MOLECULAR TRANSPORT OF MOMENTUM

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

Reynolds number, Re may be interpreted as the ratio of the convective transport to the molecular transport of momentum or as the ratio of the inertial to viscous forces ... 

The result of the mixing-length idea used to derive the expressions (16.56) and (16.57) is that the turbulent momentum and energy fluxes are related to the gradients of the mean quantities. Substitution of these relations into (16.46) leads to closed equations for the mean quantities. Thus, except for the fact that KM and Kr vary with position and direction, these models for turbulent transport are analogous to those for molecular transport of momentum and energy. 

Viscous stress is an extremely important variable, and this quantity is identified by the Greek letter r. Viscous stress represents molecular transport of momentum, analogous to heat conduction and diffusion. All molecular transport mechanisms correspond to irreversible processes that generate entropy under realistic conditions. When fluids obey Newton s law of viscosity, there is a linear relation between viscous stress and velocity gradients. All fluids do not obey Newton s law of viscosity, but almost aU gases and low-molecular-weight liquids are Newtonian. 

Introduction. In molecular transport of momentum, heat, or mass there are many similarities, which were pointed out in Chapters 2 to 6. The molecular diffusion equations of Newton for momentum, Fourier for heat, and Fick for mass are very similar and we can say that we have analogies among these three molecular transport processes. There are also similarities in turbulent transport, as discussed in Sections 5.7C and 6.1A, where the flux equations were written using the turbulent eddy momentum diffusivity e, the turbulent eddy thermal diffusivity a, and the turbulent eddy mass diffusivity. However, these similarities are not as well defined mathematically or physically and are more difficult to relate to each other. 

Now let s consider the molecular transport of momentum. The molecular mechanism is given by the stress tensor or molecular momentum flux tensor, r. Each element Ty can be interpreted as the component of momentum flux transfer in the direction. We are therefore interested in the terms tix- The rate at which the x component of momentum enters the volume element at face x is XxxAyAx Ij, the rate at which it leaves at face x + Ax is XxxAyAx i+ax, and the rate at which it enters at face y is TyxAxAz y. The net molecular contribution is therefore... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

Under mechanical equilibrium on a molecular scale, the exchange of momentum proceeds faster than the exchange of mass and heat for liquids. On the other hand, the molecular exchange of momentum, matter, and heat is on the same order as gases. The rate of exchange of transport processes is measured by the Schmidt number Sc and the Prandtl number Pr. Usually, the assumption of mechanical equilibrium in gases for heat and mass transfer is not reliable. 

We cannot finish this short introduction on the property transport problems without some observations and commentaries about the content of Figs. 3.2 and 3.3. First, we have to note that, for the generalization of the equations, only vectorial expressions can be accepted. Indeed, considering the equations given in the figures above, some particular situations have been omitted. For example, we show the case of the vector of molecular transport of the momentum that in Fig. 3.3 has been used in a simplified form by eliminating the viscous dissipation. So, in order to generalize this vector, we must complete the Tjj expression with consideration of the difference between the molecular and volume viscosities q — ri ... 

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

The first term on the left-hand side describes the variation of the fluid momentum in time and the second term describes the transport of the momentum in the flow (convective transport). The first term on the right-hand side describes the effect of gradients in the pressure p the second term, the transport of momentum due to the molecular viscosity p (diffusive transport) the third term, the effect of gravity g and in the last term, F lumps together all the other forces acting on the fluid. Techniques for solving the set of four equations (one continuity and three momentum equations) are discussed in a later section of this entry. When the flow is compressible, it is usually necessary to close the system of equations listed above using a thermodynamic equation of state (such as the ideal gas law) that calculates the density as a function of temperature and pressure. 

In a non-equilibrium gas system there are gradients in one or more of the macroscopic properties. In a mono-atomic gas the gradients of density, fluid velocity, and temperature induce molecular transport of mass, momentum, and kinetic energy through the gas. The mathematical theory of transport processes enables the quantification of these macroscopic fluxes in terms of the distribution function on the microscopic level. It appears that the mechanism of transport of each of these molecular properties is derived by the same mathematical procedure, hence they are collectively represented by the generalized property (/ ... 

The conventional parameterizations used describing molecular transport of mass, energy and momentum are the Fick s law (mass diffusion), Fourier s law (heat diffusion or conduction) and Newton s law (viscous stresses). The mass diffusivity, Dc, the kinematic viscosity, i/, and the thermal diffusivity, a, all have the same units (m /s). The way in which these three quantities are analogous can be seen from the following equations for the fluxes of mass, momentum, and energy in one-dimensional systems [13, 135] ... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

In turbulent flows, the transport of momentum, heat, and/or individual species within gradients of velocity, temperature, and concentration is caused predominantly by the chaotic motion of elements of fluid (eddies). This mixing process transports properties much more effectively than the molecular processes identified with viscosity, thermal conductivity, and diffusion. A rather complete description of these processes is given in Refs. 71-73. 

In the primary flow direction, parallel to the interface, within the mass and momentum boundary layers, molecular transport of x momentum in the x direction (i.e., ixd vxldx ) is neglected relative to convective transport of x momentum in the x direction (i.e., pvxdvx/dx). Hence, when convective, viscous, and dynamic pressure forces are equally important, the x component of the dimensionless equation of motion is... 

The laws which govern the phenomena of molecular transport of quantities of matter, of heat and of momentum, seen in the paragraphs above, can be presented in the following general form ... 

The diffusivities D, a, v characterize the ease of the molecular transports of the three quantities matter, heat and momentum, respectively. These diffusivities can be expressed in the same units, i.e. m s in basic SI units. 

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