Convective heat transfer
|
|
- Penelope Allison
- 5 years ago
- Views:
Transcription
1 CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case, the energy-conservation equation (III.36), an in particular the term representing heat conuction, was never playing a ominant role, with the exception of a brief mention of heat transport in the stuy of static Newtonian fluis (Sec. VI.1.1). The purpose of this Chapter is to shift the focus, an to iscuss motions of Newtonian fluis in which heat is transfere from one region of the flui to another. A first such type of transfer is heat conuction, which was alreay encountere in the static case. Uner the generic term convection, or convective heat transfer, one encompasses flows in which heat is also transporte by the moving flui, not only conuctively. Heat transfer will be cause by ifferences in temperature in a flui. Going back to the equations of motion, one can make a few assumptions so as to eliminate or at least suppress other effects, an emphasize the role of temperature graients in moving fluis (Sec VIII.1). A specific instance of flui motion riven by a temperature ifference, yet also controlle by the flui viscosity, which allows for a richer phenomenology, is then presente in Sec. VIII.2. VIII.1 Equations of convective heat transfer The funamental equations of the ynamics of Newtonian fluis seen in Chap. III inclue heat conuction, in the form of a term involving the graient of temperature, yet the change in time of temperature oes not explicitly appear. To obtain an equation involving the time erivative of temperature, some rewriting of the basic equations is thus neee, which will be one together with a few simplifications (Sec. VIII.1.1). Conuction in a static flui is then recovere as a limiting case. In many instances, the main effect of temperature ifferences is however rather to lea to variations of the mass ensity, which in turn trigger the flui motion. To have a more aapte escription of such phenomena, a few extra simplifying assumptions are mae, leaing to a new, close set of couple equations (Sec. VIII.1.2). VIII.1.1 Basic equations of heat transfer Consier a Newtonian flui submitte to conservative volume forces f ~ V = r ~.Itsmotionis governe by the laws establishe in Chap. III, namely by the continuity equation, the Navier Stokes equation, an the energy-conservation equation or equivalently the entropy-balance equation, which we now recall. Expaning the ivergence of the mass flux ensity, the continuity equation (III.9) becomes D (t,~r) Dt = (t,~r) ~ r ~v(t,~r). (VIII.1a) In turn, the Navier Stokes equation (III.30a) may be written in the form (t,~r) D~v(t,~r) Dt = ~ rp (t,~r) (t,~r) ~ r (t,~r)+2 ~ r (t,~r)s(t,~r) + ~ r (t,~r) ~ r ~v(t,~r). (VIII.1b)
2 120 Convective heat transfer Eventually, straightforwar algebra using the continuity equation allows one to rewrite the entropy balance equation (III.41b) as (t,~r) D apple s(t,~r) = 1 r apple(t,~r) Dt (t,~r) T (t,~r) ~ rt ~ (t,~r) + 2 (t,~r) T (t,~r) (t,~r) 2. S(t,~r): S(t,~r)+ ~r ~v(t,~r) T (t,~r) (VIII.1c) Since we wish to isolate effects irectly relate with the transfer of heat, or playing a role in it, we shall make a few assumptions, so as to simplify the above set of equations. The transport coefficients,, apple epen on the local thermoynamic state of the flui, i.e. on its local mass ensity an temperature T, an thereby inirectly on time an position. Nevertheless, they will be taken as constant an uniform throughout the flui, an taken out of the various erivatives in Eqs. (VIII.1b) (VIII.1c). This is a reasonable assumption as long as only small variations of the flui properties are consiere, which is consistent with the next assumption. Somewhat abusively, we shall in fact even allow ourselves to consier resp. apple as uniform in Eq. (VIII.1b) resp. (VIII.1c), later replace them by relate (iffusion) coefficients = / resp. = apple/ c P,anthenconsierthelatterasuniformconstantquantities. The whole proceure is only justifie in that one can check by comparing calculations using this assumption with numerical computations performe without the simplifications that it oes not lea to omitting a physical phenomenon. The flui motions uner consieration will be assume to be slow, i.e. to involve a small flow velocity, in the following sense: The incompressibility conition ~ r ~v(t,~r) =0will hol on the right han sies of each of Eqs. (VIII.1). Accoringly, Eq. (VIII.1a) simplifies to D (t,~r)/dt =0while Eq. (VIII.1b) becomes the incompressible Navier + ~v(t,~r) ~r ~v(t,~r) = in which the kinematic viscosity is taken to be constant. 1 (t,~r) ~ rp (t,~r) ~ r (t,~r)+ 4~v(t,~r), (VIII.2) The rate of shear is small, so that its square can be neglecte in Eq. (VIII.1c). Accoringly, that equation simplifies to (t,~r)t (t,~r) D apple s(t,~r) = apple4t (t,~r). (VIII.3) Dt (t,~r) The term on the left han sie of that equation can be further rewritten. As a matter of fact, one can show that the ifferential of the specific entropy in a flui particle is relate to the change in temperature by s T = c P T, (VIII.4) where c P enotes the specific heat capacity at constant pressure of the flui. In turn, this relation translates into an ientity relating the material erivatives when the flui particles are followe in their motion. The left member of Eq. (VIII.3) may then be expresse in terms of the substantial erivative of the temperature. Introucing the thermal iffusivity (lxxii) apple c P, which will from now on be assume to be constant an uniform in the flui, where c P (lxxii) Temperaturleitfähigkeit (VIII.5) is the
3 VIII.1 Equations of convective heat transfer 121 volumetric heat capacity at constant pressure, one eventually obtains DT (t,~r) + ~v(t,~r) ~r T (t,~r) = 4T (t,~r), (VIII.6) which is sometimes referre to as (convective) heat transfer equation. If the flui is at rest or if its velocity is small enough to ensure that the convective term~v ~rt remains negligible, Eq. (VIII.6) simplifies to the classical heat iffusion equation, with iffusion constant. The thermal iffusivity thus measures the ability of a meium to transfer heat iffusively, just like the kinematic shear viscosity characterizes the iffusive transfer of momentum. Accoringly, both coefficients have the same imension L 2 T 1, an can thus be meaningfully compare. Their relative strength is measure by the imensionless Prantl number Pr = c P apple (VIII.7) which in contrast to the Mach, Reynols, Froue, Ekman, Rossby... numbers encountere in the previous chapters is entirely etermine by the flui, inepenent of any flow characteristics. VIII.1.2 Boussinesq approximation If there is a temperature graient in a flui, it will lea to a heat flux ensity, an thereby to the transfer of heat, thus influencing the flui motion. However, heat exchanges by conuction are often slow except in metals, so that another effect ue to temperature ifferences is often the first to play a significant role, namely thermal expansion (or contraction), which will lea to buoyancy (Sec. IV.1.1) when a flui particle acquires a mass ensity ifferent from that of its surrounings. The simplest approach to account for this effect, ue to Boussinesq, (52) consists in consiering that even though the flui mass ensity changes, nevertheless the motion can to a very goo approximation be viewe as incompressible which is what was assume in Sec. VIII.1.1: ~r ~v(t,~r) ' 0, (VIII.8) where ' is use to allow for small relative variations in the mass ensity, which are irectly relate to the expansion rate through Eq. (VIII.1a). Denoting by T 0 a typical temperature in the flui an 0 the corresponing mass ensity (strictly speaking, at a given pressure), the effect of thermal expansion on the latter reas with ( ) = 0 (1 (V ) ), (VIII.9) T T 0 (VIII.10) the temperature ifference measure with respect to the reference value, an (V P,N (VIII.11) the thermal expansion coefficient for volume, where the erivative is taken at the thermoynamic point corresponing to the reference value 0. Strictly speaking, the linear regime (VIII.9) (53) only hols when (V ) 1, which will be assume hereafter. (52) Hence its enomination Boussinesq approximation (for buoyancy). (53)... which is obviously the beginning of a Taylor expansion.
4 122 Convective heat transfer Consistent with relation (VIII.9), the pressure term in the incompressible Navier Stokes equation can be approximate as 1 (t,~r) ~ rp (t,~r) ' ~ rp (t,~r) 0 1+ (V ) (t,~r). Introucing an effective pressure P e which accounts for the leaing effect of the potential from which the volume forces erive, P e. (t,~r) P (t,~r)+ 0 (t,~r), one fins 1 rp (t,~r) ~ (t,~r) r ~ rp ~ e. (t,~r) (t,~r) ' + (V ) (t,~r) r ~ (t,~r), 0 where a term of subleaing orer (V ) rp ~ e. has been roppe. To this level of approximation, the incompressible Navier Stokes equation + ~v(t,~r) ~r ~v(t,~r) = ~ rp e. (t,~r) 0 + (V ) (t,~r) ~ r (t,~r)+ 4~v(t,~r). (VIII.12) This form of the Navier Stokes equation emphasizes the role of a finite temperature ifference in proviing an extra force ensity which contributes to the buoyancy, supplementing the effective pressure term. Eventually, efinition (VIII.10) together with the convective heat transfer equation (VIII.6) lea at once (t,~r) + ~r (t,~r) = 4 (t,~r). (VIII.13) The (Oberbeck (ap) )Boussinesq equations (VIII.8), (VIII.12), an (VIII.13) represent a close system of five couple scalar equations for the ynamical fiels~v, which in turn yiels the whole variation of the mass ensity an P e.. VIII.2 Rayleigh Bénar convection A relatively simple example of flow in which thermal effects play a major role is that of a flui between two horizontal plates at constant but ifferent temperatures, the lower plate being at the higher temperature, in a uniform gravitational potential r ~ (t,~r) =~g, in the absence of horizontal pressure graient. The istance between the two plates will be enote by, an the temperature ifference between them by T, where T>0when the lower plate is warmer. When neee, a system of Cartesian coorinates will be use, with the (x, y)-plane miway between the plates an a vertical z-axis, with the acceleration of gravity pointing towars negative values of z, i.e. (t,~r) =gz. VIII.2.1 Phenomenology of the Rayleigh Bénar convection VIII.2.1 a :::::::: :::::::::::::::::::::::: Experimental finings If both plates are at the same temperature or if the upper one is the warmer ( T<0), the flui between them can simply be at rest, with a stationary linear temperature profile. As a matter of fact, enoting by T 0 resp. P 0 the temperature resp. pressure at a point at z =0 an 0 the corresponing mass ensity, one easily checks that equations (VIII.8), (VIII.12), (VIII.13) amit the static solution ~v st. (t,~r) =~0, st. (t,~r) = z T, P e.,st.(t,~r) =P 0 0 g z2 2 (V ) T, (VIII.14) (ap) A. Oberbeck,
5 VIII.2 Rayleigh Bénar convection 123 with the pressure given by P st. (t,~r) =P e.,st. (t,~r) 0 gz. Since z/ < 1 2 an (V ) T 1, one sees that the main part of the pressure variation ue to gravity is alreay absorbe in the efinition of the effective pressure. If T =0, one recognizes the usual linear pressure profile of a static flui at constant temperature in a uniform gravity fiel. One can check that the flui state efine by the profile (VIII.14) is stable against small perturbations of any of the ynamical fiels. To account for that property, that state (for a given temperature ifference T ) will be referre to as equilibrium state. Increasing now the temperature of the lower plate with respect to that of the upper plate, for small positive temperature ifferences T nothing happens, an the static solution (VIII.14) still hols an is still stable. When T reaches a critical value T c, the flui starts eveloping a pattern of somewhat regular cylinrical omains rotating aroun their longituinal, horizontal axes, two neighboring regions rotating in opposite irections. These omains in which warmer an thus less ense flui rises on the one sie while coler, enser flui escens on the other sie, are calle Bénar cells. (aq) 6 Figure VIII.1 Schematic representation of Bénar cells between two horizontal plates. The transition from a situation in which the static flui is a stable state, to that in which motion evelops i.e. the static case is no longer stable, is referre to as (onset of the) Rayleigh Bénar instability. Since the motion of the flui appears spontaneously, without the nee to impose any external pressure graient, it is an instance of free convection or natural convection in opposition to force convection). Remarks: Such convection cells are omnipresent in Nature, as e.g. in the Earth mantle, in the Earth atmosphere, or in the Sun convective zone. When T further increases, the structure of the convection pattern becomes more complicate, eventually becoming chaotic. In a series of experiments with liqui helium or mercury, A. Libchaber (ar) an his collaborators observe the following features [36, 37, 38]: Shortly above T c,thestablefluistateinvolve cylinrical convective cells with a constant profile. Above a secon threshol, oscillatory convection evelops: that is, unulatory waves start to propagate along the surface of the convective cells, at first at a unique (angular) frequency! 1,then as T further increases also at higher harmonics n 1! 1, n 1 2 N. Asthetemperatureifference T reaches a thir threshol, a secon unulation frequency! 2 appears, incommensurate with! 1,lateraccompaniebythecombinations n 1! 1 + n 2! 2, with n 1,n 2 2 N. Athigher T, the oscillator with frequency! 2 experiences a shift from its proper frequency to a neighboring submultiple of! 1 e.g.,! 1 /2 in the experiments with He, illustrating the phenomenon of frequency locking. Forevenhigher T, submultiplesof! 1 appear ( frequency emultiplication ), then a low-frequency continuum, an eventually chaos.? (aq) H. Bénar, (ar) A. Libchaber, born1934
6 124 Convective heat transfer Each appearance of a new frequency may be seen as a bifurcation. The ratios of the experimentally measure lengths of consecutive intervals between successive bifurcations provie an estimate of the (first) Feigenbaum constant (as) in agreement with its theoretical value thereby proviing the first empirical confirmation of Feigenbaum s theory. VIII.2.1 b ::::::::: ::::::::::::::::::::::: Qualitative iscussion Consier the flui in its equilibrium state of rest, in the presence of a positive temperature ifference T, so that the lower layers of the flui are warmer than the upper ones. If a flui particle at altitue z acquires, for some reason, a temperature that iffers from the equilibrium temperature measure with respect to some reference value (z), then its mass ensity given by Eq. (VIII.9) will iffer from that of its environment. As a result, the Archimees force acting on it no longer exactly balances its weight, so that it will experience a buoyancy force. For instance, if the flui particle is warmer that its surrounings, it will be less ense an experience a force irecte upwars. Consequently, the flui particle shoul start to move in that irection, in which case it encounters flui which is even coler an enser, resulting in an increase buoyancy an a continue motion. Accoring to that reasoning, any vertical temperature graient shoul result in a convective motion. There are however two effects that counteract the action of buoyancy, an explain why the Rayleigh Bénar instability necessitates a temperature ifference larger than a given threshol. First, the rising particle flui will also experience a viscous friction force from the other flui regions it passes through, which slows its motion. Seconly, if the rise of the particle is too slow, heat has time to iffuse by heat conuction through its surface: this tens to equilibrate the temperature of the flui particle with that of its surrounings, thereby suppressing the buoyancy. Accoringly, we can expect to fin that the Rayleigh Bénar instability will be facilitate when (V ) Tg i.e. the buoyancy per unit mass increases, as well as when the thermal iffusivity an the shear viscosity ecrease. Translating the previous argumentation in formulas, let us consier a spherical flui particle with raius R, an assume that it has some vertically irecte velocity v, while its temperature initially equals that of its surrounings. With the flui particle surface area, proportional to R 2, an the thermal iffusivity apple, one can estimate the characteristic time for heat exchanges between the particle an the neighboring flui, namely Q = C R2 with C a geometrical factor. If the flui particle moves with constant velocity v uring that uration Q, while staying at almost constant temperature since heat exchanges remain limite, the temperature ifference it acquires with respect to the neighboring flui v Q = C T R 2 v, where T/ is the temperature graient in the flui impose by the two plates. This temperature ifference gives rise to a mass ensity ifference = 0 (V ) = C 0 v R2 between the particle an its surrounings. As a result, flui particle experiences an upwars irecte buoyancy 4 3 R3 g = 4 C 3 0gv R5 (V ) T. (VIII.15) (as) M. Feigenbaum, born1944 (V ) T,
7 VIII.2 Rayleigh Bénar convection 125 On the other han, the flui particle is slowe in its vertical motion by the ownwars oriente Stokes friction force acting on it, namely, in projection on the z-axis F Stokes = 6 R v. (VIII.16) Note that assuming that the velocity v remains constant, with a counteracting Stokes force that is automatically the goo one, relies on the picture that viscous effects aapt instantaneously, i.e. that momentum iffusion is fast. That is, the above reasoning actually assumes that the Prantl number (VIII.7) is much larger than 1; yet its result is inepenent from that assumption. Comparing Eqs. (VIII.15) an (VIII.16), buoyancy will overcome friction, an thus the Rayleigh Bénar instability take place, when 4 C 3 0gv R5 (V ) T > 6 R 0 v, (V ) TgR 4 > 9 2C. Note that the velocity v which was invoke in the reasoning actually rops out from this conition. Taking for instance R = /2 which maximizes the left member of the inequality, this becomes Ra (V ) Tg 3 > 72 C =Ra c. Ra is the so-calle Rayleigh number an Ra c its critical value, above which the static-flui state is instable against perturbation an convection takes place. The value 72/C foun with the above simple reasoning on force equilibrium is totally irrelevant both careful experiments an theoretical calculations agree with Ra c = 1708 for a flui between two very large plates, the important lesson is the existence of a threshol. VIII.2.2 Toy moel for the Rayleigh Bénar instability A more refine although still crue toy moel of the transition to convection consists in consiering small perturbations ~v,, P e. aroun a static state ~v st. = ~0, st., P e.,st.,anto linearize the Boussinesq equations to first orer in these perturbations. As shown by Eq. (VIII.14), the effective pressure P e.,st. actually alreay inclues a small correction, ue to (V ) T being much smaller than 1, so that we may from the start neglect P e.. To first orer in the perturbations, Eqs. (VIII.12), projecte on the z-axis, an (VIII.13) give, after subtraction of the contributions from the static = 4v z + (V ) v z = 4. (VIII.17a) (VIII.17b) Moving the secon term of the latter equation to the right han sie increases the parallelism of this set of couple equations. In aition, there is also the projection of Eq. (VIII.12) along the x-axis, an the velocity fiel must obey the incompressibility conition (VIII.8). The proper approach woul now be to specify the bounary conitions, namely: the vanishing of v z at both plates impermeability conition, the vanishing of v x at both plates no-slip conition, an the ientity of the flui temperature at each plate with that of the corresponing plate; that is, all in all, 6 conitions. By manipulating the set of equations, it can be turne into a 6th-orer linear partial ifferential equation for, on which the bounary conitions can be impose. Instea of following this long roa, (54) we refrain from trying to really solve the equations, but rather make a simple ansatz, namely v z (t,~r) =v 0 e t cos(kx) which automatically fulfills the (54) The reaer may fin etails in Ref. [39, Chap. II].
8 126 Convective heat transfer incompressibility equation, but clearly violates the impermeability conitions, an a similar one for, with a constant. Substituting these forms in Eqs. (VIII.17) yiel the linear system v 0 = k 2 v 0 + (V ) 0 g, + k 2 v 0 g (V ) 0 =0, 0 = k T v T 0, v 0 + k 2 0 =0 for the amplitues v 0, 0. This amits a non-trivial solution only if + k 2 + k 2 (V ) T g =0. (VIII.18) This is a straightforwar quaratic equation for. It always has two real solutions, one of which is negative corresponing to a ampene perturbation since their sum is ( + )k 2 < 0; the other solution may change sign since their prouct k 4 (V ) T g is positive for T =0, yieling a secon negative solution, yet changes sign as T increases. The vanishing of this prouct thus signals the onset of instability. Taking for instance k = / to fix ieas, this occurs at a critical Rayleigh number Ra c = (V ) Tg 3 = 4, where the precise value (here 4 ) is irrelevant. From Eq. (VIII.18) also follows that the growth rate of the instability is given in the neighborhoo of the threshol by = Ra Ra c Ra c + k2, i.e. it is infinitely slow at Ra c. This is reminiscent of a similar behavior in the vicinity of the critical point associate with a thermoynamic phase transition. By performing a more rigorous calculation incluing non-linear effects, one can show that the velocity amplitue at a given point behaves like Ra Rac v / with = 1 (VIII.19) Ra c 2 in the vicinity of the critical value, an this preiction is borne out by experiments [40]. Such apowerlawbehaviorisagainreminiscentofthethermoynamicsofphasetransitions,more specifically here since v vanishes below Ra c an is finite above of the behavior of the orer parameter in the vicinity of a critical point. Accoringly the notation use for the exponent in relation (VIII.19) is the traitional choice for the critical exponent associate with the orer parameter of phase transitions. Eventually, a last analogy with phase transitions regars the breaking of a symmetry at the threshol for the Rayleigh Bénar instability. Below Ra c, the system is invariant uner translations parallel to the plates, while above Ra c that symmetry is spontaneously broken. Bibliography for Chapter VIII A nice introuction to the topic is to be foun in Ref. [41], which is a popular science account of part of Ref. [42]. Faber [1] Chapter & 9.2. Guyon et al. [2] Chapter Lanau Lifshitz [4, 5] Chapter V &
arxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationChapter 2 Governing Equations
Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement
More informationA note on the Mooney-Rivlin material model
A note on the Mooney-Rivlin material moel I-Shih Liu Instituto e Matemática Universiae Feeral o Rio e Janeiro 2945-97, Rio e Janeiro, Brasil Abstract In finite elasticity, the Mooney-Rivlin material moel
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More information6. Friction and viscosity in gasses
IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationfv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n
Chapter 11 Rossby waves Supplemental reaing: Pelosky 1 (1979), sections 3.1 3 11.1 Shallow water equations When consiering the general problem of linearize oscillations in a static, arbitrarily stratifie
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationFluid Mechanics EBS 189a. Winter quarter, 4 units, CRN Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 Veihmeyer Hall.
Flui Mechanics EBS 189a. Winter quarter, 4 units, CRN 52984. Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 eihmeyer Hall. Course Description: xioms of flui mechanics, flui statics,
More informationApplications of First Order Equations
Applications of First Orer Equations Viscous Friction Consier a small mass that has been roppe into a thin vertical tube of viscous flui lie oil. The mass falls, ue to the force of gravity, but falls more
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationCompletely passive natural convection
Early View publication on wileyonlinelibrary.com (issue an page numbers not yet assigne; citable using Digital Object Ientifier DOI) ZAMM Z. Angew. Math. Mech., 1 6 (2011) / DOI 10.1002/zamm.201000030
More informationu t v t v t c a u t b a v t u t v t b a
Nonlinear Dynamical Systems In orer to iscuss nonlinear ynamical systems, we must first consier linear ynamical systems. Linear ynamical systems are just systems of linear equations like we have been stuying
More informationME338A CONTINUUM MECHANICS
global vs local balance equations ME338A CONTINUUM MECHANICS lecture notes 11 tuesay, may 06, 2008 The balance equations of continuum mechanics serve as a basic set of equations require to solve an initial
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationMAE 210A FINAL EXAM SOLUTIONS
1 MAE 21A FINAL EXAM OLUTION PROBLEM 1: Dimensional analysis of the foling of paper (2 points) (a) We wish to simplify the relation between the fol length l f an the other variables: The imensional matrix
More information'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21
Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationApplication of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More information5-4 Electrostatic Boundary Value Problems
11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationHomework 7 Due 18 November at 6:00 pm
Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationAnalytic Scaling Formulas for Crossed Laser Acceleration in Vacuum
October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945
More informationThe effect of nonvertical shear on turbulence in a stably stratified medium
The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: 10.1063/1.869640 View online:
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationSources and Sinks of Available Potential Energy in a Moist Atmosphere. Olivier Pauluis 1. Courant Institute of Mathematical Sciences
Sources an Sinks of Available Potential Energy in a Moist Atmosphere Olivier Pauluis 1 Courant Institute of Mathematical Sciences New York University Submitte to the Journal of the Atmospheric Sciences
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the two-imensional tori are further ecompose into
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationShort Intro to Coordinate Transformation
Short Intro to Coorinate Transformation 1 A Vector A vector can basically be seen as an arrow in space pointing in a specific irection with a specific length. The following problem arises: How o we represent
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More informationEffect of Rotation on Thermosolutal Convection. in a Rivlin-Ericksen Fluid Permeated with. Suspended Particles in Porous Medium
Av. Theor. Appl. Mech., Vol. 3,, no. 4, 77-88 Effect of Rotation on Thermosolutal Convection in a Rivlin-Ericksen Flui Permeate with Suspene Particles in Porous Meium A. K. Aggarwal Department of Mathematics
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2
Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationDifferentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationTutorial Test 5 2D welding robot
Tutorial Test 5 D weling robot Phys 70: Planar rigi boy ynamics The problem statement is appene at the en of the reference solution. June 19, 015 Begin: 10:00 am En: 11:30 am Duration: 90 min Solution.
More informationThe Press-Schechter mass function
The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationObjective: To introduce the equations of motion and describe the forces that act upon the Atmosphere
Objective: To introuce the equations of motion an escribe the forces that act upon the Atmosphere Reaing: Rea pp 18 6 in Chapter 1 of Houghton & Hakim Problems: Work 1.1, 1.8, an 1.9 on pp. 6 & 7 at the
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationSection 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationLecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity,
More informationThermal Modulation of Rayleigh-Benard Convection
Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; E-mail: bsbhaauria@reiffmail.com
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationCalculus Class Notes for the Combined Calculus and Physics Course Semester I
Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation - NSF-DUE-9752485 1 Section 0 2 Contents 1 Average
More informationPHYS 414 Problem Set 2: Turtles all the way down
PHYS 414 Problem Set 2: Turtles all the way own This problem set explores the common structure of ynamical theories in statistical physics as you pass from one length an time scale to another. Brownian
More information3-dimensional Evolution of an Emerging Flux Tube in the Sun. T. Magara
3-imensional Evolution of an Emerging Flux Tube in the Sun T. Magara (Montana State University) February 6, 2002 Introuction of the stuy Dynamical evolution of emerging fiel lines Physical process working
More informationProblem 1 (20 points)
ME 309 Fall 01 Exam 1 Name: C Problem 1 0 points Short answer questions. Each question is worth 5 points. Don t spen too long writing lengthy answers to these questions. Don t use more space than is given.
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationCalculus and optimization
Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function
More informationOptimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations
Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,
More informationPartial Differential Equations
Chapter Partial Differential Equations. Introuction Have solve orinary ifferential equations, i.e. ones where there is one inepenent an one epenent variable. Only orinary ifferentiation is therefore involve.
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationThe continuity equation
Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal
More information1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity
AP Physics Multiple Choice Practice Electrostatics 1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity. A soli conucting sphere is given a positive charge Q.
More information2 The governing equations. 3 Statistical description of turbulence. 4 Turbulence modeling. 5 Turbulent wall bounded flows
1 The turbulence fact : Definition, observations an universal features of turbulence 2 The governing equations PART VII Homogeneous Shear Flows 3 Statistical escription of turbulence 4 Turbulence moeling
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationMomentum and Energy. Chapter Conservation Principles
Chapter 2 Momentum an Energy In this chapter we present some funamental results of continuum mechanics. The formulation is base on the principles of conservation of mass, momentum, angular momentum, an
More informationTOWARDS THERMOELASTICITY OF FRACTAL MEDIA
ownloae By: [University of Illinois] At: 21:04 17 August 2007 Journal of Thermal Stresses, 30: 889 896, 2007 Copyright Taylor & Francis Group, LLC ISSN: 0149-5739 print/1521-074x online OI: 10.1080/01495730701495618
More informationProblem Set 2: Solutions
UNIVERSITY OF ALABAMA Department of Physics an Astronomy PH 102 / LeClair Summer II 2010 Problem Set 2: Solutions 1. The en of a charge rubber ro will attract small pellets of Styrofoam that, having mae
More informationA simple model for the small-strain behaviour of soils
A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo 05508-900, São Paulo, Brazil, e-mail:
More information