A drag formula revisited

In this brief note, the problem of creeping flow of a micropolar fluid past an axisymmetric body is revisted. In particular, the elegant formula for the drag is rederived using a more rigorous mathematical approach.     

On a generalized system of equations in continuum mechanics

Summary A system of equations of which the equations of elasticity and the Stokes equations of hydrodynamics are particular cases, is examined. Galerkin-type representations are constructed for this system with the aid of a matrix inversion technique. These representations give rise to the fundamental singular solution which together with a derived reciprocal relationship yield integral representations for the unknown parameters in the given system of equations. The integral representations lead in a natural way to the introduction of surface potentials whose properties are stated. Some well-known cases are deduced from our general results.     

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.     

On the uniqueness and existence of Stokes flows in micropolar fluid theory

Summary The present paper addresses itself to the problem of existence and uniqueness of solutions of Stokes flows in micropolar fluid theory using the methods of potential theory. Integral representations for the velocity and microrotation vectors are derived and they lead naturally to the introduction of single-layer and double-layer potentials whose properties are stated. With the aid of these results, necessary and sufficient conditions are generated for the Stokes problem in this microcontinuum fluid mechanics theory.

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Zur Eindeutigkeit und Existenz von Stokesschen Strömungen mikropolarer Flüssigkeiten

Zusammenfassung Die gegenständliche Arbeit befaßt sich mit der Existenz und Eindeutigkeit der Lösungen Stokesscher Strömungen mikropolarer Flüssigkeiten unter Verwendung der Methoden der Potentialtheorie. Integraldarstellungen werden für die Geschwindigkeits-und Mikrorotationsvektoren abgeleitet. Diese führen zur Einführung von Einschicht-und Zweischichtpotentialen, deren Eigenschaften festgelegt werden. Mit Hilfe dieser Ergebnisse werden notwendige und hinreichende Bedingungen für das Stokes-problem der Mikrokontinuumsflüssigkeitsmechanik abgeleitet.

     

Slip flow past an approximate spheroid

Summary Creeping axisymmetric slip flow past a spheroid whose shape deviates slightly from that of a sphere is investigated. An exact solution is obtained to the first order in the small parameter characterizing the deformation. As an application, the case of flow past an oblate spheroid is considered and the drag experienced by it is evaluated. Special well-known cases are deduced and some observations made.Notation A _n, B_n, C_n, D_n, E_n, F_n, b₂, d₂ Constants - a, b radii of spheres - coefficient of sliding fraction - D drag - , _m parameters characterizing the deformation of the sphere - c a(1+) - viscosity coefficient - - dimensionless coordinate - I _n Gegenbauer function - P _n Legendre function - Stream function - U stream velocity at infinity     

Lamé-type potentials in a non-Newtonian fluid theory

Summary The existence and completeness of a set of potentials are exhibited in the case of a linear system of coupled differential equations characterizing slow incompressible flow of micropolar fluids. These potentials correspond to the Lamé's potential of classical elasticity.     

Non-Newtonian fluid sphere in a spherical container

Summary The motion of an inner non-Newtonian fluid sphere at the instant it passes the centre of a rigid spherical container is investigated. The inner and outer fluid motions have been determined and the drag on the inner sphere evaluated. Well-known cases are deduced and wall effects then examined.     

Multiple Analysis in a Reactive Distillation Column for the Synthesis of tert‐Amyl Methyl Ether

tert‐Amyl methyl ether (TAME) is produced via reactive distillation. A simulation is set up and controlled on Aspen HYSYS v 8.0 for generating the highest purity of TAME. This simulation includes a plug‐flow reactor and a reactive distillation column. Emphasis was put on finding the optimal operating conditions of the reactive distillation column in order to get the maximum purity of TAME. The operational parameters were reflux ratio, number of reactive stages in the distillation column, and condenser pressure. The results indicated the optimal reflux ratio and condenser pressure which could be adapted to industrial scale.     

Representations in hydrodynamics

Summary An elementary matrix inversion technique is employed to construct Galerkin type representations for the creeping motion equations of fluid mechanics. These representations lead to the fundamental singular solutions which together with a reciprocal theorem yield integral representations for the velocity vector and the pressure distribution. Some of the results agree with known solutions.

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Darstellungen in der Hydrodynamik

Zusammenfassung Eine elementare Matrix-Inversions-Methode wird angewendet, um Galerkin-Darstellungen für die Gleichungen der Kriechbewegung der Flüssigkeitsmechanik zu konstruieren. Diese Darstellungen führen zu den fundamentalen singulären Lösungen, die zusammen mit einem entsprechenden Theorem Integraldarstellungen für den Geschwindigkeitsvektor und die Druckverteilung liefern. Einige der erhaltenen Resultate stimmen mit den bekannten Lösungen überein.