Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

---

Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

---

Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche     

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

---

Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

---

Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

Boundary layer effect on thermal NO decomposition behind incident shock waves

The thermal decomposition of nitric oxide (diluted in Argon) has been measured behind incident shock waves by means of IR diode laser absorption spectroscopy. In two independent runs the diode laser was tuned to the=0 =1² _3/2 R(18.5)-rotational vibrational transition and the=1 =2² _3/2 R(20.5)-rotational vibrational transition of nitric oxide, respectively. These two transitions originating from the vibrational ground state (=0) and the first excited vibrational state (=1) were selected in order to probe the homogeneity along the absorption path. The measured NO decomposition could satisfactorily be described by a chemical reaction mechanism after taking into account boundary layer corrections according to the theory of Mirels. The study forms a further proof of Mirels' theory including his prediction of the laminar-turbulent transition. It also shows, that the inhomogeneities from the boundary layer do not affect the IR linear absorption markedly.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

（黄家寅）（秦圣立）ＴＨＥＰＲＯＢＬＥＭＳＯＦＮＯＮＬＩＮＥＡＲＢＥＮＤＩＮＧＦＯＲＯＲＴＨＯＴＲＯＰＩＣＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＷＩＴＨＦＯＵＲＣＬＡＭＰＥＤＥＤＧＥＳ￥ＨｕａｎｇＪｉａｙｉｎ;ＱｉｎＳｈｅｎｇｌｉ（ＱｕｆｕＮｏｒｍａｌＵｎ...     

Stationary travelling waves on a film flowing down an inclined plane

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers close to the wave number _n that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5_{n n} in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 _n that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Some self-similar solutions of the Leibenzon equation

Self-similar one-dimensional solutions of the Leibenzon equation c²_t= _zz ^k (z 0, k 2) are considered. Approximate solutions are constructed for the two cases in which the initial value = ₁ = const > 0 and on the boundary either a constant value = ₂ < ₁ is maintained or the flow (directed outwards) is given. In the first problem the dependence of the boundary flow on the governing parameters is determined. A characteristic property of the types of motion in question is the existence near the boundary of a region, expanding with time, in which the flow is almost independent of the coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1991.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Berechnung des Temperaturfeldes um eine im Erdboden verlegte Punkt- und Linienquelle bei Randbedingung III. Art an der Erdoberfläche

Zusammenfassung Für den Fall, daß sich in einem halbunendlichen Körper in der Tiefe L eine Punkt- bzw. Linienquelle befindet und daß an der Oberfläche des Körpers ein örtlich und zeitlich konstanter Wärmeübergangskoeffizient herrscht, wird das stationäre Temperaturfeld analytisch berechnet. Beim Vergleich mit einer Näherungslösung (Hilfsschicht) zeigt sich, daß nicht so sehr die Biot-Zahl Bi= · L/ als vielmehr der größte Winkel zwischen Wandnormale und Wärmestromdichte in der Hilfsschicht ein Maß für die Genauigkeit der Näherungslösung ist.

---

Calculation of the temperature field around a buried point- and linesource, respectively, when the boundary condition is Newton's law

The steady state temperature field in a semiinfinite body caused by a buried point- and linesource, respectively, has been analytically calculated. The comparison with a simple approach (additional-layer) shows that the greatest angle between the normal of the wall and the heat flux density in the additional-layer, describes the quality of the approach better than the Biot-number Bi=L/ does.

---

Formelzeichen A Fläche - Bi Biot-Zahl - C Eigenwertfunktion - E₁ Exponentialintegral - exp Exponentialfunktion - i komplexe Einheit - J₀ Besselfunktion nullter Ordnung und 1. Grades - L Verlegungstiefe der Punkt- bzw. Linienquelle - Q Quellstärke - r Radius - Re Realteil eines Ausdruckes - T Temperatur - t Integrationsvariable - x, y, z Ortskoordinaten - Wärmeübergangskoeffizienten an der Erdoberfläche - Laplace-Operator - Wärmeleitfähigkeit des Erdbodens - dimensionslose Temperatur - Integrationsvariable - dimensionsloser Radius - komplexe Ortskoordination Indizes 0 Erdoberfläche, senkrecht über der Quelle - 1 Lösung für das 1. Randwertproblem - 3 Lösung für das 3. Randwertproblem - 13 Zusatzfunktion - w Erdoberfläche - Umgebungstemperatur - Näherungslösung     

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a vertical heated plate

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a heated vertical plate is studied for high Rayleigh numbers. Similarity solutions are obtained within the frame work of boundary layer theory for a power law variation in surface temperature,T _Wx and surface injectionv _Wx(–1/2). The analysis incorporates the expression connecting porosity and permeability and also the expression connecting porosity and effective thermal diffusivity. The influence of thermal dispersion on the flow and heat transfer characteristics are also analysed in detail. The results of the present analysis document the fact that variable porosity enhances heat transfer rate and the magnitude of velocity near the wall. The governing equations are solved using an implicit finite difference scheme for both the Darcy flow model and Forchheimer flow model, the latter analysis being confined to an isothermal surface and an impermeable vertical plate. The influence of the intertial terms in the Forchheimer model is to decrease the heat transfer and flow rates and the influence of thermal dispersion is to increase the heat transfer rate.

---

Der Effekt des Oberflächenstoffaustausches bei auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt

Zusammenfassung Es wird der Effekt des Oberflächenstoffaustausches in auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt, für große Reynoldszahlen untersucht. Ähnliche Lösungen werden im Rahmen der Grenzschicht-Theorie, durch Variation des Potenzansatzes der Oberflächentemperatur,T _Wx , und der Oberflächengeschwindigkeit,v _Wx(–1/2), erreicht. Die Analyse vereinigt sowohl den Ausdruck, der Porösität und Permeabilität verbindet, als auch den Ausdruck, der Porösität und Wärmeleitfähigkeit miteinander verbindet. Der Einfluß der Temperaturverteilung auf Strömung und Wärmeübergangskennzahlen wird ebenfalls im Detail analysiert. Als Ergebnis der vorliegenden Untersuchung ergibt sich die Tatsache, daß variable Porösität Wärmeübertragungsrate und Betrag der Geschwindigkeit in Wandnähe steigert. Die bestimmenden Gleichungen, sowohl für das Darcysche Strömungsmodell als auch für das Forchheimersche Strömungsmodell, werden mit Hilfe eines implizierten Differenzenschemas gelöst. Die Berechnung wird für die beiden Fälle, isotherme Oberfläche und undurchlässige vertikale Platte, angewandt. Der Einfluß der Terme für die Trägheitskräfte im Forchheimerschen Modell senkt Wärmeübergangs- und Durchgangsrate, wogegen die Wärmeübergangsrate durch den Einfluß der Temperaturverteilung erhöht wird.

---

Nomenclature a constant defined by Eq. (12) - A constant defined by Eq. (12) - B constant defined by Eq. (3) - b _s/_f ratio of thermal conductivities - C constant defined by Eq. (1) - C _P specific heat of the convective fluid - d particle diameter - f dimensionless function defined by Eq. (14) - f _w lateral mass flux parameter - g acceleration due to gravity - k ₀ mean permeability of the mediumk ₀= ₀ ³ d ²/150 (1– ₀)² k ₀=1.75d/(1– ₀) 150 (Inertia parameter) - L length of the source or sink - m mass transfer - n constant defined in Eq. (12) - k (y) permeability of the porous medium - k (y) interial coefficient in the Ergun expression - Gr modified Grashof numberGr=(g k ₀ k ₀ (T _w–))/ ² - R _a Rayleigh number (g k ₀ x T _w–)/ - R _ad modified Rayleigh number (g k ₀ d|T _w–|)/ - N _u Nusselt number - s x/d - Q overall heat transfer rate - T temperature - T _w surface temperature - T ambient fluid temperature - u velocity in vertical direction - v velocity in horizontal direction - x vertical coordinate - y horizontal coordinate Greek symbols ₀ mean thermal diffusivity _f/ C_p - coefficient of thermal expansion - constant defined in Eq. (4) - ratio of particle to bed diameter - _e effective thermal conductivity - f thermal conductivity of fluid - _s thermal conductivity of solid - dimensionless similarity variable in Eq. (13) - value of at the edge of the boundary layer - constant defined in Eq. (1) - _e effective molecular thermal diffusivity - (y) porosity of the medium - ₀ mean porosity of the medium - viscosity of the fluid - ₀ density of the convective fluid - stream function - w condition at the wall - condition at infinity     

Waves of finite amplitude in a hot plasma

A study is made of a plane shock wave of arbitrary strength propagating in a hot rarefied plasma across the magnetic field. The question of the propagation of nonstationary waves of finite but small amplitude under these conditions is examined.Fairly detailed studies have been made of waves of finite amplitude in a cold rarefied plasma. The profile of such waves is formed as the result of nonlinear and dispersion effects, the dispersion effects being caused by electron inertia and plasma anisotropy. If the gas-kinetic pressure of the plasma is taken into account, then dispersion effects appear which are associated with the fact that the Larmor radius of the ions is finite. Stationary waves of small but finite amplitude propagating across the magnetic field in a hot plasma (when the gas-kinetic pressure p is comparable with the magnetic pressure H²/87) have been treated in [1, 2]. In [1] an isolated rarefaction wave was found in a hot plasma, instead of the compression wave characteristic of a cold plasma, and a qualitative picture of the shock wave structure was given. In [2] a study was made of a small-amplitude shock wave with the finite size of the ion Larmor radius taken into account. The present paper investigates the structure of shock waves of arbitrary strength which propagate across the magnetic field in a fairly hot rarefied plasma, and also examines nonstationary waves of finite but small amplitude excited in a plasma by a magnetic piston acting over a limited time interval.Notation p gas-kinetic pressure - H magnetic field - u, v macroscopic velocities along the x and y axes - density - m_e(m_i) mass of electron (ion) - plasma conductivity - _H ion-cyclotron frequency - V_A Alfvèn velocity - c velocity of light - adiabatic exponent - V specific volume - _0e(_0i) electron (ion) plasma frequency - S₀ velocity of sound. In conclusion the author thanks R. Z. Sagdeev and N. N. Yanenko for discussing the paper, and also R. N, Makarov for helping with the numerical computations.     

Dielectric properties of the composite system polyurethane — sodium chloride

Summary The dielectric properties of the composite system polyurethane-sodium chloride have been measured at frequencies between 10^–4 Hz and 3 · 10⁵ Hz in the temperature range from –150 °C up to +90 dgC. Three dielectric loss mechanisms have been found; they are indicated by ₁, ₂ and. The activation energy of the ₁-transition is 35 kcal/mole, that of the-transition 6.7 kcal/mole. The ₂-loss peak was only observed at frequencies of 10³ Hz and higher, forming one broad peak with the ₁-loss peak at lower frequencies. In the composite materials, the- and ₂-loss peaks measured at fixed frequencies were found at the same temperature. The ₂-loss peak, however, was shifted to a lower temperature, due to the sodium chloride filler. Comparison of experimental data of and tan with theoretical predictions concerning the dielectric properties of composite systems showed only partial agreement. The difference mainly consisted in. the temperature shift in the tan-peak of the ₁-transition.

---

Zusammenfassung Die dielektrischen Eigenschaften des Verbundssystems Kochsalz-Polyurethankautschuk wurden im Frequenzgebiet zwischen 10^–4 Hz und 3.10⁵ Hz und im Temperaturbereich von –150 °C bis +90 °C gemessen. Es wurden drei dielektrische Verlustmaxima gefunden, die mit ₁, ₂ und angedeutet werden. Die Aktivierungsenergie des ₁-Überganges beträgt 35 kcal/Mol, die des-Überganges 6.7 kcal/Mol. Das ₂-Maximum konnte nur bei Frequenzen höher als 10³Hz vom ₁-Maximum gesondert erfaßt werden. Die Lage der ₂- und-Maxima war vom Füllgrad unabhängig. Das ₁-Maximum verschiebt sich mit steigendem Füllgrad zu niedrigeren Temperaturen. Die gemessenen Werte von und tan stimmen nur teilweise mit den Aussagen einer Theorie der dielektrischen Eigenschaften von Mischkörpern überein.

     

A Capillary Microstructure of the Wetting Front

This article reports the experimental results of a study of the wetting-front microscale structure formed only by capillary forces in homogeneous and random etched glass capillary models. In the homogeneous model, water propagates through the capillary system, evenly filling the capillaries across the direction of flow. Air is trapped by the pinch-off mechanism inside the pore bodies in the form of individual bubbles. The experiments specified three consecutive steps of the pinch-off mechanism, film flow, snap-off, and interface movement. In the random model, both the bypass and pinch-off, forming bypass/cut-off mechanism, create residual air structure. Bypass traps air inside large capillary-pore aggregates which are bounded by small-diameter capillaries in where pinch-off traps air in the adjacent pores. An analysis of the residual air distribution versus depth below the surface in the homogeneous and random micromodels made it possible to identify three successive zones, namely a transition zone, a transmission zone, and a wetting-and-front zone. In the transition zone, the residual air content increases with depth from zero to the constant value in the transmission zone where it remains practically constant. The capillary processes within the wetting-and-front combined zone govern air replacement with wetting and formation of the transmission zone.     

Temperaturverteilung in tiefen und flachen Seen

Zusammenfassung Die Temperaturverteilung in tiefen und flachen Seen lÄ\t sich unter gewissen Vereinfachungen als instationares, eindimensionales WÄrmeleitproblem behandeln. Konvektion in bestimmten Bereichen des Sees kann durch Mittelung der den instabilen Temperaturprofilen entsprechenden inneren Energie berücksichtigt werden. Ma\gebliche Parameter für den örtlichen und zeitlichen Temperaturverlauf sind der vertikale Absorptionskoeffizient und das VerhÄltnis von WÄrmeaustausch an der OberflÄche einerseits zur sichtbaren Sonnen- und Himmelsstrahlung andererseits. Zeitlich verÄnderlicher Energieaustausch lÄ\t sich durch eine schrittweise Berechnung berücksichtigen. Messungen anderer Autoren an Seen sowie eigene Modellexperimente bestÄtigen die im theoretischen Modell enthaltenen Annahmen.

---

Temperature distribution in deep and shallow lakes

For certain simplifications the temperature distribution in deep as well as shallow lakes can be treated as a non-steady, one-dimensional heat conduction problem. Convection in certain regions of the lake can be taken into account by averaging the internal energy corresponding to the calculated unstable profiles. The leading parameters for the spatial and temporal temperature distribution are the vertical absorption coefficient on one hand and the ratio of heat exchange at the surface to visible solar and sky radiation on the other. Transient energy exchange can be considered by stepwise calculation. Field measurements of other authors as well as own model experiments confirm the assumptions made in the theoretical model.

---

Formelzeichen

1 Lateinische Buchstaben a Temperaturleitzahl - c spezifische WÄrmekapazitÄt - I IntensitÄt - L Seetiefe - n Brechungsindex - q Energiestromdichte - q_g Gesamtstrahlungsstromdichte an der OberflÄche - q_o sichtbare Strahlungsstromdichte an der OberflÄche - q_v WÄrmestromdichte an der OberflÄche (durch Konvektion, Verdunstung und/oder langwellige Zu- und Abstrahlung) - r Reflexionsver mögen der SeeoberflÄche - s Strahlweg - t Zeit - w volumenbezogene WÄrmequelle - z vertikale Ortskoordinate 2 Griechische Buchstaben Zenitwinkel - Zenitwinkel unter Wasser - Absorptionskoeffizient - dimensionslose vertikale Ortskoordinate - dimensionslose Variable - Temperatur - dimensionslose Temperatur - WellenlÄnge - WÄrmeleitfÄhigkeit (Index eff für turbulente WÄrmeleitfÄhigkeit) - dimensionslose Tiefe - Dichte - EnergiestromdichteverhÄltnis - dimensionslose Zeit - _k dimensionslose Variable 3 Indices e Epilimnion - h Hypolimnion - o SeeoberflÄche - spektrale Grö\e Auszug aus der vom Fachbereich Maschinenwesen der Technischen UniversitÄt München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs genehmigten Dissertation über Temperaturverteilung in Seen des Diplom-Ingenieurs Siegfried Blo\. Berichterstatter Prof. Dr.-Ing. U. Grigull und Prof. Dr. rer. nat. D. Vortmeyer.- Die Dissertation wurde am 27. Mai 1977 bei der Technischen UniversitÄt München eingereicht und durch den Fachbereich Maschinenwesen am 19. Juli 1977 angenommen. Tag der Promotion 26. Juli 1977     

Solution of regular beam equations in arbitrary emission conditions on a curvilinear surface

The regular beam equations are solved analytically for the case of emission from an arbitrary surface in conditions of total space charge (-mode) and in a given external magnetic field H (§2) for temperature-limited emission (T-mode), in an external magnetic field H (§3); and for emission with nonzero initial velocity (§4). The emitter is taken as the coordinate surface x¹=0 in an orthogonal system x¹ (i = =1,2,3), while the current density J and field on it are given functions j(x², x³), (x², x³. The solution is written as series in (x¹) with coefficients dependent on x², x³, determined from recurrence relations. For emission in the -mode and H 0, =1/3; for temperature-limited emission, =1/2; with nonzero initial velocity, =1. The results are extended to the case of a beam in the presence of a moving background of uniform density (5).     

Collision integrals of ionized air components for a screened coulomb potential

Results of a numerical evaluation of the integrals ^,s ( = 1, 2, 3, 4 and s = ... 8 -), in terms of which transport coefficients are expressed [1], are given in this paper. The results obtained are of practical use for calculating kinetic properties of ionized gases within the Chapman-Enskog theory, including third-order terms in the Sonine polynomial expansion. The integrals ^,s were calculated for singly-ionized colliding air particles at temperatures T = 10,000–40,000 °K and pressures p = 0.1, 1, and 100 atm. The screened Coulomb potential for attraction and repulsion was the model for electron-ion and ionic interactions, and the Debye screening length was chosen by taking into account screening both by electrons and by multiply-charged ions. Quantum effects are not important in the temperature and pressure ranges considered for air, and can, therefore, be neglected in calculating kinetic properties.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 168–171, September–October, 1971.     

Application of ‘Operational Quadrature Methods’ in Time Domain Boundary Element Methods

The usual time domain Boundary Element Method (BEM) contains fundamentalsolutions which are convoluted with time-dependent boundary data andintegrated over the boundary surface. Here, a new approach for theevaluation of the convolution integrals, the so-called OperationalQuadrature Methods developed by Lubich, is presented. In thisformulation, the convolution integral is numerically approximated by aquadrature formula whose weights are determined using the Laplacetransform of the fundamental solution and a linear multisep method. Tostudy the behaviour of the method, the numerical convolution of afundamental solution with a unit step function is compared with theanalytical result. Then, a time domain Boundary Element formulationapplying the Operational Quadrature Methods is derived. For thisformulation only the fundamental solutions in Laplace domain arenecessary. The properties of the new formulation are studied with anumerical example.