Pressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects


 Edward Waters
 2 years ago
 Views:
Transcription
1 Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University Benha EGYPT Abstract:The nnlinear differential equatins describing the structure f shck waves are reduced t a system f tw cupled nnlinear differential equatins. An apprimate analytical slutin fr such system is btained. This slutin enables us t btain the flw variables and the entrpy as eplicit functins f the dimensinless crdinate. The effects f viscsity and ach number n the velcity, the pressure and the entrpy acrss the shck wave are then investigated. KeyWrds: Pressure, Entrpy, Shck wave, Fluid, Apprimate slutin. 1 Intrductin The structure f nedimensinal shck relatin fr this slutin, but actually n waves is ne f the imprtant practical rigrus prcedure eists t determine such prblems in gas dynamics. This prblem inverse. interested many authrs [19] fr sme years, and the search fr slutins f Navier Stkes equatins has always been their cncern. Early, Taylr [] btained an eplicit slutin f the Navier Stkes equatins, by assuming a cnstant cefficient f viscsity while neglecting the heat cnductivity. Hamad [10] fr a variable cefficient f viscsity that is temperature dependent. The slutins btained The aim in this paper is t suggest an analytical slutin t the prblem in the frm u = u (). The ther flw variables are therefre described as eplicit functins f. Fr the purpse f cmparisn with the results that were btained by Hamad [10] and Taylr [], the cnstants f integratin in equatin (4.7) and (4.8) (Hamad, [10]) are recalculated at the inflectin pint where we set the rigin fr the variable. by Taylr and Hamad are given in the frm = Fr the distributins f velcity, (u) where is the dimensinless distance crdinate and u is the dimensinless velcity. Hwever, it is mre useful t have the inverse pressure and entrpy inside the transitin regin we prpse in the present paper the analytic epressin given in equatins (17), (19) and
2 (1). The effect f the viscsitytemperature relatin n the flw acrss the shck wave can therefre be studied and discussed easily. Basic Equatins The fundamental equatins describing the steady, ne dimensinal flw, parallel t  ais, f viscus, heat cnducting cmpressible fluid may be written in the dimensinal frm (cf. Pai, [11]). d ( ρ u ) = 0, (1) d du dp 4 d du ρ u + ( µ ) = 0, () d d 3 d d d u 4 = 0. d kh u ρ u h + µ (3) d d 3 µ cp Fr a perfect gas, the equatin f state is given by p = ρ RT, (4) where ρ is the density, u the velcity, h the enthalpy, p the pressure, T the abslute temperature, µ the cefficient f viscsity, k the heat cnductivity, R the universal gas cnstant, c p and c v the specific heats at cnstant pressure and cnstant vlume, respectively. It is nw cnvenient t intrduce the unprimed nndimensinal variables as ρ u ρ =, u =, ρ µ µ =, µ u h h =, h p p =, p = where L = 1.55 L µ γ ρ u is the mean free path ahead f the shck frnt and while u = is the ach number as , c γp c = is the sund velcity, ρ cp γ = is c the rati f specific heats and P r is the Prandtl number. It shuld be mentined that the subscript crrespnds t  subscript 1 crrespnds t +. v and If we integrate equatins (1)(4) we find in dimensinless frm that ρ u = 1, (5) 4 du γ u ( ) γ µ u = 1.55 γ, (6) 3 d 1 + γ u + h µ dh = 1.55 d P r 4 du ( γ 1) µ u. 3 d ( γ 1) ( γ 1) γ h+ u 1( + ). (7) We shall assume the viscsity temperature relatin in the frm µ = h. (8) Far in frnt f the wave and far behind the wave all gradients f the variables f state becme zer. T find the velcity and the enthalpy as du d dh = d = 0 +, we cnsider in equatins (6) and (7). Cnsequently ( γ 1) 1 + u 1 = A =, (9) u ( γ + 1)
3 3 ( γ 1) ( γ 1) 1 + γ h 1 = B = (10) h γ + 1 Fr P r = equatins (6) (8), with k = 0, are reduced t γ( γ 1) ( γ 1) h = u ( γ 1)( 1+γ ) u+γ 1, + (11) ( γ + 1) du ( u 1)( u A) =. γ d uh 3 Shck Wave Thickness (1) Discussin f shck wave structure has been based n a single parameter characteristic f the prfile. This parameter δ is the maimum slpe thickness f the prfile f the velcity u and it is defined by Prandtl (cf. rduchw and Libby, [5]) as du δ = (1 A)/ (13) d ma It is cnvenient t take the rigin at the inflectin pint which is essential fr the eistence f a shck wave, where we have d u = 0, at = 0. (14) d The relatin between the velcity u c at the inflectin pint ( = 0) and the velcity A at plus infinity (cf. eyrhff, [6]) is u c = A, (15) fr µ = cnstant. If µ = h, the velcity u c at the rigin is the slutin f the quartic algebraic equatin u 4 + a 1 u 3 + a u + a 3 u + a 4 = 0, (16) where a 1 =(A+1), a = + (1 α) (3 + α) A ( γ 1) a, 4 4A a 3 =, ( 1 + γ ) = α γ ( γ 1) A 1 + =. ( γ 1) α, Equatin (16) can be slved eactly psitin t evaluate the shck wave thickness δ as given by equatin (13). The calculatins fr the case µ = cnstant and the case µ cnstant with different values f are then given in Table I. Table I: The shck wave thickness δ µ = cnstant µ cnstant Structure f Weak Shck Wave If instead f cnsidering the shck wave as a surface f discntinuity we shall lk at it in the present wrk as a transitin regin in which the variatin f the flw variables is cntinuus. The width f the transitin regin is
4 4 cnsidered t etend frm minus infinity t plus infinity, but it culd be shwn that the main change takes place in a regin with finite length. Althugh n eact analytical slutin f equatin (9) and (10) in the frm u = u () is knwn, we prpse an apprimate analytical slutin similar t that cnsidered by Thmpsn et al. ([1]). The frmula t be suggested fr the velcity is the fllwing u = β1 + β tanh ( β 4 ) + β 3 sec h ( β 4), (17) where β 1, β, β 3 and β 4 are fur unknwn cnstants t be determined frm the bundary cnditins n u at ± and at = 0. These cnditins then give β 1 = (A + 1)/, β = (A1)/, β 3 = (u c  β 1 ), du c d β 4 =, (18) β where u c is the eact slutin f Equatin (16) and du c /d is given frm Equatins (11) and (1) by putting u= u c. Table II Cmparisn f the prpsed slutin and that btained eactly by Taylr. 0 = = 1. 0 = 1.3 Eact Prpsed Eact Prpsed slutin slutin slutin slutin Eact slutin Prpsed slutin
5 5 T cmpare ur results with that btained by Talyr [] and Hamad [10], the cnstants f integratins are nt taken t be zer, but we shuld determine them such that the rigin f the crdinate is t be lcated where the velcity prfile has an inflectin pint. A cmparisn between the velcity distributin due t ur suggested slutin in Equatin (17) and the eact slutin btained by Taylr [] are tabulated in Table II. Fr µ cnstant, a cmparisn between the velcity distributin due t ur suggested analytical slutin (17) and that btained by Hamad [10]are tabulated in Table III. Table III Cmparisn f the prpsed slutin and that btained by Hamad = 1.1 = 1. = 1.3 Eact slutin Prpsed slutin Eact slutin Prps ed sluti n Eact sluti n Prpsed slutin
6 6 Frm these Tables it is clear that the velcity distributins inside the transitin regin are identical with that btained by Taylr [] and Hamad [10]. The advantage f having u frm Equatin (17) is that we can directly find the density ρ frm Equatin (5) and the enthalpy h frm equatin (11) where bth are functins f the crdinate. The pressure can als be calculated where it is given by p=ρh (19) The velcity distributins inside the transitin regin fr =1., γ = 5/3 with µ = cnstant and µ cnstant are shwn in Fig. 1. The velcity distributins inside the transitin regin fr =1.5, γ = 5/3 with µ = cnstant and µ cnstant are shwn in Fig.. Als the velcity distributins inside the transitin regin fr the case µ cnstant with =1. and =1.5 are shwn in Fig. 3. The pressure distributins are presented in the fllwing cases : (i) = 1., γ= 5/3 with µ = cnstant and µ cnstant as shwn in Fig. 4. (ii) = 1.4, γ= 5/3 with µ = cnstant and µ cnstant as shwn in Fig The Entrpy Distributin It will be f interest t determine the entrpy S as an eplicit functin f. In general, fr a perfect gas, S = c p ln ( T / T `)  R ln ( p / p `). (0) In a nndimensinal frm, equatin (0) may be written as S S = = ln( hu c v γ 1 ) (1) Using equatins (11) and (17), Equatin (1) gives the entrpy as an eplicit functin f. Fig. (7) shws the distributin f the entrpy when = 1.5 fr µ =cnstant and µ cnstant. T shw the effect f ach numbers n the entrpy distributins inside the transitin regin we carried the calculatins fr the = 1. and = 1.5 cases and these are shwn in Fig Results The apprimate analytical slutin which we have established in this paper can prvide us directly with all calculatins f the shck wave. In fact, we summarize the main results btained as fllws: 1 The shck wave thickness becmes greater when µ cnstant fr cnstant as shwn in Table I.  The shck wave thickness decreases with the increase f as shwn in Table I. 3 The prpsed analytical slutin (17) gives results identical with thse btained via the eact slutin frµ =cnstant r µ cnstant and fr different values f as shwn in Tables II and III. 4 The transitin regin fr µ cnstant is greater than that in case µ = cnstant with cnstant as shwn in Figures 1and fr
7 7 velcity, Figures 4 and 5 fr pressure and Figure 7 fr entrpy. 5 The greater initial ach number will prduce a smaller transitin regin as shwn in Fig. 3 fr velcity, Fig 6 fr entrpy and Figures 4 and 5 fr pressure. 6 The increase in viscsity causes the transitin regin f the entrpy t increase fr a given as shwn in Fig The entrpy increases with the increase f initial ach number as shwn in Fig. 4. References [1] Rankine, W.J..: On the thermdynamic thery f waves f finite lngitudinal disturbance. Transactins f the Ryal Sciety 160,1870, [] Taylr, G.I.: The cnditins necessary fr discntinuus mtin in gases. Prc. R. Sc. Lnd. A 84,1910, 371. [3] Becker, R.: Stsswelle und Detnatin, hysik 8, 19, [4] Thmas, I.H. :Nte n Becker's thery f the shck frnt, Jurnal f Chemical Physics 1,1944, 449. [5] rduchw,. and Libby, P.A. : J. Aernaut. Sci.,1949, 16: 674. [6] eyerhff, L. : J. Aernaut. Sci., 17,1950, 775. [7] ises, R. Vn : On the thickness f a steady shck wave, J. Aernaut. Sci., 17,1955, 551. [8] Wang Chang, C.S. : On the thery f the thickness f weak shck waves. University f ichigan, Dept. f Eng. Research APL/JHU, C [9] Gilbarg, D.& Palucci, D.T.: the structure f shck waves in the cntinuum thery f fluid, J. Rat. ech. Anal., 1953, [10] Hamad, H.: Effect f viscsity n the structure f shck waves. Prceedings f the Ryal Sciety A 45,1996, [11] Pai, S.I.:Intrductin t the thery f cmpressible flw,van Nstr and Reinhld C., New Yrk, Lndn, Trnt, elburne, P [1] Thmpsn, Philip, A., Strck, Thmas W., and Lim, David S. : Phys. Fluids 6, 48, 1983.
8 8
9 9
Effects of piezoviscous dependency on squeeze film between circular plates: Couple Stress fluid model
Turkish Jurnal f Science & Technlgy Vlume 9(1), 97103, 014 Effects f piezviscus dependency n squeeze film between circular plates: Cuple Stress fluid mdel Abstract U. P. SINGH Ansal Technical Campus,
More informationIntroduction: A Generalized approach for computing the trajectories associated with the Newtonian N Body Problem
A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering
More information7.0 Heat Transfer in an External Laminar Boundary Layer
7.0 Heat ransfer in an Eternal Laminar Bundary Layer 7. Intrductin In this chapter, we will assume: ) hat the fluid prperties are cnstant and unaffected by temperature variatins. ) he thermal & mmentum
More informationThermodynamics and Equilibrium
Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,
More informationLim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?
THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques,
More information(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f
1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
More informationCompressibility Effects
Definitin f Cmpressibility All real substances are cmpressible t sme greater r lesser extent; that is, when yu squeeze r press n them, their density will change The amunt by which a substance can be cmpressed
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationExaminer: Dr. Mohamed Elsharnoby Time: 180 min. Attempt all the following questions Solve the following five questions, and assume any missing data
Benha University Cllege f Engineering at Banha Department f Mechanical Eng. First Year Mechanical Subject : Fluid Mechanics M111 Date:4/5/016 Questins Fr Final Crrective Examinatin Examiner: Dr. Mhamed
More informationENGINEERING COUNCIL CERTIFICATE LEVEL THERMODYNAMIC, FLUID AND PROCESS ENGINEERING C106 TUTORIAL 5 THE VISCOUS NATURE OF FLUIDS
ENGINEERING COUNCIL CERTIFICATE LEVEL THERMODYNAMIC, FLUID AND PROCESS ENGINEERING C106 TUTORIAL 5 THE VISCOUS NATURE OF FLUIDS On cmpletin f this tutrial yu shuld be able t d the fllwing. Define viscsity
More informationOn Boussinesq's problem
Internatinal Jurnal f Engineering Science 39 (2001) 317±322 www.elsevier.cm/lcate/ijengsci On Bussinesq's prblem A.P.S. Selvadurai * Department f Civil Engineering and Applied Mechanics, McGill University,
More informationEHed of Curvature on the Temperature Profiles
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1967 EHed f Curvature n the Temperature Prfiles in Cnduding Spines J. E. FRANCIS add R. V. KASER, University f Oklahma, Nrman and GORDON SCOFIELD, University f Missuri,
More informationA Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture
Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu
More informationOF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 50 DECEMBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
More informationSupporting information
Electrnic Supplementary Material (ESI) fr Physical Chemistry Chemical Physics This jurnal is The wner Scieties 01 ydrgen perxide electrchemistry n platinum: twards understanding the xygen reductin reactin
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More informationSOLUTION OF THREECONSTRAINT ENTROPYBASED VELOCITY DISTRIBUTION
SOLUTION OF THREECONSTRAINT ENTROPYBASED VELOCITY DISTRIBUTION By D. E. Barbe,' J. F. Cruise, 2 and V. P. Singh, 3 Members, ASCE ABSTRACT: A twdimensinal velcity prfile based upn the principle f maximum
More informationThe air wave surrounding an expanding sphere. [For summary see p. 292.] INTRODUCTION
The air wave surrunding an expanding sphere By G.. TAYLOR, F.R.S. (Received 5 December 1939) [Fr summary see p. 292.] NTRODUCTON When the surfce f a sphere vibrates in any assigned manner the spherical
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT
17 Kragujevac J. Sci. 8 (006) 174. ON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT Hazem Ali Attia
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References  J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (AddisnWesley, Reading, 966) T study neutrn diffusin
More informationChapter 5: Diffusion (2)
Chapter 5: Diffusin () ISSUES TO ADDRESS... Nnsteady state diffusin and Fick s nd Law Hw des diffusin depend n structure? Chapter 51 Class Eercise (1) Put a sugar cube inside a cup f pure water, rughly
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More information1 The limitations of Hartree Fock approximation
Chapter: PstHartree Fck Methds  I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More information3. Mass Transfer with Chemical Reaction
8 3. Mass Transfer with Chemical Reactin 3. Mass Transfer with Chemical Reactin In the fllwing, the fundamentals f desrptin with chemical reactin, which are applied t the prblem f CO 2 desrptin in ME distillers,
More informationPreCalculus Individual Test 2017 February Regional
The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal MassSpring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal MassSpring System A Hrizntal MassSpring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationOn Huntsberger Type Shrinkage Estimator for the Mean of Normal Distribution ABSTRACT INTRODUCTION
Malaysian Jurnal f Mathematical Sciences 4(): 74 () On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin Department f Mathematical and Physical Sciences, University f Nizwa, Sultanate
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal MassSpring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal MassSpring System A Hrizntal MassSpring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationSurface and Contact Stress
Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated
More informationLecture 17: Free Energy of Multiphase Solutions at Equilibrium
Lecture 17: 11.07.05 Free Energy f Multiphase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTIPHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical
More informationSGP  TR  30 PROCEEDINGS FOURTH WORKSHOP GEOTHERMAL RESERVOIR ENGINEERING. Editors. December1315, , 1978 SGP  TR  30 CONF
SGP  TR  30 SGP  TR  30 CON78122226 PROCEEDINGS OURTH WORKSHOP GEOTHERMAL RESERVOIR ENGINEERING Paul Paul Krugerand and Henry.. Ramey, Ramey., r. r. Editrs December1315, 1315., 1978 DISTRIBUTION
More informationMore Tutorial at
Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers,
More informationTheoretical study of third virial coefficient with Kihara potential
Theretical study f third virial cefficient with Kihara ptential Jurnal: Manuscript ID cjp0170705.r Manuscript Type: Article Date Submitted by the Authr: 6Dec017 Cmplete List f Authrs: Smuncu E.; Giresun
More informationEXPERIMENTAL STUDY ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSSVENTILATION FLOW RATE
EXPERIMENTAL STUD ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSSVENTILATION FLOW RATE Tmnbu Gt, Masaaki Ohba, Takashi Kurabuchi 2, Tmyuki End 3, shihik Akamine 4, and Tshihir Nnaka 2
More informationSEISMIC STABILITY ANALYSIS OF FOOTING ADJACENT TO SLOPES BY SLIP LINE METHOD
The 4 th Wrld Cnference n Earthquake Engineering Octber 7, 008, Beijing, China SEISMIC STABILITY ANALYSIS OF FOOTING ADJACENT TO SLOPES BY SLIP LINE METHOD ABSTRACT : M. Jahanandish and M.R. Arvin Assciate
More informationMODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:
MODULE FOUR This mdule addresses functins SC Academic Standards: EA3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA3.2 Use
More information(Communicated at the meeting of January )
Physics.  Establishment f an Abslute Scale fr the hermelectric Frce. By G. BOR ELlUS. W. H. KEESOM. C. H. JOHANSSON and J. O. LND E. Supplement N0. 69b t the Cmmunicatins frm the Physical Labratry at
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More informationCHAPTER 3 INEQUALITIES. Copyright The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationModeling the Nonlinear Rheological Behavior of Materials with a HyperExponential Type Function
www.ccsenet.rg/mer Mechanical Engineering Research Vl. 1, N. 1; December 011 Mdeling the Nnlinear Rhelgical Behavir f Materials with a HyperExpnential Type Functin Marc Delphin Mnsia Département de Physique,
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationStudy Group Report: Platefin Heat Exchangers: AEA Technology
Study Grup Reprt: Platefin Heat Exchangers: AEA Technlgy The prblem under study cncerned the apparent discrepancy between a series f experiments using a plate fin heat exchanger and the classical thery
More informationExact Solution for Thermal StagnationPoint Flow with Surface Curvature and External Vorticity Effects
Jurnal f Applied Mathematics and Physics, 17, 5, 966989 http://www.scirp.rg/jurnal/jamp ISSN Online: 374379 ISSN Print: 37435 Exact Slutin fr Thermal StagnatinPint Flw with Surface Curvature and External
More information(2) Even if such a value of k was possible, the neutrons multiply
CHANGE OF REACTOR Nuclear Thery  Curse 227 POWER WTH REACTVTY CHANGE n this lessn, we will cnsider hw neutrn density, neutrn flux and reactr pwer change when the multiplicatin factr, k, r the reactivity,
More informationMass transport with varying diffusion and solubility coefficient through a catalytic membrane layer
Mass transprt with varying diffusin and slubility cefficient thrugh a catalytic membrane layer Prceedings f Eurpean Cngress f Chemical Engineering (ECCE6) Cpenhagen, 60 September 007 Mass transprt with
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
More informationELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322
ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private
More informationChE 471: LECTURE 4 Fall 2003
ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.
More informationAn Introduction to Complex Numbers  A Complex Solution to a Simple Problem ( If i didn t exist, it would be necessary invent me.
An Intrductin t Cmple Numbers  A Cmple Slutin t a Simple Prblem ( If i didn t eist, it wuld be necessary invent me. ) Our Prblem. The rules fr multiplying real numbers tell us that the prduct f tw negative
More information( ) ( ) PreCalculus Team Florida Regional Competition March PreCalculus Team Florida Regional Competition March α = for 0 < α <, and
Flrida Reginal Cmpetitin March 08 Given: sin ( ) sin π α = fr 0 < α
More informationWAVE RESISTANCE AND LIFT ON CYLINDERS BY A COUPLED ELEMENT TECHNIQUE. R. Eatock Taylor and G.X. Wu
E Lab.. Sh,essbu&kunde Technic. r. WAVE RESISTANCE AND LIFT ON CYLINDERS BY A COUPLED ELEMENT TECHNIQUE R. Eatck Taylr and G.X. Wu Lndn Centre fr Marine Technlgy Department f Mechanical Engineering University
More informationMaterials Engineering 272C Fall 2001, Lecture 7 & 8 Fundamentals of Diffusion
Materials Engineering 272C Fall 2001, Lecture 7 & 8 Fundamentals f Diffusin Diffusin: Transprt in a slid, liquid, r gas driven by a cncentratin gradient (r, in the case f mass transprt, a chemical ptential
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 13 weeks 610 Majr Learning Indicatrs Identify situatins relevant
More informationLHS Mathematics Department Honors PreCalculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Prealculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationFIELD QUALITY IN ACCELERATOR MAGNETS
FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series
More informationDINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2.
INGWLL EMY Mathematics Higher Prelim Eaminatin 00/0 Paper ssessing Units & NTIONL QULIFITIONS Time allwed  hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin  Questins  0 (0
More information^YawataR&D Laboratory, Nippon Steel Corporation, Tobata, Kitakyushu, Japan
Detectin f fatigue crack initiatin frm a ntch under a randm lad C. Makabe," S. Nishida^C. Urashima,' H. Kaneshir* "Department f Mechanical Systems Engineering, University f the Ryukyus, Nishihara, kinawa,
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationCompressibility and collisional effects on thermal instability of a partially ionized medium
Pram~na, Vl. I0, N. 3, March 978, pp. 267272, printed in India. Cmpressibility and cllisinal effects n thermal instability f a partially inized medium R C SHARMA and K C SHARMA Department f Mathematics,
More informationUnit code: H/ QCF level: 5 Credit value: 15 OUTCOME 3  STATIC AND DYNAMIC FLUID SYSTEMS TUTORIAL 3  VISCOSITY
Unit 43: Plant and Prcess Principles Unit cde: H/601 44 QCF level: 5 Credit value: 15 OUTCOME 3  STATIC AND DYNAMIC FLUID SYSTEMS TUTORIAL 3  VISCOSITY 3 Understand static and namic fluid systems with
More informationCopyright 1979, by the author(s). All rights reserved.
Cpyright 1979, by the authr(s). All rights reserved. Permissin t make digital r hard cpies f all r part f this wrk fr persnal r classrm use is granted withut fee prvided that cpies are nt made r distributed
More information1/2 and e0 e s ' 1+ imm w 4 M s 3 πρ0 r 3 m. n 0 ktr. .Also,since n 0 ktr 1,wehave. 4 3 M sπρ 0 r 3. ktr. 3 M sπρ 0
Chapter 6 6.1 Shw that fr a very weak slutin drplet (m 4 3 πr3 ρ 0 M s ), (6.8) can be written as e 0 ' 1+ a r b r 3 where a σ 0 /n 0 kt and b imm w / 4 3 M sπρ 0. What is yur interpretatin f thecnd and
More informationQuantum Harmonic Oscillator, a computational approach
IOSR Jurnal f Applied Physics (IOSRJAP) eissn: 784861.Vlume 7, Issue 5 Ver. II (Sep.  Oct. 015), PP 3338 www.isrjurnals Quantum Harmnic Oscillatr, a cmputatinal apprach Sarmistha Sahu, Maharani Lakshmi
More informationLi. AEDCTR INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWODIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM
Li. «B ~»»» INFLUENCE OF INITIAL BOUNDARY LAYER ON THE TWODIMENSIONAL TURBULENT MIXING OF A SINGLE STREAM R. C. Bauer and R. J. Matz ARO, Inc. April 1971 Apprved fr public release; distributin unlimited.
More informationAnalysis of Hydrodynamics and Heat Transfer in a Thin Liquid Film Flowing Over a Rotating Disk by the Integral Method
S. Basu B. M. Cetegen 1 Fellw ASME Mechanical Engineering Department, University f Cnnecticut, Strrs, CT 062693139 Analysis f Hydrdynamics and Heat Transfer in a Thin Liquid Film Flwing Over a Rtating
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationRelationship Between Amplifier Settling Time and PoleZero Placements for SecondOrder Systems *
Relatinship Between Amplifier Settling Time and PleZer Placements fr SecndOrder Systems * Mark E. Schlarmann and Randall L. Geiger Iwa State University Electrical and Cmputer Engineering Department Ames,
More informationHubble s Law PHYS 1301
1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and
More information3. Design of Channels General Definition of some terms CHAPTER THREE
CHAPTER THREE. Design f Channels.. General The success f the irrigatin system depends n the design f the netwrk f canals. The canals may be excavated thrugh the difference types f sils such as alluvial
More informationA PLETHORA OF MULTIPULSED SOLUTIONS FOR A BOUSSINESQ SYSTEM. Department of Mathematics, Penn State University University Park, PA16802, USA.
A PLETHORA OF MULTIPULSED SOLUTIONS FOR A BOUSSINESQ SYSTEM MIN CHEN Department f Mathematics, Penn State University University Park, PA68, USA. Abstract. This paper studies travelingwave slutins f the
More informationImplementation of Spur Dikes to Reduce Bank Erosion of Temporary Diversion Channels During Barrages Construction
Australian Jurnal f Basic and Applied Sciences, 3(4): 31903205, 2009 ISSN 19918178 Implementatin f Spur Dikes t Reduce Bank Ersin f Temprary Diversin Channels During Barrages Cnstructin 1 2 3 4 Ali Talaat,
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints  Definitins In this sectin we cnsider nth rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationNUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED CONCRETE STRUCTURES WITH CRACKS
VIII Internatinal Cnference n Fracture Mechanics f Cnete and Cnete Structures FraMCS8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) NUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED
More informationGeneral Chemistry II, Unit II: Study Guide (part 1)
General Chemistry II, Unit II: Study Guide (part 1) CDS Chapter 21: Reactin Equilibrium in the Gas Phase General Chemistry II Unit II Part 1 1 Intrductin Sme chemical reactins have a significant amunt
More informationChapter 4. Unsteady State Conduction
Chapter 4 Unsteady State Cnductin Chapter 5 Steady State Cnductin Chee 318 1 41 Intrductin ransient Cnductin Many heat transfer prblems are time dependent Changes in perating cnditins in a system cause
More information5.4 Measurement Sampling Rates for Daily Maximum and Minimum Temperatures
5.4 Measurement Sampling Rates fr Daily Maximum and Minimum Temperatures 1 1 2 X. Lin, K. G. Hubbard, and C. B. Baker University f Nebraska, Lincln, Nebraska 2 Natinal Climatic Data Center 1 1. INTRODUCTION
More information39th International Physics Olympiad  Hanoi  Vietnam Theoretical Problem No. 1 /Solution. Solution
39th Internatinal Physics Olympiad  Hani  Vietnam  8 Theretical Prblem N. /Slutin Slutin. The structure f the mrtar.. Calculating the distance TG The vlume f water in the bucket is V = = 3 3 3 cm m.
More informationEDA Engineering Design & Analysis Ltd
EDA Engineering Design & Analysis Ltd THE FINITE ELEMENT METHOD A shrt tutrial giving an verview f the histry, thery and applicatin f the finite element methd. Intrductin Value f FEM Applicatins Elements
More informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2Space and 3Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
More informationx x
Mdeling the Dynamics f Life: Calculus and Prbability fr Life Scientists Frederick R. Adler cfrederick R. Adler, Department f Mathematics and Department f Bilgy, University f Utah, Salt Lake City, Utah
More informationEstimation of Thermodynamic Properties and Ionic Equilibria of Cobalt Chloride Solution at 298 K
Materials Transactins, Vl., N. () pp. 117 t 11 # The Japan Institute f Metals Estimatin f Thermdynamic Prperties and Inic Equilibria f Cbalt Chlride Slutin at 98 K Manseung Lee 1 and Yungj Oh 1 Department
More informationA solution of certain Diophantine problems
A slutin f certain Diphantine prblems Authr L. Euler* E7 Nvi Cmmentarii academiae scientiarum Petrplitanae 0, 1776, pp. 858 Opera Omnia: Series 1, Vlume 3, pp. 0517 Reprinted in Cmmentat. arithm. 1,
More informationand for compressible flow
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 9.5 Qantitative Relatins r the Trblent Bndary ayer Descriptin Trblent Flw V and p are randm nctins
More informationAircraft Performance  Drag
Aircraft Perfrmance  Drag Classificatin f Drag Ntes: Drag Frce and Drag Cefficient Drag is the enemy f flight and its cst. One f the primary functins f aerdynamicists and aircraft designers is t reduce
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationFigure 1a. A planar mechanism.
ME 5  Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,
More informationTechnology, Dhauj, Faridabad Technology, Dhauj, Faridabad
STABILITY OF THE NONCOLLINEAR LIBRATION POINT L 4 IN THE RESTRICTED THREE BODY PROBLEM WHEN BOTH THE PRIMARIES ARE UNIFORM CIRCULAR CYLINDERS WITH EQUAL MASS M. Javed Idrisi, M. Imran, and Z. A. Taqvi
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationAP Statistics Notes Unit Two: The Normal Distributions
AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (zscres).
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (RobbinsMiller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (RbbinsMiller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More information4th Indian Institute of Astrophysics  PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics  PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
More informationBASIC DIRECTCURRENT MEASUREMENTS
Brwn University Physics 0040 Intrductin BASIC DIRECTCURRENT MEASUREMENTS The measurements described here illustrate the peratin f resistrs and capacitrs in electric circuits, and the use f sme standard
More informationTHE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALLBOUNDED FLOWS
THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALLBOUNDED FLOWS M.S. Chng Department f Mechanical Engineering The University f Melburne Victria 3010 AUSTRALIA min@unimelb.edu.au J.P. Mnty
More informationNumerical Simulation of the Thermal Resposne Test Within the Comsol Multiphysics Environment
Presented at the COMSOL Cnference 2008 Hannver University f Parma Department f Industrial Engineering Numerical Simulatin f the Thermal Respsne Test Within the Cmsl Multiphysics Envirnment Authr : C. Crradi,
More informationKey words Shock waves. Dusty gas. Solid particles. Shock jump relations. Mach number
Shck jum relatins fr a dusty gas atmshere Shck jum relatins fr a dusty gas atmshere R. K. Anand Deartment f Physics, University f Allahabad, Allahabad00, India Email: anand.rajkumar@rediffmail.cm Abstract
More information