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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

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. Key-Wrds:- Pressure, Entrpy, Shck wave, Fluid, Apprimate slutin. 1 Intrductin The structure f ne-dimensinal 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 [1-9] 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 viscsity-temperature 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 = (A-1)/, 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 nn-dimensinal 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

(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

(1.1) 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 information

Thermodynamics and Equilibrium

Thermodynamics 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 information

, which yields. where z1. and z2

, 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 information

February 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 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 information

OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION

OF 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 information

Supporting information

Supporting 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 information

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion

22.54 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 (Addisn-Wesley, Reading, 966) T study neutrn diffusin

More information

3. Mass Transfer with Chemical Reaction

3. 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 information

Lecture 5: Equilibrium and Oscillations

Lecture 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 information

Lecture 17: Free Energy of Multi-phase Solutions at Equilibrium

Lecture 17: Free Energy of Multi-phase Solutions at Equilibrium Lecture 17: 11.07.05 Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical

More information

Surface and Contact Stress

Surface 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 information

Study Group Report: Plate-fin Heat Exchangers: AEA Technology

Study Group Report: Plate-fin Heat Exchangers: AEA Technology Study Grup Reprt: Plate-fin 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 information

FIELD QUALITY IN ACCELERATOR MAGNETS

FIELD 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 information

Relationship Between Amplifier Settling Time and Pole-Zero Placements for Second-Order Systems *

Relationship Between Amplifier Settling Time and Pole-Zero Placements for Second-Order Systems * Relatinship Between Amplifier Settling Time and Ple-Zer Placements fr Secnd-Order Systems * Mark E. Schlarmann and Randall L. Geiger Iwa State University Electrical and Cmputer Engineering Department Ames,

More information

Unit code: H/ QCF level: 5 Credit value: 15 OUTCOME 3 - STATIC AND DYNAMIC FLUID SYSTEMS TUTORIAL 3 - VISCOSITY

Unit 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 information

and for compressible flow

and 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 information

A solution of certain Diophantine problems

A solution of certain Diophantine problems A slutin f certain Diphantine prblems Authr L. Euler* E7 Nvi Cmmentarii academiae scientiarum Petrplitanae 0, 1776, pp. 8-58 Opera Omnia: Series 1, Vlume 3, pp. 05-17 Reprinted in Cmmentat. arithm. 1,

More information

Technology, Dhauj, Faridabad Technology, Dhauj, Faridabad

Technology, Dhauj, Faridabad Technology, Dhauj, Faridabad STABILITY OF THE NON-COLLINEAR 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 information

AP Statistics Notes Unit Two: The Normal Distributions

AP 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 (z-scres).

More information

Bootstrap 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) >

Bootstrap 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 information

Lyapunov Stability Stability of Equilibrium Points

Lyapunov Stability Stability of Equilibrium Points Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),

More information

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic. Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage

More information

Numerical Simulation of the Flow Field in a Friction-Type Turbine (Tesla Turbine)

Numerical Simulation of the Flow Field in a Friction-Type Turbine (Tesla Turbine) Numerical Simulatin f the Flw Field in a Frictin-Type Turbine (Tesla Turbine) Institute f Thermal Pwerplants Vienna niversity f Technlgy Getreidemarkt 9/313, A-6 Wien Andrés Felipe Rey Ladin Schl f Engineering,

More information

( ) kt. Solution. From kinetic theory (visualized in Figure 1Q9-1), 1 2 rms = 2. = 1368 m/s

( ) kt. Solution. From kinetic theory (visualized in Figure 1Q9-1), 1 2 rms = 2. = 1368 m/s .9 Kinetic Mlecular Thery Calculate the effective (rms) speeds f the He and Ne atms in the He-Ne gas laser tube at rm temperature (300 K). Slutin T find the rt mean square velcity (v rms ) f He atms at

More information

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell 6.5 Natural Cnvectin in Enclsures Enclsures are finite spaces bunded by walls and filled with fluid. Natural cnvectin in enclsures, als knwn as internal cnvectin, takes place in rms and buildings, furnaces,

More information

" 1 = # $H vap. Chapter 3 Problems

 1 = # $H vap. Chapter 3 Problems Chapter 3 rblems rblem At 1 atmsphere pure Ge melts at 1232 K and bils at 298 K. he triple pint ccurs at =8.4x1-8 atm. Estimate the heat f vaprizatin f Ge. he heat f vaprizatin is estimated frm the Clausius

More information

Dynamic strain softening of concrete in compression under rapid loading K. Fujikake*-, J. Mizuno*, A. Suzuki*, T. Ohno" & T.

Dynamic strain softening of concrete in compression under rapid loading K. Fujikake*-, J. Mizuno*, A. Suzuki*, T. Ohno & T. Transactins n the Built Envirnment vl 2, 1998 WIT Press, www.witpress.cm, ISSN 174-59 Dynamic strain sftening f cncrete in cmpressin under rapid lading K. Fujikake*-, J. Mizun*, A. Suzuki*, T. hn" & T.

More information

Chapters 29 and 35 Thermochemistry and Chemical Thermodynamics

Chapters 29 and 35 Thermochemistry and Chemical Thermodynamics Chapters 9 and 35 Thermchemistry and Chemical Thermdynamics 1 Cpyright (c) 011 by Michael A. Janusa, PhD. All rights reserved. Thermchemistry Thermchemistry is the study f the energy effects that accmpany

More information

Optimization of frequency quantization. VN Tibabishev. Keywords: optimization, sampling frequency, the substitution frequencies.

Optimization of frequency quantization. VN Tibabishev. Keywords: optimization, sampling frequency, the substitution frequencies. UDC 519.21 Otimizatin f frequency quantizatin VN Tibabishev Asvt51@nard.ru We btain the functinal defining the rice and quality f samle readings f the generalized velcities. It is shwn that the timal samling

More information

Kinetics of Particles. Chapter 3

Kinetics of Particles. Chapter 3 Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between

More information

Sensible Performance Analysis of Multi-Pass Cross Flow Heat Exchangers

Sensible Performance Analysis of Multi-Pass Cross Flow Heat Exchangers 108, 11002 (2017) DOI: 101051/ mateccnf/201710811002 Sensible Perfrmance nalysis f Multi-Pass Crss Flw Heat Exchangers 1 Karthik Silaipillayarputhur, awfiq l-mughanam 2, bdulelah I l-niniya 2 1 PO Bx 380,

More information

Homology groups of disks with holes

Homology groups of disks with holes Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.

More information

L a) Calculate the maximum allowable midspan deflection (w o ) critical under which the beam will slide off its support.

L a) Calculate the maximum allowable midspan deflection (w o ) critical under which the beam will slide off its support. ecture 6 Mderately arge Deflectin Thery f Beams Prblem 6-1: Part A: The department f Highways and Public Wrks f the state f Califrnia is in the prcess f imprving the design f bridge verpasses t meet earthquake

More information

Journal of Chemical and Pharmaceutical Research

Journal of Chemical and Pharmaceutical Research Available n line www.jcpr.cm Jurnal f Chemical and Pharmaceutical Research ISSN N: 0975-7384 CODEN(USA): JCPRC5 J. Chem. Pharm. Res., 010, (6):301-305 Electrical cnductivity f s-acetylthichline halides

More information

INVESTIGATION OF B-JUMP NEGATIVE STEP IN RADIAL STILLING BASINS

INVESTIGATION OF B-JUMP NEGATIVE STEP IN RADIAL STILLING BASINS Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 INVESTIGATION OF B-JUMP NEGATIVE STEP IN RADIAL STILLING BASINS A.M. Negm 1, G. M. Abdel-Aal 1, T.M. Owais and A.A. Habib 3 1 Assciate

More information

rcrit (r C + t m ) 2 ] crit + t o crit The critical radius is evaluated at a given axial location z from the equation + (1 , and D = 4D = 555.

rcrit (r C + t m ) 2 ] crit + t o crit The critical radius is evaluated at a given axial location z from the equation + (1 , and D = 4D = 555. hapter 1 c) When the average bld velcity in the capillary is reduced by a factr f 10, the delivery f the slute t the capillary is liited s that the slute cncentratin after crit 0.018 c is equal t er at

More information

Department of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets

Department of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0

More information

arxiv:hep-ph/ v1 2 Jun 1995

arxiv:hep-ph/ v1 2 Jun 1995 WIS-95//May-PH The rati F n /F p frm the analysis f data using a new scaling variable S. A. Gurvitz arxiv:hep-ph/95063v1 Jun 1995 Department f Particle Physics, Weizmann Institute f Science, Rehvt 76100,

More information

COASTAL ENGINEERING Chapter 2

COASTAL ENGINEERING Chapter 2 CASTAL ENGINEERING Chapter 2 GENERALIZED WAVE DIFFRACTIN DIAGRAMS J. W. Jhnsn Assciate Prfessr f Mechanical Engineering University f Califrnia Berkeley, Califrnia INTRDUCTIN Wave diffractin is the phenmenn

More information

CONSTRUCTING STATECHART DIAGRAMS

CONSTRUCTING STATECHART DIAGRAMS CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6

More information

IN a recent article, Geary [1972] discussed the merit of taking first differences

IN a recent article, Geary [1972] discussed the merit of taking first differences The Efficiency f Taking First Differences in Regressin Analysis: A Nte J. A. TILLMAN IN a recent article, Geary [1972] discussed the merit f taking first differences t deal with the prblems that trends

More information

37 Maxwell s Equations

37 Maxwell s Equations 37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut

More information

Chem 75 February 16, 2017 Exam 2 Solutions

Chem 75 February 16, 2017 Exam 2 Solutions 1. (6 + 6 pints) Tw quick questins: (a) The Handbk f Chemistry and Physics tells us, crrectly, that CCl 4 bils nrmally at 76.7 C, but its mlar enthalpy f vaprizatin is listed in ne place as 34.6 kj ml

More information

Finding the Earth s magnetic field

Finding the Earth s magnetic field Labratry #6 Name: Phys 1402 - Dr. Cristian Bahrim Finding the Earth s magnetic field The thery accepted tday fr the rigin f the Earth s magnetic field is based n the mtin f the plasma (a miture f electrns

More information

Zeros of Sections of the Zeta Function. I

Zeros of Sections of the Zeta Function. I Zers f Sectins f the Zeta Functin. I By Rbert Spira 1. Intrductin. The sectins f the title are the Dirichlet plynmials: (1) U(s) = n"5 n = l We write s = a -f- it, and take M ^ 3. Turan [1], [2] shwed

More information

**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!**

**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!** Tpics lists: UV-Vis Absrbance Spectrscpy Lab & ChemActivity 3-6 (nly thrugh 4) I. UV-Vis Absrbance Spectrscpy Lab Beer s law Relates cncentratin f a chemical species in a slutin and the absrbance f that

More information

Spontaneous Processes, Entropy and the Second Law of Thermodynamics

Spontaneous Processes, Entropy and the Second Law of Thermodynamics Chemical Thermdynamics Spntaneus Prcesses, Entrpy and the Secnd Law f Thermdynamics Review Reactin Rates, Energies, and Equilibrium Althugh a reactin may be energetically favrable (i.e. prducts have lwer

More information

Lead/Lag Compensator Frequency Domain Properties and Design Methods

Lead/Lag Compensator Frequency Domain Properties and Design Methods Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin

More information

Example 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W

Example 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W Eample 1 rbt has a mass f 60 kg. Hw much des that rbt weigh sitting n the earth at sea level? Given: m Rbt = 60 kg ind: Rbt Relatinships: Slutin: Rbt =589 N = mg, g = 9.81 m/s Rbt = mrbt g = 60 9. 81 =

More information

Two-Dimensional COMSOL Simulation of Heavy-Oil Recovery by Electromagnetic Heating

Two-Dimensional COMSOL Simulation of Heavy-Oil Recovery by Electromagnetic Heating Excerpt frm the Prceedings f the COMSOL Cnference 9 Bstn w-imensinal COMSOL Simulatin f Heavy-Oil Recvery by Electrmagnetic Heating M. Carrizales *, and Larry W. Lake he University f exas at Austin, *Crrespnding

More information

An analysis of single stage axial-flow turbine performance using three-dimensional calculating methods.

An analysis of single stage axial-flow turbine performance using three-dimensional calculating methods. Calhun: The NPS Institutinal Archive DSpace Repsitry Theses and Dissertatins Thesis and Dissertatin Cllectin 967-09 An analysis f single stage axial-flw turbine perfrmance using three-dimensinal calculating

More information

Design Constraints for Liquid-Protected Divertors

Design Constraints for Liquid-Protected Divertors Design Cnstraints fr iquid-prtected Divertrs S. Shin, S. I. Abdel-Khalik, M. Yda, and the ARIES Team G. W. Wdruff Schl f Mechanical Engineering, Gergia Institute f Technlgy, Atlanta, GA 3033-0405 USA [seungwn.shin@me.gatech.edu,

More information

ABSORPTION OF GAMMA RAYS

ABSORPTION OF GAMMA RAYS 6 Sep 11 Gamma.1 ABSORPTIO OF GAMMA RAYS Gamma rays is the name given t high energy electrmagnetic radiatin riginating frm nuclear energy level transitins. (Typical wavelength, frequency, and energy ranges

More information

On the microwave background anisotropy produced by big voids in open universes

On the microwave background anisotropy produced by big voids in open universes Mn. Nt. R. Astrn. Sc. 80, 1181-1189 (1996) On the micrwave backgrund anistrpy prduced by big vids in pen universes M. J. Fullana,t J. V. Arnau and D. Saez 1 * [Departament d'astrnmia i Astrftsica, Universitat

More information

Reduction of the Adjoint Gradient Formula in the Continuous Limit

Reduction of the Adjoint Gradient Formula in the Continuous Limit Reductin f the Adjint Gradient Frmula in the Cntinuus Limit Antny Jamesn and Sangh Kim Stanfrd University, Stanfrd, CA 9435, U.S.A Abstract We present a new cntinuus adjint methd fr aerdynamic shape ptimizatin

More information

SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis

SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical mdel fr micrarray data analysis David Rssell Department f Bistatistics M.D. Andersn Cancer Center, Hustn, TX 77030, USA rsselldavid@gmail.cm

More information

Determining the Accuracy of Modal Parameter Estimation Methods

Determining the Accuracy of Modal Parameter Estimation Methods Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system

More information

CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS

CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS 1 Influential bservatins are bservatins whse presence in the data can have a distrting effect n the parameter estimates and pssibly the entire analysis,

More information

Physics 2010 Motion with Constant Acceleration Experiment 1

Physics 2010 Motion with Constant Acceleration Experiment 1 . Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin

More information

Design and Analysis of Gas Turbine Blade by Potential Flow Approach

Design and Analysis of Gas Turbine Blade by Potential Flow Approach V. Vijaya kumar et al Int. Jurnal f Engineering Research and Applicatins RESEARCH ARTICLE OPEN ACCESS Design and Analysis f Gas Turbine Blade by Ptential Flw Apprach V. Vijaya Kumar 1, R. Lalitha Narayana

More information

Computational modeling techniques

Computational modeling techniques Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ Mdeling with differential equatins Mdeling strategy Fcus

More information

Comparison of two variable parameter Muskingum methods

Comparison of two variable parameter Muskingum methods Extreme Hydrlgical Events: Precipitatin, Flds and Drughts (Prceedings f the Ykhama Sympsium, July 1993). IAHS Publ. n. 213, 1993. 129 Cmparisn f tw variable parameter Muskingum methds M. PERUMAL Department

More information

Copyright Paul Tobin 63

Copyright Paul Tobin 63 DT, Kevin t. lectric Circuit Thery DT87/ Tw-Prt netwrk parameters ummary We have seen previusly that a tw-prt netwrk has a pair f input terminals and a pair f utput terminals figure. These circuits were

More information

Short-Circuit Current Interruption in a Low-Voltage Fuse with Ablating Walls. By S. Ramakrishnan and w.m.c. van den Heuvel

Short-Circuit Current Interruption in a Low-Voltage Fuse with Ablating Walls. By S. Ramakrishnan and w.m.c. van den Heuvel Shrt-Circuit Current Interruptin in a Lw-Vltage Fuse with Ablating Walls By S. Ramakrishnan and w.m.c. van den Heuvel EUT Reprt 85-E-151 ISBN 90-6144-151-X ISSN 0167-9708 August 1985 Eindhven University

More information

Vane geometry effect on lubrication conditions between vane tip and cam-ring in hydraulic vane machines

Vane geometry effect on lubrication conditions between vane tip and cam-ring in hydraulic vane machines Internatinal Jurnal f Mechanical Engineering and Applicatins 015; 3(1-): 1-10 Published nline Nvember 3, 014 (http://www.sciencepublishinggrup.cm/j/ijmea) di: 10.11648/j.ijmea.s.01503010.11 ISSN: 330-03X

More information

Lecture 7 Further Development of Theory and Applications

Lecture 7 Further Development of Theory and Applications P4 Stress and Strain Dr. A.B. Zavatsk HT08 Lecture 7 Further Develpment f Ther and Applicatins Hke s law fr plane stress. Relatinship between the elastic cnstants. lume change and bulk mdulus. Spherical

More information

LCAO APPROXIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (cation, anion or radical).

LCAO APPROXIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (cation, anion or radical). Principles f Organic Chemistry lecture 5, page LCAO APPROIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (catin, anin r radical).. Draw mlecule and set up determinant. 2 3 0 3 C C 2 = 0 C 2 3 0 = -

More information

Autumn 2012 CHEM452B Bruce H. Robinson 322 Gould Hall HW 10(A) Homework 10A KEY (there will not be a 10B) 2

Autumn 2012 CHEM452B Bruce H. Robinson 322 Gould Hall HW 10(A) Homework 10A KEY (there will not be a 10B) 2 Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) Hmewrk 0A KEY (there will nt be a 0B) QA) Let c be the speed f sund in air. he square f the speed f sund, () f the gas with respect t the change in the

More information

Chaotic behavior of the piccolo

Chaotic behavior of the piccolo Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century PROCEEDINGS f the 22 nd Internatinal Cngress n Acustics Numerical Cmputatin in Musical Acustics: Paper ICA2016-54 Chatic behavir f the piccl

More information

AMERICAN PETROLEUM INSTITUTE API RP 581 RISK BASED INSPECTION BASE RESOURCE DOCUMENT BALLOT COVER PAGE

AMERICAN PETROLEUM INSTITUTE API RP 581 RISK BASED INSPECTION BASE RESOURCE DOCUMENT BALLOT COVER PAGE Ballt ID: Title: USING LIFE EXTENSION FACTOR (LEF) TO INCREASE BUNDLE INSPECTION INTERVAL Purpse: 1. Prvides a methd t increase a bundle s inspectin interval t accunt fr LEF. 2. Clarifies Table 8.6.5 Als

More information

ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn

More information

STUDIES OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACHES TO VISCOUS AUTOMATIC AERODYNAMIC SHAPE OPTIMIZATION

STUDIES OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACHES TO VISCOUS AUTOMATIC AERODYNAMIC SHAPE OPTIMIZATION STUIES OF THE CONTINUOUS AN ISCRETE AJOINT APPROACHES TO VISCOUS AUTOMATIC AEROYNAMIC SHAPE OPTIMIZATION Siva K. Nadarajah and Antny Jamesn epartment f Aernautics and Astrnautics Stanfrd University Stanfrd,

More information

Synchronous Motor V-Curves

Synchronous Motor V-Curves Synchrnus Mtr V-Curves 1 Synchrnus Mtr V-Curves Intrductin Synchrnus mtrs are used in applicatins such as textile mills where cnstant speed peratin is critical. Mst small synchrnus mtrs cntain squirrel

More information

ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE

ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE ?wsw *>*> NASA CONTRACTOR REPORT M NASA CR-1985 ON ADDITIONAL RESEARCH ON INSTABILITIES IN ATMOSPHERIC FLOW SYSTEMS ASSOCIATED WITH CLEAR AIR TURBULENCE by Richard C. Steffler Prepared by UNITED AIRCRAFT

More information

CHEM 116 Electrochemistry at Non-Standard Conditions, and Intro to Thermodynamics

CHEM 116 Electrochemistry at Non-Standard Conditions, and Intro to Thermodynamics CHEM 116 Electrchemistry at Nn-Standard Cnditins, and Intr t Thermdynamics Imprtant annuncement: If yu brrwed a clicker frm me this semester, return it t me at the end f next lecture r at the final exam

More information

Chebyshev Pseudospeetral Method of Viscous Flows with Corner Singularities

Chebyshev Pseudospeetral Method of Viscous Flows with Corner Singularities Jurnal f Scientific Cmputing, Vl. 4, N. 1, 1989 Chebyshev Pseudspeetral Methd f Viscus Flws with Crner Singularities W. W. Schaltz, 1 N. Y. Lee, 1'2 and J. P. Byd 3 Received February 3, 1989 Chebyshev

More information

Direct Monte Carlo Simulation of Time- Dependent Problems

Direct Monte Carlo Simulation of Time- Dependent Problems the Technlgy Interface/Fall 007 Direct Mnte Carl Simulatin f Time- Depent Prblems by Matthew. N. O. Sadiku, Cajetan M. Akujubi, Sarhan M. Musa, and Sudarshan R. Nelatury Center f Excellence fr Cmmunicatin

More information

Lecture 24: Flory-Huggins Theory

Lecture 24: Flory-Huggins Theory Lecture 24: 12.07.05 Flry-Huggins Thery Tday: LAST TIME...2 Lattice Mdels f Slutins...2 ENTROPY OF MIXING IN THE FLORY-HUGGINS MODEL...3 CONFIGURATIONS OF A SINGLE CHAIN...3 COUNTING CONFIGURATIONS FOR

More information

Aerodynamic Shape Optimization Using the Adjoint Method

Aerodynamic Shape Optimization Using the Adjoint Method Aerdynamic Shape Optimizatin Using the Adjint Methd Antny Jamesn epartment f Aernautics & Astrnautics Stanfrd University Stanfrd, Califrnia 94305 USA Lectures at the Vn Karman Institute, russels Febuary

More information

LECTURE 4 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS: 1.0 INTRODUCTION 2.0 SOURCES OF HEAT IN MINES 3.0 STRATA HEAT

LECTURE 4 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS: 1.0 INTRODUCTION 2.0 SOURCES OF HEAT IN MINES 3.0 STRATA HEAT LECTURE 4 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS: 1.0 INTRODUCTION 2.0 SOURCES OF HEAT IN MINES 3.0 STRATA HEAT 3.1 Gethermal Step and Gethermal Gradient 3.2 Thermal Cnductivity f Rcks 3.3 Heat Flux

More information

Chapter 17 Free Energy and Thermodynamics

Chapter 17 Free Energy and Thermodynamics Chemistry: A Mlecular Apprach, 1 st Ed. Nivald Tr Chapter 17 Free Energy and Thermdynamics Ry Kennedy Massachusetts Bay Cmmunity Cllege Wellesley Hills, MA 2008, Prentice Hall First Law f Thermdynamics

More information

Displacement and Deflection Sensitivity of Gas-coupled Laser Acoustic. Detector

Displacement and Deflection Sensitivity of Gas-coupled Laser Acoustic. Detector 1st Internatinal Sympsium n Laser Ultrasnics: Science, echnlgy and Applicatins July 16-18 008, Mntreal, Canada Displacement and Deflectin Sensitivity f Gas-cupled Laser Acustic Detectin James N. CARON

More information

The calculation method of small-scale water injection multiple in water drive reservoirs

The calculation method of small-scale water injection multiple in water drive reservoirs Available nline.jcpr.cm Jurnal f Chemical and Pharmaceutical Research, 04, 6(5):04-09 Research Article ISSN : 0975-7384 CODEN(USA) : JCPRC5 The calculatin methd f small-scale ater injectin multiple in

More information

1. INTRODUCTION. In many polymer processing operations, molten polymers. emerge from dies into a stress field which deforms the

1. INTRODUCTION. In many polymer processing operations, molten polymers. emerge from dies into a stress field which deforms the . NTRODUCTON. Histrical ntes f melt spinning prcess n many plymer prcessing peratins, mlten plymers emerge frm dies int a stress field which defrms the melt int a final fabricated shape. This is the case

More information

We need to do review for some thermodynamic relations: Equation of state P=ρ R T, h= u + pv = u + RT dh= du +R dt Cp dt=cv dt + R dt.

We need to do review for some thermodynamic relations: Equation of state P=ρ R T, h= u + pv = u + RT dh= du +R dt Cp dt=cv dt + R dt. Cmpressible Flw Cmpressible flw is the study f fluids flwing at speeds cmparable t the lcal speed f sund. his ccurs when fluid speeds are abut 30% r mre f the lcal acustic velcity. hen, the fluid density

More information

Einstein's special relativity the essentials

Einstein's special relativity the essentials VCE Physics Unit 3: Detailed study Einstein's special relativity the essentials Key knwledge and skills (frm Study Design) describe the predictin frm Maxwell equatins that the speed f light depends nly

More information

0606 ADDITIONAL MATHEMATICS

0606 ADDITIONAL MATHEMATICS PAPA CAMBRIDGE CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge Internatinal General Certificate f Secndary Educatin MARK SCHEME fr the Octber/Nvember 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maimum

More information

NATURAL CONVECTION HEAT TRANSFER FROM A HEAT SINK WITH FINS OF DIFFERENT CONFIGURATION

NATURAL CONVECTION HEAT TRANSFER FROM A HEAT SINK WITH FINS OF DIFFERENT CONFIGURATION Internatinal Jurnal f Innvatin and Applied Studies ISSN 2028-9324 Vl. 9 N. 3 Nv. 2014, pp. 1043-1047 2014 Innvative Space f Scientific Research Jurnals http://www.ijias.issr-jurnals.rg/ NATURAL CONVECTION

More information

Chapter 32. Maxwell s Equations and Electromagnetic Waves

Chapter 32. Maxwell s Equations and Electromagnetic Waves Chapter 32 Maxwell s Equatins and Electrmagnetic Waves Maxwell s Equatins and EM Waves Maxwell s Displacement Current Maxwell s Equatins The EM Wave Equatin Electrmagnetic Radiatin MFMcGraw-PHY 2426 Chap32-Maxwell's

More information

Ray tracing equations in transversely isotropic media Cosmin Macesanu and Faruq Akbar, Seimax Technologies, Inc.

Ray tracing equations in transversely isotropic media Cosmin Macesanu and Faruq Akbar, Seimax Technologies, Inc. Ray tracing equatins in transversely istrpic media Csmin Macesanu and Faruq Akbar, Seimax Technlgies, Inc. SUMMARY We discuss a simple, cmpact apprach t deriving ray tracing equatins in transversely istrpic

More information

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y= Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents

More information

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is

I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,

More information

INVERSE PROBLEMS IN AERODYNAMICS AND CONTROL THEORY

INVERSE PROBLEMS IN AERODYNAMICS AND CONTROL THEORY INVERSE PROBLEMS IN AERODYNAMICS AND CONTROL THEORY Antny Jamesn Department f Aernautics and Astrnautics Stanfrd University, CA Internatinal Cnference n Cntrl, PDEs and Scientific Cmputing Dedicated t

More information

Supplement 8: Conservative and non-conservative partitioned systems: equivalences and interconversions

Supplement 8: Conservative and non-conservative partitioned systems: equivalences and interconversions Research The quantitatin f buffering actin. I. A frmal and general apprach. Bernhard M. Schmitt Supplement 8: Cnservative and nn-cnservative partitined systems: equivalences and intercnversins The aims

More information

DEDICATED TO THE MEMORY OF R.J. WEINSHENK 1. INTRODUCTION

DEDICATED TO THE MEMORY OF R.J. WEINSHENK 1. INTRODUCTION CONVOLUTION TRIANGLES FOR GENERALIZED FIBONACCI NUMBERS VERNER E. HOGGATT, JR. San Jse State Cllege, San Jse, Califrnia DEDICATED TO THE MEMORY OF R.J. WEINSHENK. INTRODUCTION The sequence f integers Fj

More information

The Sputtering Problem James A Glackin, James V. Matheson

The Sputtering Problem James A Glackin, James V. Matheson The Sputtering Prblem James A Glackin, James V. Mathesn I prpse t cnsider first the varius elements f the subject, next its varius parts r sectins, and finally the whle in its internal structure. In ther

More information

Supplemental Material

Supplemental Material Supplemental Material uman NF-α amin acid sequence: 1 VRSSSRPS PVAVVANP QAEQLQWLN RRANALLAN VELRNQLVV PSELYLIYS 61 QVLFQP SVLLI SRIAVSYQ VNLLSAISP QREPEAE APWYEPIYL 121 VFQLE RLSAEINRP YLFAESQV YFIIAL

More information

UNIV1"'RSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION

UNIV1'RSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION UNIV1"'RSITY OF NORTH CAROLINA Department f Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION by N. L. Jlmsn December 1962 Grant N. AFOSR -62..148 Methds f

More information

COMP 551 Applied Machine Learning Lecture 4: Linear classification

COMP 551 Applied Machine Learning Lecture 4: Linear classification COMP 551 Applied Machine Learning Lecture 4: Linear classificatin Instructr: Jelle Pineau (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted

More information

Time-domain lifted wavelet collocation method for modeling nonlinear wave propagation

Time-domain lifted wavelet collocation method for modeling nonlinear wave propagation Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August Time-dmain lifted wavelet cllcatin methd fr mdeling nnlinear wave prpagatin Kelvin Chee-Mun Lee and Wn-Seng Gan Digital Signal

More information