Generalization of the Prandtl problem for a creep model

An approximate solution to the problem of compression of an infinite layer of material between rough parallel plates is constructed with the creep equations being fulfilled. Constitutive relations in accordance with which the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity are used. The behavior of the solution in the neighborhood of the maximum friction surface is studied. It is shown that the existence of the solution depends on one of the parameters included in the constitutive equations. If the solution exists, the equivalent strain rate tends to infinity in the neighborhood of the maximum friction surface, and the asymptotic behavior of the solution depends on the same parameter. It is established that there is a range of this parameter in which the nature of the change in the equivalent strain rate near the maximum friction surface is the same as in the solutions for rigid plastic materials.     

Effect of a Dispersed Admixture on the Flow Pattern and Heat Transfer in a Supersonic Dusty-Gas Flow Around a Cylinder

A mathematical model of two-phase (gas-solid particle) flow which takes into account particle-particle collisions and the feedback effect of the admixture on the gas parameters is proposed. The dispersed phase is described by a kinetic equation of the Boltzmann type and the carrier gas by modified Navier-Stokes equations. Using this model, a supersonic uniform dusty-gas flow past a cylinder is calculated. The fields of the macroparameters of the admixture and the carrier medium are obtained. The dependence of the heat transfer at the stagnation point on the relative particle size and the free-stream admixture concentration is studied in detail. The ranges of these parameters on which particle collisions and the feedback effect of the admixture on the carrier-gas flow are important are found.     

Steady Film Flow of a Highly Viscous Heavy Fluid with Mass Supply

Asymptotic models of a thin layer of highly viscous heavy incompressible Newtonian fluid are constructed for steady axisymmetric (plane) flow on a curved rigid surface with distributed or point mass supply on a surface section near the axis (plane) of symmetry. Examples of analytical and numerical investigations of the free-surface shape and hydrodynamic-parameter fields are given. The models constructed are generalized for the case of a viscoplastic fluid and solutions which can be used for describing extrusive volcanic eruptions are obtained.     

Effect of Rotation on the Stability of Advective Flow in a Horizontal Fluid Layer at a Small Prandtl Number

The stability of advective flow in a rotating infinite horizontal fluid layer with rigid bound-aries is investigated for a small Prandtl number Pr = 0.1 and various Taylor numbers for perturbations of the hydrodynamic type. Within the framework of the linear theory of stability, neutral curves describing the dependence of the critical Grashof number on the wave number are obtained. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 29–38.Original Russian Text Copyright © 2005 by Schwarz.     

Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model

A semianalytic solution of the problem on the compression of an annular layer of a plastic material obeying the double shear model on a cylindrical mandrel is obtained. The approximate statement of boundary conditions, which cannot be satisfied exactly in the framework of the constructed solution, is based on the same assumptions as the statement of the classical plasticity problem of compression of a material layer between rough plates (Prandtl’s problem). It is assumed that the maximum friction law is satisfied on the inner surface of the layer. The solution is singular near this surface. The strain rate intensity factor is calculated, and its dependence on the process and material parameters is shown.     

Stability of Advective Flow in a Rotating Horizontal Incompressible Fluid Layer Heat-Insulated from Below with Rigid Boundaries at Low Prandtl Number

Fluid Dynamics - The stability of advective flow in a rotating plane horizontal incompressible fluid layer with rigid boundaries is investigated. The linear temperature distribution is specified on...     

Unsteady heat transfer from a rotating disk to fluids with low Prandtl numbers

Summary An analysis is presented concerning unsteady heat transfer from a rotating disk to a low Prandtl number fluid under the condition of a step change in surface temperature with time. Entire time history results for the surface heat flux are given for Prandtl numbers up to 0.04, and these results are obtained by means of a first-order perturbation about the solution for zero Prandtl number. Steady-state heat transfer predicted by this method agrees almost precisely with exact values.     

Energy separation of gases with low and high Prandtl numbers

The process of energy separation of a gas with the Prandtl number which is not equal to unity is investigated. The gas flows through a heat exchanger consisting of two coaxial axisymmetric pipes with sub- and supersonic velocities. Heat exchange between the gas streams takes place as a result of the fact that the recovery factor is not equal to unity. The flow is described by one-dimensional gas dynamic equations for averaged quantities.     

Erosion Profile by a Global Model for Granular Flow

In this paper we study an integro-differential equation that models the erosion of a mountain profile caused by small avalanches. The equation is in conservative form, with a non-local flux involving an integral of the mountain slope. Under suitable assumptions on the erosion rate, the mountain profile develops several types of singularities, which we call kinks, shocks and hyper-kinks. We study the formation of these singularities and derive admissibility conditions. Furthermore, entropy weak solutions to the Cauchy problem are constructed globally in time, taking limits of piecewise affine approximate solutions. Entropy and entropy flux functions are introduced, and a Lax entropy condition is established for the weak solutions.     

Experimental Investigation of Transitional Flow in Porous Media with Usage of a Pore Doublet Model

The transition from laminar to turbulent flow in porous media is studied with a new method. To mimic interconnected pores, a simplified geometry consisting of a pipe with a relatively large diameter that is split into two parallel pipes with different diameters is studied. This is a pore-doublet setup and the pressure drops over the parallel pipes are recorded by pressure transducers for different flow rates. Results show that the flow in the parallel pipes is redistributed when turbulent slugs pass through one of them. The presence of the slugs is revealed by positive skewness in the pressure signals as well as an increase of the standard deviation of the pressure drops and correlation between the pressure drops of the pipes. A frequency analysis of the pressure drops show that lower band frequency pressure variations in one pipe are communicated to the other pipe while higher band frequencies are filtered out.     

Characterization of Single-Phase Flow Through Carbonate Rocks: Quantitative Comparison of NMR Flow Propagator Measurements with a Realistic Pore Network Model

We report here the quantitative comparisons between the measured NMR flow propagator of a carbonate rock and the flow propagator calculated with a porous network extracted from the micro-CT image of the twin plug. We developed a numerical model based on a particle tracking algorithm in pore space. The particle tracking in throats is described using the first arrival time distribution. As pores have an important volume fraction in the sample considered, we implemented a time-delay mechanism for particle transport in the pores. We consider that the nodes have volume and there is a transport of the tracking particles inside the nodes, which leads to an “apparent” time-delay. Simulations of flow propagator show good agreement with low field NMR experiments performed on the twin plug of the sample used for pore network extraction with a single adjustable parameter (that describes the dynamics in the pores). These results lead us to a better understanding of the connection between pore structure and the behavior of NMR flow propagator in fluid-saturated rocks and are essential in interpreting the experimental data and correlating NMR parameters to petrophysical properties.     

The Slow Erosion Limit in a Model of Granular Flow

We study a 2 × 2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler (Granul Matter 2:9–18, 1999). The Cauchy problem for this system has been studied by the authors in recent papers (Amadori and Shen in Commun Partial Differ Equ 34:1003–1040, 2009; Shen in J Math Anal Appl 339:828–838, 2008). In this paper, we first consider an initial-boundary value problem. The boundary condition is given by the flow of the incoming material. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded but possibly large total variations. We then study the “slow erosion (or deposition) limit”. We show that, if the thickness of the moving layer remains small, then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing the rate at which the material slides down. More precisely, in the limit as the thickness of the moving layer tends to zero, the slope of the mountain is provided by an entropy solution to a scalar integro-differential conservation law.     

The Relevance of a Back-Scatter Model for Depth-Averaged Flow Simulation

This study demonstrates the importance of a sophisticated sub-grid model when performing a depth-averaged unsteady RANS simulation of a shallow flow. The reduction of resolution and the spatial dimensions exclude important physical processes as present in three-dimensional turbulence. Especially the effect of the bottom turbulence on the formation of horizontal eddies appears of key importance. A method is proposed to incorporate these effects by means of a kinematic simulation that mimics the residual turbulent fluctuations in a straight channel flow after depth-averaging. This method is developed in the context of the evolution of large eddies in a shallow mixing layer. A comparison with experiments shows that the proposed method works satisfactory. Naturally, it does not fully account for the omission of all 3D-effects.     

Anatomy of a laminar starting thermal plume at high Prandtl number

We present an experimental study of the dynamics of a plume generated from a small heat source in a high Prandtl number fluid with a strongly temperature-dependent viscosity. The velocity field was determined with particle image velocimetry, while the temperature field was measured using differential interferometry and thermochromic liquid crystals. The combination of these different techniques run simultaneously allows us to identify the different stages of plume development, and to compare the positions of key-features of the velocity field (centers of rotation, maximum vorticity locations, stagnation points) respective to the plume thermal anomaly, for Prandtl numbers greater than 10³. We further show that the thermal structure of the plume stem is well predicted by the constant viscosity model of Batchelor (Q J R Met Soc 80: 339–358, 1954) for viscosity ratios up to 50.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

Mixed convection from a horizontal plate to fluids of any Prandtl number

Laminar mixed convection over a horizontal plate with uniform wall temperature or uniform wall heat flux is analyzed by introducing proper buoyancy parameters and transformation variables for fluids of any Prandtl number between 0.001 and 10,000. Both cases of buoyancy assisting and opposing flow conditions are investigated. For the buoyancy-assisting case, the obtained numerical results are very accurate over the entire range of mixed convection intensity from pure forced convection limit to pure free convection limit. For the buoyancy-opposing case, solutions are obtained from the forced convection limit to the point of breakdown.

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Mischkonvektion an einer horizontalen Platte für Fluide mit beliebiger Prandtl-Zahl

Zusammenfassung Es wurde laminare Mischkonvektion an einer horizontalen Platte mit einheitlicher Wandtemperatur oder einheitlicher Wandwärmestromdichte bei Einführung zweckmäßiger Auftriebsparameter und Transformationsvariablen für Fluide mit beliebiger Prandtl-Zahl zwischen 0,001 und 10 000 untersucht. Es wurden die Fälle der Strömung entgegen und in Richtung der Auftriebskraft untersucht. Für den Fall der Strömung in Richtung der Auftriebskraft wurden sehr genaue numerische Ergebnisse für den gesamten Bereich der gemischten Konvektion von rein erzwungener Konvektion bis zu rein freier Konvektion erhalten. Für den Fall der Strömung entgegen der Auftriebsrichtung wurden Lösungen für erzwungene Konvektion bis zum Umkehrpunkt erhalten.

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Nomenclature C _f local friction coefficient - f reduced stream function - g gravitational acceleration - Gr local Grashof number for UWT,g (T _w –T )x ³/ ² - Gr* local Grashof number for UHF,g q _w x ⁴/k ² - m =10 for UWT; and =6 for UHF - n =5 for UWT; and =3 for UHF - Nu local Nusselt number - p pressure - Pr Prandtl number,/ - q _w wall heat flux - Ra local Rayleigh number for UWT,Gr Pr - Ra* local Rayleigh number for UHF,Gr*Pr - Re local Reynolds number,u x/ - T fluid temperature - T _w wall temperature - T free-stream temperature - u velocity component inx-direction - u free-stream velocity - v velocity component iny-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - =0 for UWT; and =1 for UHF - buoyancy parameter, =( Ra)^1/5/( Re)^1/2 for UWT; and =( Ra*)^1/6/( Re)^1/2 for UHF - pseudo-similarity variable, (y/x) - dimensionless temperature, =(T–T )/(T _w –T ) for UWT; and =(T–T )/(q _w x/k) for UHF - =[( Re)^1/2+( Ra)^1/5] for UWT; and =[( Re)^1/2+( Ra*)^1/6] for UHF - dynamic viscosity - kinematic viscosity - /(1+) - dimensionless pressure - density - Pr/(1+Pr) - _w wall shear stress,(u/y) _y=0 - stream function - Pr/(1+Pr)^1/3