Model Prandtl

The following flow model is provided by the Prandtl-Reuss law (see Sadovskii, 1992, 1997) ... 

The flow model of Prandtl-Reuss for elastoplastic plate is as follows ... 

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

Temam R. (1986) A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity. Arch. Rat. Mech. Anal. 95, 137-181. 

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

Model experiments where free convection is the important part of the flow are expressed by the Grashof number instead of the Archimedes number, as in Eq. (12.61). The general conditions for scale-model experiments are the use of identical Grashof number, Gr, Prandtl number, Pr, and Schmidt number,, Sc, in the governing equations for the room and in the model. 

A proper representation of the effective viscosity is often problematic. Based on the Prandtl mixing length model for turbulence, Bloor and Ingham-suggest that the variation in p, should be of the form... 

Although Eq. (6-18) can be used to eliminate the stress components from the general microscopic equations of motion, a solution for the turbulent flow field still cannot be obtained unless some information about the spatial dependence and structure of the eddy velocities or turbulent (Reynolds) stresses is known. A classical (simplified) model for the turbulent stresses, attributed to Prandtl, is outlined in the following subsection. 

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

For turbulent flow in smooth tubes, the semiempirical Prandtl-von Karman/Nikuradse or Blasius models represent the friction factor quite well. Whether a tube is hydraulically smooth or rough depends upon... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

In a system with both heat and mass transfer, an extra turbulent factor, kx, is included which is derived from an adapted energy equation, as were e and k. The turbulent heat transfer is dictated by turbulent viscosity, pt, and the turbulent Prandtl number, Prt. Other effects that can be included in the turbulent model are buoyancy and compressibility. 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

Figure 2 Illustration of an instability in the Prandtl-Tomlinson model. The sum of the substrate potential and the elastic energy of the spring is shown at various instances in time. The energy difference between the initial and the final point of the thick line will be the dissipated energy when temperature and sliding velocities are very small.

In the example shown in Figure 5, c is positive and the exponent y is unity however, neither of these statements are universal. For example, the Prandtl-Tomlinson model can best be described with y = 2/3 in certain regimes,26 27 whereas confined boundary lubricants are best fit with y = l.25 28 Moreover, the constant c can become negative, in particular when junction growth is important, where the local contact areas can grow with time as a result of slow plastic flow of the opposed solids or the presence of adhesive interactions that are mediated by water capillaries.29,30... 

Making assumptions regarding the dissipation of heat can also influence solid friction, although typically it is less of an issue. This can be explored most easily within the Prandtl-Tomlinson model however, the lessons to be learned... 

Figure 10 Friction velocity relationship Fk( o) in the Prandtl-Tomlinson model at...

In this section, we give a brief overview of theoretical methods used to perform tribological simulations. We restrict the discussion to methods that are based on an atomic-level description of the system. We begin by discussing generic models, such as the Prandtl-Tomlinson model. Below we explore the use of force fields in MD simulations. Then we discuss the use of quantum chemical methods in tribological simulations. Finally, we briefly discuss multiscale methods that incorporate multiple levels of theory into a single calculation. 

Parameter for molecular diffusion model In a moving zone, equivalent to the reciprocal of Peclet number, dispersion number Reynolds number, Re Prandtle number, Pr Schmidt number Sc... 

The four observations, listed previously, were enough for Ludwig Prandtl (1925) to hypothesize a simple model for describing turbulent transport that works surprisingly well, considering the complexity of turbulent flow. 

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