Viscosity length scale

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... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

The ratio Zt/Zp is in technical reactors much higher than 1. It becomes, e.g. also for a small scale reactor of V-IOOL (H/D = 2 D = 0.4 m) equipped with three turbines (d/D = 0.3) and working at a average impeller power per mass of only = lmVs in media with water like viscosity to Zt/Zp>36...72. The maximal energy dissipation in the impeller zones, required for the calculation of length scale of turbulence here taken from Eq. (20). 

The ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

Figure 7. Experimental data (symbols) for TNB s viscosity [78] superimposed on the results of the fitting procedure (line) from Lubchenko and Wolynes [47] are shown. Ta is diown by a tickmark. (TNB = trinaphthyl benzene). The temperature Ter signifies a crossover from activated to collisional viscosity, dominant at the lower and higher temperatures, respectively (see text). The temperature is varied between the boiling point and the glass transition. The right-hand side panel depicts the temperature dependence of the length scales of cooperative motions in the liquid. The thick solid and dashed lines are the critical radius and the cooperativity length respectively. Taken from Ref. [47] with permission.

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

The relationship between the various length scales can be best understood by looking at their dependence on the turbulence Reynolds number defined in terms of the turbulent kinetic energy k, the turbulent dissipation rate e, and the kinematic viscosity v by... 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

The major difficulty in predicting the viscosity of these systems is due to the interplay between hydrodynamics, the colloid pair interaction energy and the particle microstructure. Whilst predictions for atomic fluids exist for the contribution of the microstructural properties of the system to the rheology, they obviously will not take account of the role of the solvent medium in colloidal systems. Many of these models depend upon the notion that the applied shear field distorts the local microstructure. The mathematical consequence of this is that they rely on the rate of change of the pair distribution function with distance over longer length scales than is the case for the shear modulus. Thus... 

Consider a fluidized bed operated at an elevated temperature, e.g. 800°C, and under atmospheric pressure with ah. The scale model is to be operated with air at ambient temperature and pressure. The fluid density and viscosity will be significantly different for these two conditions, e.g. the gas density of the cold bed is 3.5 times the density of the hot bed. In order to maintain a constant ratio of particle-to-fluid density, the density of the solid particles in the cold bed must be 3.5 times that in the hot bed. As long as the solid density is set, the Archimedes number and the Froude number are used to determine the particle diameter and the superficial velocity of the model, respectively. It is important to note at this point that the rale of similarity requires the two beds to be geometrically similar in construction with identical normalized size distributions and sphericity. It is easy to prove that the length scales (Z, D) of the ambient temperature model are much lower than those in the hot bed. Thus, an ambient bed of modest size can simulate a rather large hot bed under atmospheric pressure. 

The reference scale for the axial coordinate is denoted by zs and for the radial coordinate is given by rs. However, both length scales are arbitrary at this point in the derivation. The scale for the axial velocity (in the principal flow direction) u is mo, a uniform inlet velocity. The density and viscosity can also be scaled by their values in the inlet flow stream, po and p,o, respectively. The scale for the v velocity, vs, is an unknown for now. The nondimensional variables can then be written as... 

We imagine a distribution of a which is characterized by an amplitude o0 and a length scale L which exceeds the maximum scale of the turbulent pulsation, l. We denote the pulsating velocity by u the turbulent coefficient of diffusion, the coefficient of thermal conductivity and the effective turbulent kinematic viscosity are all expressed by the formula k = ul. For an initial uniform distribution of a, obviously,... 

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