Molecular momentum flux

As for the Prandtl number, we consider the heat transfer flux which can be written with the use of the fluid enthalpy (Eq. (6.164)) and the molecular momentum flux given by Eq. (6.165) ... 

Forces Due to Viscous Momentum Flux (i.e., 3). A molecular momentum flux mechanism exists which relates viscous stress to Unear combinations of velocity gradients via Newton s law of viscosity if the fluid is Newtonian. Viscous... 

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... 

Launder and Spalding [94] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of... 

Nevertheless, the given momentum flux formula (2.368) is not useful before the unknown average velocity after the collisions v( has been determined. For elastic molecular collisions this velocity can be calculated, in an averaged sense, from the classical momentum conservation law and the definition of the center of mass velocity as elucidated in the following. 

VISCOSITY AND MOMENTUM FLUX. Although Eq. (3.3) serves to define the viscosity of a fluid, it can also be interpreted in terms of momentum flux. The moving fluid just above plane C in Fig. 3.1 has slightly more momentum in the X direction than the fluid just below this plane. By molecular collisions momentum is transferred from one layer to the other, tending to speed up the slower moving layer and to slow down the faster moving one. Thus, momentum passes across plane C to the fluid in the layer below this layer transfers momentum to the next lower layer, and so on. Hence x-direction momentum is transferred in the y direction all the way to the wall bounding the fluid, where = 0, and is delivered to the wall as wall shear. Shear stress at the wall is denoted by t ,. 

To start we set up a momentum balance in the z direction over a system Ax thick, bounded in the z direction by the planes z = 0 and z = L, and extending a distance W in the y direction. First, we consider the momentum flux due to molecular transport. The rate of momentum out-rate of momentum in is the momentum flux at point x -t- Ax minus that at x times the area LW. 

Molecular weight of species A Avogadro s number Number of particles of species. A Hypothetical number of activated complexes Density of states for the ath degree(s) of freedom of an activated molecule with energy in the specified degree(s) of freedom Momentum flux in x direction Probability density for states of energy e Enthalpy release per mole of reaction Molar partition function for species A Reduced molar partition function for the activated complex... 

An alternative NEMD method has been developed that is much simpler to implement than is the SLLOD method, particularly for charged systems such as ionic liquids. The method is called reverse nonequilibrium molecular dynamics (RNEMD) and was first developed as a means for computing thermal conductivity but has also been applied to viscosity. It differs from conventional equilibrium and nonequilibrium methods where the cause is an imposed shear rate and the measured effect is a momentum flux/stress. RNEMD does the opposite it imposes the difficult to compute quantity (the momentum flux or stress) and measures the easy to compute property (the shear rate or velocity profile). The method is very simple to implement because it only requires periodic swapping of momenta between atoms at different positions in the box. These swaps set up a velocity profile in the system (i.e., a shear rate). By tracking the frequency and amount of momentum... 

We can write the tangential momentum flux on a surface s located near the wall as equal to Here, is the number density of molecules crossing surface Y m is the molecular mass is the tangential (slip) velocity on the surface and Vg is the mean thermal speed of the molecule. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

For practical applications, second-order closure models are required for the third-order diffusion correlations, the pressure-strain correlation and the dissipation rate correlation as described by Launder and Spalding [95] and Wilcox ([185], Sect. 6.3). Launder and Spalding [95] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of the terms in the equation. These terms can thus be sufficiently approximated by a gradient... 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [109] or the average species velocity is used [148]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. Nevertheless, the given momentum flux formula (2.537) is not useful before the unknown average velocity after the collisions v has been determined. For elastic molecular collisions this velocity can be calculated, in... 

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. 

MOMEN- TUM BALANCE Rate of change of momentum per unit volume Rale of change of momenium by convection per unit volume Rale of change of momentum by molecular transfer (viscous transfer) per volume Generation per volume (External forces) (Ex gravity) Empirically determined flux specified (3)< Velocity specified (1.2b) ... 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

The desorption flux is so low under these conditions that no gas phase collisions occurred between molecular desorption and LIF probing. Phase space treatments " of final-state distributions for dissociation processes where exit channel barriers do not complicate the ensuing dynamics often result in nominally thermal distributions. In the phase space treatment a loose transition state is assumed (e.g. one resembling the products) and the conserved quantities are total energy and angular momentum the probability of forming a particular flnal state of ( , J) is obtained by analyzing the number of ways to statistically distribute the available (E, J). 

The present approach to the prediction of thermal transport in turbulent flow neglects the effect of thermal flux and temperature distribution upon the relationship of thermal to momentum transport. The influence of the temperature variation upon the important molecular properties of the fluid in both momentum and thermal transport may be taken into account without difficulty if such refinement is necessary. 

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