382-401 (1980). 

A. K Kapila and B. J. Matkowsky, "Reactive-diffusive system 
with Arrhenius Kinetics: The Robin Problem", SIAM J. AppI . 

Hath.,39, 21 36 (1980). 

A. K. Kapila. "Evolution of Deflagration in a cold combustible 
subjected to a uniform energy flux", Int. J. Engng. Sci. . 19 . 

495 509 (1981). 

A. K. Kapila, "Arrhenius systems: dynamics of jump due to slow 
passage through criticality", SIAM J. AppI. Math. . 41 . 27 42 
(1981). 

A. K. Kapila and A. B. Poore, "Steady response of a nonadiabatic 
tubular reactor", Chem. Engng. Sci .. 37, 57-68 (1981). 













A. K. Kapila, D. S. Stewart and G. S. S. Ludford, “Deflagrations 
and detonations in the limit of small heat release*’, Journal de 
Mecanique Theoretique et Appliquee, 3, 105 (1984). 

A. K. Kapila, B. J. Matkowsky and A. van Barten, "An asymptotic 
theory of deflagrations and detonations. Part I: The Steady 
Solutions", SIAM Journal on Applied Mathematics, 43, 491-519 
(1983). 

A. K. Kapila, "Combustion of a fuel droplet". Proceedings of the 
27th Conference of Army Mathematicians (1981). 

A. K. Kapila, "On Stability Results for Premixed Flows, Based on 
Concentrated-Source Models of Arrhenius Kinetics", Submitted to 
Combustion Science and Technology. 

T. L. Jackson and A. K. Kapila, "Effect of thermal expansion on 
the stability of plane, freely propagating flames," Combustion 
Science and Technology, 41, 191 (1984). 

A. van Elarten, A. K. Kapila and B. J. Matkowsky, "Acoustic 
coupling of flames," SIAM Journal on Applied Mathematics, 44, 

982 995 (1984). 

T. L. Jackson and A. K. Kapila, "Shock-induced thermal runaway," 
accepted for publication by the SIAM Journal on Applied 
Mathematics. 

T. L. Jackson and A. K. Kapila, "Thermal expansion effects on 
perturbed premixed flames," in Reacting Flows (Lectures in 
applied mathematics, v. 24), Proceedings of the ’85 AMS/Siam 
Summer Seminar in Applied Mathematics, G. S. S. Ludford, ed , 

325 (1986). 

P. Hall, "Centrifugal Instabilities in Finite Boundaries: A 
Periodic Model", Accepted for publication by the Journal of 
Fluid Mechanics. 

P Hall and G. Seminara, "Nonlinear Stability of Cavitation 
Bubbles in Sound Fields", Submitted to the Journal of Fluid 
Mechanics. 

P. Hall, "Centrifugal Instability of a Stokes Layer Subharmonie 
Destabilization of the Taylor Vortex", Journal of Fluid 
Mechanics, 105 . 523-530 (1981). 

P. Hall, "Centrifugal Instibilities of Circumferential Flows in 
Finite Cylinders: The Wide Gap Problem", Proceedings of the 
Royal Society of London, A384 . 359-379 (1982). 

A. K. Kapila, "An Asymptotic Theory of Deflagrations and 
Detonations, Part I: The Steady Solutions", Accepted by the SIAM 
Journal on Applied Mathematics. 














A. K. Kapila, "Response of a Plane Flame to a Normally Incident 
Acoustic Wave", Accepted for publication in the Proceedings of 
the International Chemical Reaction Engineering Conference, 
Pune, India. 

D. A. Drew and R. T. Lahey, Jr., "The Virtual Mass and Lift 
Force on a Sphere in Rotating and Straining Flow", accepted by 
International Journal of Multiphase Flow. 

D. A. Drew, "Effect of a Wall on the Lift Force", accepted by 
Chemical Engineering Science 

D. A. Drew, "Hindered Settling of a Fluid Fluid Suspension", 
Proceedings of the ARRADCOM Research and Technology Conference, 
Vol. 1, 347 357 (1983). 

D. A. Drew, "One dimensional burning wave in a bed of 
monopropellant particles," Combustion Science and Technology, 
47, 139 164 (1986) 

J. Schonberg, D. A. Drew, and G. Belfort, "Viscous interactions 
of many neutrally buoyant spheres in Poiseuille flow," J. Fluid 
Mech. 167 415 426 (1986) 

D. A. Drew and G. H. Ganser, "Nonlinear periodic waves in a 
two phase flow model," accepted for publication by SIAM J. 
Applied Mathematics. 

M. Bentrcia and D. A. Drew, "Investigation of the fouling layer 
growth and distribution at the interface of pressure driven 
membranes.- Perturbation method," submitted to Chem. Eng Sci 

M. Bentrcia and D A. Drew, "Investigation of the fouling layer 
growth and distribution at the interface of pressure driven 
membranes. Integral method," submitted to Chem. Eng. Sci. 

E. F. Pate and G. M. Odell, "A Computer Simulation of Chemical 
Signaling During the Aggregation Phase of Dictyostelium 
discoideum", Journal of Theoretical Biology, 88, 201-239 (1981)