# Full text of "DTIC ADA175423: Analysis of Nonlinear Problems in Hydrodynamics and Reaction-Diffusion."

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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 ^UNCLASSIFIED _ AD-A175 423 REPORT DOCUMENTATION PAGE security classification authority 4 PERFORMING ORGANIZATION REPORT NUMBER(S) IB ac STRlCTIVE MARKINGS 3 DISTRIBUTION’AVAILABILITY OF REPOR; Approved public reiiuse; distribution uni iniitcd. 5 MONITORING ORGANIZATION REPORT NUMBER(S) ARO 11490.16-MA: 14063.23-MA; 16753.24-MA :>a NAME OF PERFORM.NG ORGANIZATION Rensselaer Polytechnic Inst, 6b OFFICE SYMBOL ?a NAME OF VON TORiNG ORGANIZATION (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. .. 7709-221 rsa NAME OF .NDING / SPONSORING ORGANIZATION A. S. Am-. Research Oft ice A0D a tSS (C/ty, State, and ZIP Code) 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 10 SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK Park, NC 37709-2211 ELEMENT NO NO NO- WORK UNIT ACCESSION NO ’ 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 SECURITY CLASSIFIC 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) *»'■ * ‘ « ' H " s' « * * * # % V % J* 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)