Viscoelastic jet drawn under conditions of heat exchange with the ambient medium

The effect of a constant or variable pull on a heated (cooled) viscoelastic jet is investigated. The problem is resolved into the subproblems of the extension of a viscoelastic rod and the motion determined by the rate of the process. The formation of polystyrene fibers is treated as an example.Mekhanika Polimerov, Vol. 4, No. 3, pp. 540–546, 1968     

The existence of minimal surfaces of given topological structure under prescribed boundary conditions

The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently. Nos. [9], [10], [11], [12], [13], [14] of the bibliography at the end of the paper. References to this list are made in square brackets throughout this paper.     

Homotopy analysis method for the asymmetric laminar flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of the disks contracting, rotation, heat transfer and different permeability on the viscous fluids and temperature distribution between two heated contracting rotating disks. Two cases are considered. For the first case, we neglect the viscous dissipation effects in the energy equation and reduce the Navier-Stokes equations and energy equation into nonlinear coupled ODEs by introducing the Von Kármán type similarity transformations. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number and rotation ratio on the velocity and temperature are discussed in detail. The second and more general case is that we consider the viscous dissipation in the energy equation. Under this assumption, the energy equation is reduced to a ordinary differential equation including the Eckert number, whose solution also is solved by HAM.     

Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

The creeping flow around several spherical particles moving on a line perpendicular to a plane wall is calculated numerically using the boundary integral method. The locations of the point forces on the surfaces of the spheres are chosen so as to describe precisely the lubrication regions when the surfaces are close to one another. Earlier results are recovered for the cases of a single sphere and a wall and of two equal spheres far from a wall. New results are presented for two (equal or unequal) spheres close to a plane wall and several equal spheres far from a wall.     

On the visualization of laminar boundary layer oscillations and the transition to turbulent flow

Zusammenfassung Eine Methode für die Sichtbarmachung periodischer Grenzschichtph?nomene ist beschrieben und angewandt auf die Beobachtung des Umschlages Laminar-Turbulent an einem zweidimensionalen Flügel mit NACA-0012-Profil. Str?mungsbilder, die die verschiedenen Stadien der laminaren Grenzschichtschwingungen und ihren endgültigen Zusammenbruch in Turbulenz zeigen, sind hinzugefügt.      

Slow rotation of two spheres placed about the axis of a circular cylinder in a direction perpendicular to their line of centers

Thesimultaneous effects of a cylinder wall and of a neighboring sphere on the frictional force and torque experienced by a sphere is theoretically calculated, at small Reynolds numbers, for the case when two spheres symmetrically placed about the axis of a circular cylinder rotate slowly with equal angular speeds, in opposite directions, about axes perpendicular to their line of centers. Numerical values are provided which enable the frictional force and torque on the reference sphere to be computed from the fundamental geometrical parameters of the system.

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Résumé Les effects simultanés d'une paroi cylindrique et d'une sphère voisine sur la force de frottement et le couple subis par une sphère sont calculés théoriquement pour des nombres de Reynolds petits. On considère le cas où deux sphères placées symétriquement autour de l'axe d'un cylindre circulaire tournent en sens contraire, les axes de rotation étant perpendiculaires à la ligne de centre. Des valeurs numériques qui permettent de calculer la force de frottement et le couple pour la sphère de référence à partir des paramètres géométriques fondamentaux du système, sont données.

     

Numerical solution of the momentum and heat transfer equations for a hydromagnetic flow due to a stretching sheet of a non-uniform property micropolar liquid

A study of the hydromagnetic flow due to a stretching sheet and heat transfer in an incompressible micropolar liquid is made. Temperature-dependent thermal conductivity and a non-uniform heat source/sink render the problem analytically intractable and hence a numerical study is made using the shooting method based on Runge-Kutta and Newton-Raphson methods. The two problems of horizontal and vertical stretching are considered to implement the numerical method. The former problem involves one-way coupling between linear momentum and heat transport equations and the latter involves two-way coupling. Further, both the problems involve two-way coupling between the non-linear equations of conservation of linear and angular momentums. A similarity transformation arrived at for the problem using the Lie group method facilitates the reduction of coupled, non-linear partial differential equations into coupled, non-linear ordinary differential equations. The algorithm for solving the resulting coupled, two-point, non-linear boundary value problem is presented in great detail in the paper. Extensive computation on velocity and temperature profiles is presented for a wide range of values of the parameters, for prescribed surface temperature (PST) and prescribed heat flux (PHF) boundary conditions.     

Towards the computation of minimum drag profiles in viscous laminar flow

A numerical method is given for the solution of certain optimum design problems of fluid mechanics. The profile of given area and smallest drag in a uniform laminar flow is computed. This profile is long and slim, its front end is shaped like a wedge of angle 90° and its rear end is shaped like a cusp. Owing to the numerical complexity of the problem the precision of the results is average (around 5%). However, this work is a good illustration of the theoretical method exposed previously and it shows how good precision can be obtained if one is prepared to pay for it. A numerical solution of the adjoint system of the stationary Navier-Stokes equation is also given; this equation will play an important role in optimum design in fluid mechanics.     

On the numerical modelling of the transitional flow in rarefied gases

The aim of this work is to simulate rarefied gas flow in complex geometries, under flow conditions that range from the hydrodynamic, through the transitional, to the molecular regimes. Existing computational models apply to molecular or viscous flow, but the treatment of the transitional flow is still underdeveloped.To deal with the difficult transitional flow, two models with overlapping ranges of applicability are introduced. A direct simulation Monte Carlo (DSMC) type model, which can be used in the molecular and up to the lower transitional flow, has been designed. For the viscous to the upper transitional flow, a numerical model using a particle method is proposed. The objective is to obtain a smooth transition between the probabilistic simulation of particle histories and the deterministic approach of the solution of partial differential equations.The DSMC model has been successfully applied to molecular and lower transitional flow in a complex geometry with stationary and moving boundaries. The test results agree well with published data. The particle method was tested using simplified Navier-Stokes equations in a channel. Preliminary results in the low viscous range seem to indicate that the approach is viable.     

Problem with nonequilibrium boundary conditions in the kinetic theory of gases

The Boltzmann kinetic equation is considered in a new formulation with nonequilibrium distribution functions on free boundaries, which makes it possible to simulate nonequilibrium superand subsonic flows. Transport processes for such flows are analyzed. The possibility of anomalous transport is determined, in which case the heat flux, temperature gradient, and the corresponding components of the nonequilibrium stress tensor and the velocity gradient have the same sign.     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

Viscosimetric flow of an incompressible elastoviscoplastic material under the presence of a lubricant on the boundary surfaces

Generalizing the model of large deformations by accounting for the viscous properties of materials, we obtain the analytical solutions of some quasistatic boundary value problems concerning the viscosimetric flows of an elastoviscoplastic material in the gap between the rigid coaxial cylindrical surfaces when, in the neighborhood of one of the rigid cylinders (either internal or external), there is a layer of an elastic non-Newtonian lubricant, and the rigid adhesion conditions are satisfied on boundary surfaces. The conditions are studied of origination of a flow in the lubricant layer and in the basic material. The values of the maximum velocity are specified under which the flow does not tresspass the lubricant layer.     

The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe

Summary An exact solution of pulsating laminar flow superposed on the steady motion in a circular pipe is presented under the assumption of parallel flow to the axis of pipe. Total mass of flow on time average is found to be identified with that given byHagen-Poiseuille's low calculated on the steady component of pressure gradient. The phase lag of velocity variation from that of pressure gradient increases from zero in the steady motion to 90° in the pulsation of infinite frequency. Integration of work for changing kinetic energy of fluid through one period is vanished, while that of dissipation of energy by internal friction remains finite and excess amount caused by the components of periodic motion is added to the components of steady flow.It is found that the given rate of mass flow is attained in pulsating motion by giving the same amount of average gradient of pressure as in steady flow, but that excess works to the steady case are necessary for maintenance of this motion.

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Zusammenfassung Eine exakte Lösung der pulsierenden laminaren Strömung in einem Kreisrohr wird angegeben mit der Annahme, dass die Richtung dem Geschwindigkeitsvektor der Rohrachse parallel ist. Die Durchflussmenge stimmt überein mit der aus der stationären Druckgefällekomponente gerechneten Menge. Für die Erhaltung der Bewegung dagegen ist die der Dissipation entsprechende Extraarbeit notwendig. Die Quantität dieser Arbeit hängt ab von den Frequenzen der Stromschwingungen.

     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

Plastic flow in a conical channel: Qualitative features of the solutions under different yield conditions

The problem of the convergence of the solutions of problems of plasticity theory, with a yield condition which depends on the hydrostatic stress, to solutions based on classical plasticity theory with von Mises or Tresea conditions is considered, with a particular choice of the parameters of the material model. For the case of axisymmetric flow of material in a channel with converging and diverging walls, solutions according to two plasticity theories with a yield condition which depends on the hydrostatic stress are compared with the classical solution. It is shown that only the solution using Spencer's model shows all the main features of the classical solution. As the internal criterion of the choice of the preferred plasticity theory when examining a special class of problems, it is suggested that the criterion of the convergence of the solutions to the solutions of classical plasticity theory should be used.