Diffusivity of momentum

The creation terms embody the changes in momentum arising from external forces in accordance with Newton s second law (F = ma). The body forces arise from gravitational, electrostatic, and magnetic fields. The surface forces are the shear and normal forces acting on the fluid diffusion of momentum, as manifested in viscosity, is included in these terms. In practice the vector equation is usually resolved into its Cartesian components and the normal stresses are set equal to the pressures over those surfaces through which fluid is flowing. 

Pj. 22 Molecular diffusivity of momentum A K Molecular diffusivitv of heat... 

By analogy, equation 1.68 shows that a shear stress component can be interpreted as a flux of momentum because it is proportional to the gradient of the concentration of momentum. In particular, ryx is the flux in the y-direction of the fluid s x-component of momentum. Furthermore, the kinematic viscosity v can be interpreted as the diffusivity of momentum of the fluid. 

It has been assumed that the density is constant in writing these equations, which are therefore strictly valid only for incompressible flow. ed is called the eddy diffusivity and eh the eddy thermal diffusivity. Although s can be interpreted as the eddy diffusivity of momentum, it is usually called the eddy viscosity and sometimes by the better name eddy kinematic viscosity. 

Diffusion of momentum of the velocity fluctuations (or dissipation of turbulent kinetic energy) occurs at the Kolmogorov scale, which is estimated as... 

Momentum can be transported in an analogous manner to mass or heat. The diffusion of momentum is described with a kinematic viscosity, v, which has SI units of m /s ... 

PRANDTL NUMBER. A dimensionless number equal to the ratio of llie kinematic viscosity to the tlienuoiiielric conductivity (or thermal diffusivity), For gases, it is rather under one and is nearly independent of pressure and temperature, but for liquids the variation is rapid, Its significance is as a measure of the relative rates of diffusion of momentum and heat m a flow and it is important m the study of compressible flow and heat convection. See also Heat Transfer. 

The Prandtl number via has been found to be the parameter which relates the relative thicknesses of the hydrodynamic and thermal boundary layers. The kinematic viscosity of a fluid conveys information about the rate at which momentum may diffuse through the fluid because of molecular motion. The thermal diffusivity tells us the same thing in regard to the diffusion of heat in the fluid. Thus the ratio of these two quantities should express the relative magnitudes of diffusion of momentum and heat in the fluid. But these diffusion rates are precisely the quantities that determine how thick the boundary layers will be for a given external flow field large diffusivities mean that the viscous or temperature influence is felt farther out in the flow field. The Prandtl number is thus the connecting link between the velocity field and the temperature field. 

The dissipation (or diffusion ) of momentum and matter is also described by equations similar to (XIV.2.1). For momentum, the diffusivity is given by the ratio ly/p, called the kinematic viscosity, where 17 is the coefficient of viscosity. For gases at STP all those coefficients have about the same order of magnitude, namely, between... 

B Eddy diffusivity 8 for eddy diffusivity of momentum 8 for eddy diffusivity of heat mVs fft/h... 

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

Molecular diffusivity of momentum j, MCp Moleculai diffusivily of heal O k... 

When the molecular diffusivities of momentum, heat, and mass are equal to each other, the velocity, temperature, and concentration boundary layers coincide. 

During food engineering operations, many fluids deviate from laminar flow when subjected to high shear rates. The resulting turbulent flow gives rise to an apparent increase in viscosity as the shear rate increases in laminar flow, i.e., shear stress = viscosity x shear rate. In turbulent flow, it would appear that total shear stress = (laminar stress + turbulent stress) x shear rate. The most important part of turbulent stress is related to the eddies diffusivity of momentum. This can be recognized as the atomic-scale mechanism of energy conversion and its redistribution to the dynamics of mass transport processes, responsible for the spatial and temporal evolution of the food system. 

The thermodynamic inequality may also guide the selection of general constitutive laws governing the diffusion of momentum, energy, and species mass in non-equilibrium chemical reacting mixtures. 

The Prandtl number, Pr = v/a, that determine the ratio between the diffusivity of momentum and the diffusivity of heat. 

S = ntul/f V can also be interpreted as a dimensionless relaxation time r, where tn/f is a characteristic time for particle motion and v/u] h a characteristic time for the turbulent fluctuations. Hence S" " = r". The viscous sublayer is the region near a smooth wall where momentum transport is dominated by the viscous forces, which are large compared with eddy diffusion of momentum. Fol lowing the usual practice and taking the sublayer thickness to extend to y = 5, particles with a slop distance < 5 would not reach the wall if the sublayer were truly stagnant. 

The DE (3-95) is identical in form to the familiar heat equation for radial conduction of heat in a circular cylindrical geometry. Thus we see that the evolution in time of the steady Poiseuille velocity profile is completely analogous to the conduction of heat starting with an initial parabolic temperature profile -(1 - r2)/4. In our problem, the final steady velocity profile is established by diffusion of momentum from the wall of the tube so that the initial profile for w eventually evolves to the asymptotic value uT - 0 as 1 oo. The characteristic time scale for any diffusion process (whether it is molecular diffusion, heat conduction, or the present process) is (f y cli fl iisivity ), where tc is the characteristic distance over which diffusion occurs. In the present process, tc = R and the kinematic viscosity v plays the role of the diffusivity so that... 

In retrospect, it should perhaps have been evident from (3-95) that this would be the appropriate characteristic time scale.13 However, without the preceding discussion, the important observation of an analogy between the diffusion of momentum in start-up of a unidirectional flow and the conduction of heat would not have been evident. We shall discuss the nature of this process in more detail after we have solved (3-95) and (3-96) to obtain the time-dependent velocity profile. 

The physical, intuitive approach is very simple. We first recall that, as t increases, the effect of the wall motion is propagated farther and farther out into the fluid as a consequence of fhe diffusion of momentum normal to the wall. Further, we have seen in the previous section that diffusion will occur across a region of characteristic dimension lc in a characteristic time increment,... 

However, in this case, it is straightforward to nondimensionalize. The characteristic velocity scale is uc = U, and an appropriate characteristic length scale is lc = d. Because the velocity field is established by means of diffusion of momentum, the characteristic time scale is tc = d2 j v. Using these characteristic quantities, we find that the problem in dimensionless form is... 

In the preceding sections of this chapter, we have considered several examples of transient unidirectional flows. In each case, it was assumed that the flow started from rest with the abrupt imposition of either a finite pressure gradient or a finite boundary velocity, and we saw that the flow evolved toward steady state by means of diffusion of momentum with a time scale tc = i2Jv. Here we consider a final example of a transient unidirectional flow problem in which time-dependent motion is produced in a circular tube by the sudden imposition of a periodic, time-dependent pressure gradient ... 

The first two are obvious. The time scale R2/ v is the time for diffusion of momentum across the tube and is certainly relevant to the initial start-up portion of the problem. We recognize the existence of a second time scale in this problem, however, and that is the period 1 / > of the imposed pressure gradient. We may anticipate that this second time scale will be especially relevant for larger times after the initial transients have died out. For the moment, we retain tc as defined in (3-253). Then, introducing dimensionless variables... 

In Chapters 2 and 3 we have already introduced the concept of penetration depth for an approximate solution of conduction problems (recall Section 2.4.1, and Exs. 2.11 and 3.9). This concept, which we utilized to determine the steady or unsteady penetration depth of heat (or thermal boundary layer) in solids and stagnant fluids, actually applies to all diffusion processes, such as diffusion of momentum, mass, electricity, and neutrons, as well as diffusion (or conduction) of heat It is a convenient tool for an approximate solution of conduction problems and is indispensable for convection problems, which are considerably more complicated than conduction problems. 

This law on the diffusion of momentum and the Fourier law of conduction (on the diffusion of heat) are special cases of diffusion phenomena. 

This is consistent with our assumption that Id > V only when u/D < 1. The ratio of the diffusivity of momentum and of the diffusivity of concentration defines the Schmidt number Sc = u/D. This description also applies to temperature fluctuations in turbulent flows, when the temperature field can be approximated as a passive scalar. In that case the diffusion coefficient is replaced by the heat conduction coefficient n and the non-dimensional ratio Pr = u/n is known as the Prandtl number. 

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