Theory of oscillation type viscometers: The oscillating cup

This is the first of a series of papers dealing with the theory of oscillation type viscometers with particular emphasis on those in which a finite right circular cylinder (cup or disk) oscillates in contact with a fluid. The purpose is to obtain formulae which relate the density and viscosity of the fluid to the frequency and logarithmic decrement of the oscillation and the various physical properties of the suspension system. The study is to include an accurate analysis of the fluid motion in the vicinity of the edges since this is the main source of error in most existing theories. The first paper gives a general formulation of the problem and an exact solution (including transients) for the oscillating cup viscometer with the fluid inside the cup.

---

Zusammenfassung Diese Abhandlung ist der erste Bericht aus einer Veröffentlichungsreihe, die sich mit der Theorie schwingender Viskosimeter befasst. Dabei sind besonders die behandelt, bei denen ein endlicher Kreiszylinder (Becher oder Scheibe) von einem flüssigen Medium gefüllt oder umgeben schwingt. Die Aufgabe der Untersuchung ist es, die Abhängigkeit der Frequenz und des logarithmischen Dekrementes der Schwingung von der Dichte und der Viskosität der Flüssigkeit und von den verschiedenen physikalischen Grössen des Aufhängungssystems zu finden. Weiterhin ist eine genaue Analyse der Flüssigkeitsbewegung in den Randzonen eingeschlossen, da diese die Hauptursache der Fehler in den meisten der vorhandenen Theorien war. Die erste Abhandlung gibt die grundlegende Formulierung des Problemes und eine exakte Lösung (einschliesslich der Einschwingzeit) für das Viskosimeter, bei dem ein mit der zu messenden Flüssigkeit gefüllter schwingender Becher verwendet wird.

---

List of Symbols The following is a partial list of symbols including only those that are used repeatedly: A surface of contact between fluid and cylinder - D(s) defined in equation (16) - f() transient, equation (42) - h half height of cylinder or height of liquid in an open cup - I moment of inertia of suspension system - I moment of inertia of fluid inside cup - I _k modified Bessel function of orderk - J _k Bessel function of orderk - M() frictional moment exerted by the fluid - n normal toA outward from the fluid - r radial coordinate in the fluid This work was supported by the United States Air Force through the Air Force Office of Scientific Research and Development Command under contract No. AF 18(600)1548. Part of the work was done while one of the authors (G. F. N.) was being supported by a grant from the Alfred P. Sloan Foundation.     

Theory of oscillation type viscometers IV: A thick disk

The viscous drag exerted by a fluid on an oscillating disk is determined by methods similar to those used in Part III. The present paper differs from Part III in that the boundary layer thickness is assumed here to be small compared with both the thickness and the radius of the disk. An extrapolation of the formulae derived here agree sufficiently well, however, with those derived for the thin disk (boundary layer thickness large compared with the thickness but small compared with the radius) that it is possible to make an interpolation between the present formulae and those of Part III that will permit the evaluation of the drag for arbitrary shape disks provided the boundary layer thickness is small compared with the radius.

---

Zusammenfassung Der Einfluss der Zähigkeitskräfte auf eine in einer Flüssigkeit schwingende Scheibe wird nach den gleichen Methoden wie in Tiel III behandelt. Jedoch wird im Unterschied dazu vorausgesetzt, dass die Grenzschichtdicke klein ist im Verhältnis zur Dicke und zum Radius der Scheibe. Die abgeleiteten Formeln lassen sich mit gutem Ergebnis für die dünne Scheibe extrapolieren (hier ist die Grenzschichtdicke jedoch gross, verglichen mit der Scheibendicke, aber klein gegenüber dem Radius), so dass es möglich ist, zwischen diesen Formeln und denen des Teils III zu interpolieren. Damit wird die Berechnung des Reibungsmomentes für Scheiben unterschiedlichster Form ermöglicht, vorausgesetzt, dass die Grenzschichtdicke klein ist gegenüber dem Radius der Scheibe.

---

---

This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract No. AF 18 (600) 1548. Reproduction in whole or in part is permitted for any purpose of the United States Government. Part of the work was done while one of the authors (G. F. N.) was being supported by a grant from the Alfred P. Sloan Foundation.     

Theory of oscillation type viscometers III: A thin disk

The small amplitude oscillation of a thin disk of finite radius suspended in an infinite fluid is considered with a view toward its use as a viscometer. The fluid motion is analyzed under the assumption that the boundary layer thickness of the fluid is small compared with the radius of the disk but large compared with the thickness of the disk. Formulas are obtained relating the frequency and decrement of oscillation to the density and viscosity of the fluid.

---

Zusammenfassung Es wird untersucht, inwieweit eine dünne aufgehängte Scheibe endlicher Abmessung, wenn sie mit kleiner Amplitude schwingt, sich zur Messung der Zähigkeit verwenden lässt. Die Bewegung der Flüssigkeit wird unter der Voraussetzung analysiert, dass die Grenzschichtdicke klein ist im Vergleich zum Radius der Scheibe, aber gross verglichen mit der Scheibendicke. Die Formeln zeigen die Abhängigkeit der Frequenz und des logarithmischen Dekrements der Schwingung von der Dichte und der Zähigkeit der Flüssigkeit.

---

---

This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract No. AF 18 (600) 1548. Reproduction in whole or in part is permitted for any purpose of the United States Government. Part of the work was done while one of the authors (G. F. N.) was being supported by a grant from the Alfred P. Sloan Foundation.     

Theory of micropolar plates

Zusammenfassung Auf der Basis der linearen Theorie der mikropolaren Elastizität wird eine Theorie für Dehnung und Biegung mikropolarer Platten entwickelt. Die Platten werden aus Hantel-Elementen zusammengesetzt angenommen. Die Theorie berücksichtigt Drehungs- und Schiebungseffekte sowohl klassischen wie auch mikropolaren Ursprungs. Es entstehen symmetrische und antimetrische Spannungsverteilungen als Bestandteile einer ungekoppelten Feldtheorie. Es werden Sätze bezüglich der inneren Energie und der Eindeutiekeit der Lösung aufgestellt. Einige Näherungstheorien sowie die Natur mikropolarer Biegewellen in unendlichen Platten werden diskutiert.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

Hypergraphs induced by algebras of fixed type

A characterization of the weak subalgebra lattice of a partial algebra of a fixed type is a natural algebraic problem. In Pióro (2000, 2002) [13] and [15] we have shown that this algebraic problem is equivalent to the following hypergraph question, interesting in itself: When can edges of a hypergraph be directed to form a partial algebra of a fixed type (equivalently, to form a directed hypergraph of a fixed type)? This problem will be solved in the present paper.     

On fixed point theorems of Leray-Schauder type

In this paper we prove a few fixed points theorems of Leray-Schauder type in hyperconvex metric spaces.

     

Theory of Integral Equations for Axisymmetric Scattering by a Disk

Computational Mathematics and Mathematical Physics - A theory of integral equations for radial currents in the axisymmetric problem of scattering by a disk is constructed. The theory relies on the...     

On fixed point and almost fixed point problems of lower semicontinuous type multivalued mappings

In the present paper, some new almost fixed point theorems and fixed point theorems for lower semicontinuous type multivalued mappings are obtained in metrizable H-spaces.     

Bounded oscillation of second order unstable neutral type difference equations

In this paper the authors present sufficient conditions for all bounded solutions of the second order neutral difference equation $$\Delta ^2 (y_n - py_{n - k} ) - q_n f(y_{n - \ell } ) = 0, n \in \mathbb{N}$$ to be oscillatory. Examples are provided to illustrate the results.     

On oscillation of second order neutral type delay differential equations

Oscillation criteria are obtained by using the so called H-method for the second order neutral type delay differential equations of the form

(r(t)ψ(x(t))z^′(t))^′+q(t)f(x(σ(t)))=0, tt₀,

where z(t)=x(t)+p(t)x(τ(t)), r, p, q, τ, σ, C([t₀,∞),R) and f,ψC(R,R).

The results of the paper contains several results obtained previously as special cases. Furthermore, we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations.     

Zero-divisors among finite structures of a fixed type

Conditions are established which guarantee that a finite structure be a zero-divisor relative to the class of all finite structures of its type. These conditions are concerned with reducts, disjointness conditions on ranges and/or projections of operations and/or relations, and diophantine equations induced by a relation. Presented by R. McKenzie.     

The connection between spectral and oscillation properties of differential-operator equations of arbitrary order. I. Sturm type theorems

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 48, pp. 101–111, 1987.