DTIC ADA175423: Analysis of Nonlinear Problems in Hydrodynamics and Reaction-Diffusion.

1/1

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

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AD-A175 423 REPORT DOCUMENTATION PAGE

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

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Rensselaer Polytechnic Inst,

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1. S. Arm; Research Oft ice

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

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

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’ ii title (Include Security Classification)

Analysis of Nonlinear Problems in Hydrodynamics and Reaction-Diffusion

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

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

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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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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)