Helmut F. Bauer

Diffusion, convection and chemical reaction in a channel

Abstract For laminar flow between a flat conduit exact solutions of the problem of steady-state diffusion and convection are presented for constant wall concentration, constant flux across the wall and a heterogeneous chemical reaction of the first order. The initial conditions were considered to be arbitrary functions of the coordinate across the channel. The same was performed for the cases of simultaneous diffusion, convection and homogeneous chemical reaction for constant wall concentration and heterogeneous chemical reaction. The eigenvalues are presented in tables and the eigenfunctions are presented in analytical form, which contains the confluent hypergeometric function.

Free oscillations and response of a viscous liquid in a rigid circular cylindrical tank

Abstract The natural damped frequencies and the response to translational excitation of a viscous liquid in a circular cylindrical container are obtained. The response of the liquid surface elevation above its equilibrium position, as well as the viscous liquid force in x-direction due to translational excitation in this direction have been numerically evaluated. In this analysis the side wall adhesive boundary conditions have been satisfied, while only the normal bottom condition has been observed. This makes the results of the analysis applicable to liquid height ratio h/a>1.

The response of a non-linear system to pulse excitation

Abstract The response of a non-linear single degree of freedom system subjected to a pulse excitation is investigated. By proper transformation of the displacement variable, such that the additional transformation function will be a solution of the linear system disturbed by the same pulse, the equation of motion is brought into a form for which Lighthills extension of Poincares perturbation method can be employed for the solution of the problem. Both the new displacement variable and the time are expanded into power series in ϵ with unknown coefficients of a third variable. The method is applied to various pulses for a vibrator with a cubic Duffing type restoring force. The pulses being treated are the exponentially decaying pulse, the exponential asymptotic step and the step function. The analytical results obtained are compared with the linear solution and illustrate the influence of the various parameters, such as pulse parameter, forcing function amplitude, and non-linearity.

Oscillations of a viscous liquid in a cylindrical container

Abstract The natural damped frequencies of a viscous liquid in a circular cylindrical container are obtained for slipping and anchored contact line at the container wall r = a. In addition the response to translational forced excitation has been determined. The results may also be applied to viscous liquid in a micro-gravity environment, as long as the contact angle of the liquid surface to the cylindrical container wall is in the vicinity of π 2 , indicating, that the free liquid surface equilibrium position remains a plane surface. It could be found that there exists in contrast to frictionless liquid a small filling range, in which the liquid performs only an aperiodic motion. The adherence condition at the side wall was replaced by the somewhat weaker condition of an anchored contact line.

Axisymmetric viscous liquid oscillations in a cylindrical container

The axisymmetric natural damped frequencies (m=0) of a viscous liquid in a cylindrical container are obtained for slipping and anchored contact line at the container wall. The results may also be applied to viscous liquid in a micro gravity environment, if the contact angle of the contact line to the container wall is in the vicinity of π/2, indicating that the free liquid surface remains a plane surface in equilibrium. It was found that there exists, in contrast to frictionless liquid, a range where only aperiodic motion of the liquid is possible. This appears for small liquid height ratios h/a.ZusammenfassungEs werden die axialsymmetrischen gedämpften Eigenfrequenzen einer viskosen Flüssigkeit im zylindrischen Behälter bestimmt. Dabei werden die Fälle der gleitenden und verankerten Kontaktlinie an der Zylinderwand untersucht. Die Ergebnisse können auch für den Fall der Schwerelosigkeit benutzt werden, wenn der Kontaktwinkel der Kontaktlinie an der Behälterwand in der Nähe von neunzig Grad liegt und somit die Gleichgewichtslage der freien Flüssigkeitsoberfläche eine ebene Oberfläche bildet. Zum Unterschied zu reibungsfreien Flüssigkeitsschwingungen gibt es Gebiete, in denen die viskose Flüssigkeit nur aperiodische Bewegungen ausführen kann. Dies gilt für kleine Flüssigkeitshöhenverhältnisse h/a.

The response of a non-linear n-degree-of-freedom system to pulse excitation

Abstract The response of a non-linear n-degree-of-freedom system subjected to pulse excitation is investigated. After the system is transformed such that the coupling takes place only in the non-linear terms, it can by proper transformation of the displacement variable be brought into a form for which Lighthills extension of Poincares perturbation method can be employed for the solution. The additive transformation function is a solution of the linear problem disturbed by the same pulses. The new displacement variables and the time for each coordinate are expanded into power series in ϵ with unknown coefficients of other variables ξλ. The method is applied to a non-linear two degree-of-freedom system of which one mass is subjected to a step function while the other mass is disturbed by an exponentially decaying pulse. The analytical results obtained are compared with the linear solution of the problem and illustrate the influence of the non-linearities.

On the shape of a rotating fluid system consisting of a gas bubble enclosed in a liquid globe

ZusammenfassungVorliegende Arbeit untersucht die Gleichgewichtsfiguren eines im schwerelosen Zustand rotierenden Systems, bestehend aus einer Gasblase und einem endlichen Flüssigkeitskörper. Das Gas wird von der Flüssigkeit eingeschlossen. Es wird die kritische Winkelgeschwindigkeit bestimmt, bei welcher die Oberfläche der Gasblase diejenige des umgebenden Flüssigkeitskörpers berührt. Das Problem ist von Bedeutung für Experimente in der für die Zukunft geplanten Orbit-Werkstatt.

Zeitliche Temperaturschwankungen an der Oberfläche einer Flüssigkeitssäule

ZusammenfassungEs wird gezeigt, daß durch zeitliche Temperaturgradientenschwankungen, die eine Variation der Oberflächenspannung erzeugen, Oberflächenwellen induziert werden. Bei Annäherung der zeitlichen Temperaturschwankungsfrequenz an die Resonanzen des Flüssigkeitssystems können dabei große Auslenkungen der freien Flüssigkeitsoberfläche erzeugt werden, die zur Disintegration des Systems führen können.Es wird die analytische Lösung für angular- und axialabhängigen Temperaturgradienten, der zeitliche Oszillation aufweist, für eine unendlich lange zylindrische Flüssigkeitssäule im schwerelosen Raum behandelt, wobei Vergrößerungsfunktionen und Phasen der Geschwindigkeits- und Druckverteilung angegeben werden.SummaryIt will be shown that surface waves are induced by time-varying temperature gradients imposed on the free liquid surface of a floating zone in zero-gravity. At or near the resonances of the liquid column large amplitudes of the liquid surface elevation may lead to the disintegration of the liquid system. An analytical solution for angularly and axially depending temperatures is presented for an infinite liquid bridge, for which magnification functions and phases are determined for the velocity distribution, surface elevation and the pressure distribution.

NONLINEAR PROPELLANT SLOSHING IN A RECTANGULAR CONTAINER OF INFINITE LENGTH

ABSTRACT An approximate solution is given for the problem of nonlinear motion of liquid with a free fluid surface in a rectangular container of infinite length. The solution for free oscillations is derived with a perturbation method and the second-order nonresonant solution is presented. In the free oscillation case the system was detected to exhibit a hardening characteristic for fluid heights in the range 0 0.2. Results for surface elevation, fluid force, and liquid moment are presented for harmonic translatory excitation of the container. In addition to the deviations in magnitude due to large amplitude motion as a result of large excitation amplitude, a superharmonic resonance phenomenon which does not occur in linearized theory was detected. The magnification functions show the influence of the superharmonic content of the solution. While in linearized theory the fluid force and moment due to translatory excitation exhibit only odd vibration modes, the nonlinear results clearly show coupling with the even modes. Furthermore, the range at which the free fluid surface begins to splash is presented versus forcing frequency for various excitation amplitudes.

Vibrational behavior of non-linear systems to pulse excitations of finite duration

Abstract The response of a non-linear system subjected to a pulse excitation of finite duration is investigated for the region during which the pulse is acting as well as for the region where the pulse has ceased to act. The original differential equation is transformed into a form for which Lighthills extension of Poincares perturbation method can be employed. The method is exhibited for arbitrary pulse load and was in the numerical evaluation applied to a sinusoidal, a one-minus-cosine and a triangular load. It illustrates the influence of the non-linear parameter for hardening and softening restoring characteristics. The application of the method of solution is also indicated for non-linear many-degree-of-freedom systems.