Solve The Differential Equation Y 39; 39;-y 39;-6y 0

[Marine Farinha]

The characteristic equation for this differential equation. Ourexamples demonstrated how to solve it if we have two distinct real roots. For complex or repeated roots, a somewhat different strategy is needed. Wewill discuss these other cases later on. For real distinct roots we canuse the quadratic formula an obtain a general solution

In this chapter, the numerical solution of ordinary differential equations (odes) will be described. There is a direct connection between this area and that of partial differential equations (pdes), as noted in, for example, [1]. The ode field is large; but here we restrict ourselves to those techniques that appear again in the pde field. Readers wishing greater depth than is presented here can find it in the great number of texts on the subject, such as the classics by Lapidus and Seinfeld [2], Gear [3], Jain [4] or the very detailed volumes by Hairer et al. [5, 6]. There is a very clear chapter in Gerald [7].

THE UN I VE RSITY OF MI CHI GANCOLLEGE OF ENGINEERINGDepartment of Mechanical EngineeringHeat Transfer and Thermodynamics LaboratoryTechnical Report No. 2THE INFLUENCE OF LOCALIZED, NORMA.L SURFACE OSCILLA.TIONSON THE STEA.DY, LAMINAR FLOW OVER A. FLAT PLATETsung Yen NaVedat S. ArpaciJohn A.. ClarkORA Project 05065under contract with:AERONA.UTICAL RESEARCH LABORATORY, OARAERONAUTICAL SYSTEMS DIVISIONA.IR FORCE SYSTEMS COMMANDCONTRA.CT No. AF 33 (657)-8368WRIGHT-PATTERSON AIR FORCE BASE, OHIOadministered through:OFFICE OF RESEARCH ADMINISTRATION ANN ARBORApri 1 1964

ACKNOWLEDGMENTSThe author is indebted to the many individuals who have contributed tothe success of this work. In particular, he would like to express his sincereappreciation to his thesis adviser, Professor Vedat S. Arpaci, for his continued advice and encouragement and to Professors John A. Clark, Arthur G.Hansen, Wen Jei Yang, and Young King Liu, who served as members of the thesiscommittee.This research is supported by the Aeronautical Research Laboratory, Officeof Aerospace Research,Aeronautical Systems Division, Air Force Systems Command,Wright-Patterson Air Force Base, Ohio. The assistance of Dr. Max G. Scherbergof this laboratory is appreciated.ii

TA.BILE OF CONTENTSPageLIST OF ILLUSTRATIONS ivNOMENCLATURE viABSTRACT xCHAPTERI. INTRODUCTION 1II. THEORETICAL ANALYSIS 5Statement of the Problem 5The Formulation of the Problem 5Kinematic Relations 8Formulation of the Problem in the AcceleratingCoordinate System 15Simplification of the Formulation 20Perturbation of the Differential Equations 28Solutions of the Differential Equations 36Zeroeth-order approximation 36First-order approximation 38Second-order approximation 43Results of the Analysis 53III. CONCLUSIONS 77APPENDIXI. DERIVATION OF EQUATIONS (43) AND (44) 79II. SOLUTION OF EQUATION (114) 85III. A. NUMERICAL EXAMPLE 89Limitations of Xf 89Order-of-Magnitude 91The First-Order Velocity 91Phase Angles 94REFERENCES 96iii

NOMENCLATURE (Continued0( ) order of magnitudep pressurePr Prandtl numberr distance vectorR distance vectorRe Reynolds numberR( 3 real part of a complex quantityt time in (x,y) coordinate systemT time in (X,Y) coordinate systemu velocity in x-directionU velocity in X-directionUO velocity in the mainstreamv velocity in the y-directionV velocity in the Y-directionx abscissa in (x,y) coordinate systemx' abscissa in (xl,yl) coordinate systemX abscissa in (X,Y) coordinate systemy ordinate in (x,y) coordinate systemy' ordinate in (xl,yT) coordinate systemY ordinate in (X,Y) coordinate systemA inverse of the cubic root of Prvii

NOMENCLATURE (Continued)Greek Letters0a phase angleP an angle6 boundary layer thicknessC maximum amplitude of oscillationr dimensionless distance, Y*/6It dimensionless distance, Y*/StO temperaturedimensionless parameter, e'%'4/ReXT dimensionless parameter, e'2't2/Ret1 viscosityp density of fluida normal stressT shearing stressdisturbance function, exp[-m(X-L)2+iutt]Os steady disturbance function, exp [-m'(X'-1)2]a function of X, defined by Eqo (152)X frequencySubscripts0,1,2 zeroth, first and second approximation20 steady component of the second approximationcondition at infinityviii

NOMENCLATURE (Concluded)c complex functionI imaginary part of a complex quantityR real part of a complex quantityw condition at wallx',y' local Cartesian coordinatesX, Y accelerating coordinatesSuperscriptsdimensionless quantities* dimensionless quantities multiplied by Re /2ix

ABSTRACTThe steady, laminar flow of an incompressible fluid overa semi-infiniteflat plate with a localized vibration on the surface of the plate is analyzed.Formulation of the problem is made with reference to a set of coordinateframes fixed on the surface of the plate. The complete momentum, continuity,and energy equations are then simplified by the boundary layer assumptions.The simplified equations are then linearized by the perturbation proceduresand the first three approximations are considered. The zeroth-order approximation is the well-known Blasius' solution. The first- and second-order approximations are solved by the integral method. Integration of the solutions withrespect to time gives a steady component of both skin friction and heat transfer. This steady change depends orn (e'2'4)/Re in the case of skin frictionand (E'2uw'2)/Re in the case of heat transfer as well as the distance from theleading edge of the flat plate. The phase of skin friction is approximatelyin phase with the input disturbance, while that of heat transfer lags approximately t/2 radians. Integration of the steady change over the distance alongthe surface of the plate gives a net increase in skin friction and decreasein heat transfer.x

CHAPTER IINTRODUCTIONIn the last two decades the unsteady boundary layer literature onthe periodic and starting-ending type transients has grown rapidly. Theexisting literature may be divided into two classes, the problems dealing with the acoustic streaming and the problems having vibration. Actually, when an object vibrates, a sound wave is emitted. Therefore,these two areas are very closely related.The phenomenon of acoustic streaming was first observed by Andrade2when he made a photographic study of isothermal streaming in a standingwave tube using tobacco smoke for visual identification. The existenceof these steady streaming (secondary) flows was mathematically established24by Schlichting4 using a successive approximation technique. He predictedthe existence of two regions of streaming in each quadrant around the cylinder; a thin layer next to the cylinder, called the d-c boundary layer,with streaming directed toward the surface along the propagation axis,and an outer streaming with direction away from the surface along thisaxis. The outer region forms a vortex when the fluid is bounded, butin unlimited space the outer core moves to infinity.West28 in his experiments on streaming patterns around small cylinders vibrating in air and in water, observed a very interesting phenomenono Beginning with a feeble motion, streaming is increased graduallywith increasing frequency and increasing amplitude to a certain point1

2where suddenly a new type of circulation is started. Beyond this point,if the frequency is held constant and the amplitude is increased, thevortex system shrinks to a region near the cylinder surrounded by a vigorous circulation of the opposite sense. This phenomena has also been22 26observed by Raney, et al., and Skavlem and Tjotta; however, it hasnot been able to be predicted theoretically. Other experimental and analytical work on streaming in its general aspects include those of West e 29,. 30 20 o 20tervelt,30 Nyborg, and Holtsmark, et al.10The study of the effect of small oscillations in the main stream onthe boundary layer is initiated by Lighthill.l8 He considered the problem of two-dimensional flow about a fixed cylindrical body when fluctuations in the external flow are produced by harmonic fluctuations inmagnitude, but not in the direction of the oncoming streamo. von KarmanPolhausen s technique was used to solve the resulting equations. Healso discussed the temperature fluctuations using successive approximations. Rott and Rosenzweig23 extended Lighthill's analysis in severalways. A practical method for obtaining the response to the larinarboundary layer to an impulsive change in velocity is presented. Hilland Stenning9 consider the case of the boundary layer flow over a cylinder which undergoes a rotational oscillation. The amplitudes and frequencies of oscillation.in all three papers are assumed to be small andthe results are presented in terms of universal functions.Lemlichl7 investigated the effect of transverse vibrations upon theheat transfer rates from horizontal heated wires to air. Anatanarayanan

3and Ramachandranl experimentally studied the effect of vibration on heattransfer from wires to air in forced parallel flow. An increase in heattransfer rates of 130% are observed.Kubanskil4'l5 has studied experimentally the influence of stationary sound fields on free convection from an electrically heated horizontal cylinder in airo The direction of sound wave in his experimentswas longitudinal, ioe., parallel to the axis. For a test cylinder of204 cm in diameter and 32.5 cm long subjected to an intensity of radiated vibrations in the center of the beam from 0.03 to 0.16 W/cm2 and afrequency range of 8 to 30 kc, the free convective heat transfer coefficient is increased by approximately 75%. He also obtained heat transfer data for the case of a horizontal cylinder in a standing sound fieldwith a superimposed horizontal cross flow. 6 The sound wave is perpendicularto the direction of forced flow and also to the axis of the cylinder. Fand and Kaye6'7 performed photographic studies of the boundarylayer flow near a horizontal cylinder in the presence of sound fieldswhose direction of propagation was horizontal and perpendicular to theaxis of the cylinder. A new type of streaming was identified called"thermoacoustic streaming" which is characterized by the formation oftwo vortexes above the cylinder when the sound pressure level reaches acertain critical valueoSchoenhals and Clark5 considered the response of velocity andtemperature of a laminar incompressible fluid to a semiinfinite flatplate oscillating harmonically in a horizontal direction. The method of

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