Momentum transfer

The summation of the momentum flux over the surface of a spherical particle by 

Throughout this discussion the restriction Ma. l is applied. This is reasonable for many ultrafine-particle systems. Relaxation of this restriction is possible, but this leads to difficult problems outside the free-molecular regime. 

For nonequilibrium host gas, a number of particle forces, arise. These include isothermal drag force, thermal force, photophoretic force, diffusion force, stress force, and other additional cross effects in combined flows of heat, mass, and momentum. 

Of these various forces, only the isothermal drag force and the thermal force will be discussed here. The isothermal drag force presents perhaps the simplest of the nonequilibrium, noncontinuum phenomena. Yet, as will be shown, the current state of knowledge of the drag force is inadequate. The thermal force provides an example of the complexity inherent in particle motion in nonequilibrium host gas. 

Numerous experimental and theoretical investigations have been made of the isothermal drag force on a particle in the noncontinuum regime. Much of the work is summarized in several references [2.5-7]. As will be evident in this discussion, there exists even now no satisfactory comprehensive description of this phenomenon. 

When the flow characteristics of the fluid are Newtonian, the shear stress Ry in a fluid is proportional to the velocity gradient and to the viscosity. 

The shear stress Ry within the fluid, at a distance y from the boundary surface, is a measure of the rate of transfer of momentum per unit area at right angles to the surface. 

Since pux) is the momentum per unit volume of the fluid, the rate of transfer of momentum per unit area is proportional to the gradient in the T-direction of the momentum per unit volume. The negative sign indicates that momentum is transferred from the fast-to the slow-moving fluid and the shear stress acts in such a direction as to oppose the motion of the fluid. 

From the definition of thermal conductivity, the heat transferred per unit time through unit area at a distance y from the surface is given by  

Real liquids are viscous and are characterized by internal shear stresses and viscous dissipation of energy. The processes occurring in a viscous liquid are thermodynamically irreversible and have spatial heterogeneity. 

Consider a liquid conforming to the Newtonian law of proportionality between the shear stress and the shear velocity, giving rise to so-called Couette flow (Fig. 4.1). 

Couette flow represents a stationary shear flow between two infinite plates separated from each other by a distance h. One of the plates is immovable, while the other moves translationally with a constant velocity U along the x-axis. The pressure p in the liquid is constant. The velocity of the liquid has one component w(y) along the x-axis, and satisfies the conditions of adherence of liquid to surfaces, y = 0 and y = h, that is, u(0) = 0 and u h) = U. Consider an arbitrary in- 

It follows from Newton s second law that for the considered liquid element  

p is the coefficient of dynamic viscosity of the liquid, which depends on temperature and to a lesser degree on pressure. 

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The second temi is proportional to the optical quadnipole transition moment, and so on. For small values of momentum transfer, only the first temi is significant, thus... 

The momentum-transfer or diflfiision cross section is and the viscosity cross section is... 

When a pure gas flows through a channel the accompanying fall in pressure is accounted for partly by acceleration of the flowing stream and partly by momentum transfer to the stationary walls. Since a porous medium may be regarded as an assembly of channels, similar considerations apply to flow through porous media, but in the diffusional situations of principal interest here accelerational pressure loss can usually be neglected. If more than one molecular species is present, we are also interested in the relative motions of the different species, so momentum transfers by collisions between different types of molecules are also important. 

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

Clearly the general situation is very complicated, since all three mechanisms operate simultaneously and might be expected to interact in a complex manner. Indeed, this problem has never been solved rigorously, and the momentum transfer arguments we shall describe circumvent the difficulty by first considering three simple situations in which each of the three separate mechanisms in turn operates alone. In these circumstances Che relations between fluxes and composition and/or pressure gradients can be found without too much difficulty. Rules of combination, which are essea-... 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

Expressions (2.7) and (2.8) differ by a factor 311/8, or about 1.18, and this is typical of the order of error associated with simple momentum transfer arguments. The flux relation (2.5) can also be written in the alternative form... 

A good exposition of the momentum transfer arguments is given by Present [9], but for our purpose it is necessary only to quote the result. For a binary mixture this takes the form... 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

Equations (2.10), (2.18) and (2.24) provide the flux relations in situations where each of the three separate mechanisms of momentum transfer dominates. However, there remains the problem of finding the flux relations in "intermediate" situations where all three mechanisms may be of comparable importance. This has been discussed by Mason and Evans [7], who assumed first that the rates of momentum transfer due to mechanisms (i) and (ii) should be combined additively. If we write equation (2.10) in the form... 

The right hand side represents the rate of momentum transfer from species r by mechanism (i) and, combining this with the rate of transfer by mechanism (ii) as given by equation (2.IS), we obtain... 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

These are the flux relations associated with the dusty gas model. As explained above, they would be expected to predict only the diffusive contributions to the flux vectors, so they should be compared with equations (2.25) obtained from simple momentum transfer arguments. Equations (3,16) are then seen to be just the obvious vector generalization of the scalar equations (2.25), so the dusty gas model provides justification for the simple procedure of adding momentum transfer rates. 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

Kramers and Kistemaker evaluated the momentum transfer Co the wall by molecular impacts in a frame of reference moving with the mole mean velocity This has some algebraic advantage over working in the rest frame of the tube... 

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