Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

The influence of vibration on Poiseuille flow of a non-Newtonian fluid,I.

Summary The effect is calculated of a superposed sinusoidal vibration on the rate of discharge of an incompressible, isotropic, non-Newtonian fluid, which is in plane Poiseuille flow, under a constant pressure gradient, between two parallel plates. The calculations are made both when the vibration is longitudinal and when it is transverse. Calculations are also made in the case of Poiseuille flow through a straight pipe of circular cross-section, when a sinusoidal vibration, either longitudinal ar rotational, about the axis of the pipe is superposed.

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Zusammenfassung Es wird die Wirkung einer überlagerten Sinusschwingung auf den Durchsatz einer inkompressiblen, isotropen, nicht-newtonschen Flüssigkeit berechnet, die in einer ebenen Poiseuille-Strömung zwischen zwei parallelen Platten unter der Wirkung eines konstanten Druckgefälles ausfließt. Dabei werden sowohl Longitudinal- als auch Transversal-Schwingungen betrachtet. Anschließend werden analoge Rechnungen für eine Poiseuille-Strömung in einem geraden zylindrischen Rohr durchgeführt, wobei eine in bezug auf die Rohrachse entweder longitudinale oder rotatorische Sinusschwingung überlagert wird.

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With 1 figure     

The evolution of small rotational perturbations to a uniform rotation of a viscoelastic fluid contained between parallel plates

The transfer mechanism of momentum from the boundaries into the interior of a viscoelastic fluid is examined in the case when the boundary consists of two parallel plates.Special emphasis is given to the case of a Maxwellian fluid. It is seen that an increase in the azimuthal angular velocity at the boundaries progresses into the interior of the liquid in the form of a velocity discontinuity wave. Simultaneously, an axial circulatory flow pumps liquid from the low rotation region near the middle plane into the high rotation region near the boundaries. Detailed calculations are given for the intial stage of evolution.     

The influence of vibration on Poiseuille flow of a non-Newtonian fluid. II

Summary An incompressible, isotropic, non-Newtonian fluid undergoes plane Poiseuille flow between two parallel plates, or through a pipe of circular cross-section, as a result of a uniform time-independent pressure gradient. The effect is studied of superposed vibrations of the boundaries, which are not necessarily purely sinusoidal, on the mean rate of discharge of the fluid. The calculations are carried out in detail for a particular constitutive equation of the Rivlin-Ericksen type, with the assumption that the fluid is slightly non-Newtonian. It is seen that in addition to a change in the mean rate of discharge which arises as an interaction of the vibration with the pressure gradient, there may also occur a change in the mean rate of discharge which arises from the interaction of the harmonic components of the vibration and may be independent of the pressure gradient.

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Zusammenfassung Gegenstand der Untersuchung ist eine inkompressible, isotrope, nicht-newtonsche Flüssigkeit entweder in einer ebenen Poiseuille-Strömung zwischen zwei parallelen Platten oder einer solchen durch ein kreiszylindrisches Rohr unter dem Einfluß eines zeitunabhängigen Druckgradienten. Es wird der Einfluß von überlagerten Vibrationen der Wände, die nicht rein sinusförmig zu sein brauchen, auf den mittleren Flüssigkeitsdurchsatz untersucht. Die Rechnungen werden im einzelnen für eine spezielle Stoffgleichung vom Rivlin-Ericksen-Typ durchgeführt, wobei die Flüssigkeit als nur schwach nicht-newtonisch angenommen wird. Man findet, daß außer der Änderung des mittleren Durchsatzes als Folge der Wechselwirkung von Vibration und Druckgradient eine solche Änderung auch als Folge der Wechselwirkung der harmonischen Komponenten der Vibration auftreten kann, die u. U. vom Druckgradienten unabhängig ist.

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With 1 figure     

Heat-conducting micropolar fluids

Summary In this paper heat-conducting micropolar fluids are introduced as an extension of the theory of micropolar fluids. Constitutive equations appropriate to describe the thermal and mechanical response of micropolar fluids are constructed. The heat conduction equation is derived and the field equations are obtained. The solution to the problem ofPoiseuille flow through a channel with flat walls is given.

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Zusammenfassung In dieser Arbeit werden wärmeleitende mikropolare Flüssigkeiten als Erweiterung der Theorie mikropolarer Flüssigkeiten eingeführt. Es werden geeignete Zustandsgleichungen zur Beschreibung der thermischen und mechanischen Empfindlichkeit mikropolarer Flüssigkeiten abgeleitet. Die Wärmeleitungsgleichung und die Feldgleichungen werden ermittelt. Für das Problem derPoiseuille-Strömung durch einen Kanal mit glatten Wänden wird die Lösung angegeben.

     

Flow of a viscoelastic fluid between eccentric rotating cylinders and related problems

Summary Results obtained in a previous paper byBallal andRivlin on the forces associated with the slow flow of an incompressible non-Newtonian fluid, contained in the annular region between two infinite eccentric rotating cylinders, are applied to the calculation of the forces associated with the planetary motion of the inner cylinder about the axis of the outer cylinder. Either, neither, or both of the cylinders may rotate about their axes with constant angular velocities. As a limiting case the motion of an infinite cylinder is considered, in a half-space of incompressible non-Newtonian fluid bounded by a rigid plane, when the cylinder moves with constant velocity perpendicular to its length and parallel to the plane.

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Zusammenfassung Die in einer früheren Veröffentlichung vonBallal undRivlin erhaltenen Ergebnisse bezüglich der Kräfte, die beim langsamen Strömen einer inkompressiblen nicht-newtonschen Flüssigkeit im Ringspalt zwischen zwei unendlich langen exzentrisch rotierenden Zylindern auftreten, werden für die Berechnung der Kräfte angewendet, die bei der Planetenbewegung des inneren Zylinders um die Achse des äußeren Zylinders auftreten. Dabei dürfen beide Zylinder entwedet ruhen oder mit konstanter Winkelgeschwindigkeit um ihre Achsen rotieren. Als Grenzfall wird die Bewegung eines unendlich langen Zylinders in einem Halbraum betrachtet, der durch eine feste Ebene begrenzt wird, wobei der Zylinder eine konstante Geschwindigkeit senkrecht zur Achse und parallel zur Ebene besitzt.

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With 21 figures and 1 table     

Some superposition theorems for second-order fluids

Theorems are derived, within the framework of the second-order Rivlin-Ericksen theory, relating to the superposition of longitudinal flows on plane flows in an incompressible isotropic viscoelastic fluid. Also, some theorems are obtained concerning the effects of inertia on the solution of plane flow problems in such viscoelastic fluids.     

A model for adiabatic supersonic turbulent boundary layers

An asymptotic analysis of the equations describing supersonic turbulent flow over an adiabatic wall is carried out for high Reynolds numbers, Re, and mainstream Mach numbers, M _e=O(1). A general expression for the adiabatic-wall temperature is derived. The asymptotic theory constrains the types of turbulence models that are suitable to represent the effects of viscous dissipation. A simple algebraic turbulence model is proposed and comparisons with measured total enthalpy profile data show good agreement, capturing the overshoot observed in total enthalpy near the boundarylayer edge.This work was supported by NASA Langley Research Center under Grant NAG-1-832 and the Air Force Office of Scientific Research under Grants AFOSR-91-0069 and F49620-93-0130; Dr. Ruban was supported by a grant from United Technologies Corporation.     

Run-up and spin-up in a viscoelastic fluid. I.

Summary An incompressible viscoelastic fluid is contained between parallel rigid plates which at some instant of time are subjected to a translational velocity which is then held constant. The dependence on time of the resulting flow field in the fluid is calculated. Similar calculations are carried out when the fluid is contained in an infinitely long circular cylinder which is set in longitudinal motion or in rotation.It is seen that for a certain broad class of viscoelastic fluids the transfer of momentum, or angular momentum, from the boundary to the interior of the fluid takes place by a mechanism which is essentially diffusive in character. For another broad class of fluids, of which the Maxwellian fluid is a special case, the transfer of momentum results from the reflection back and forth of a velocity shock wave. These reflections take place at the boundaries in the case of runup between parallel plates and at the boundary and axis in the case of run-up, or spin-up, in a circular cylinder.With 4 figures