Molecular momentum

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... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

Equation (2.16) consists of two contributions the molecular momentum flow tensor, it, and the convective momentum flow tensor, pvv. The term p8 represents the pressure effect, while the contribution t, for a Newtonian fluid, is related to the velocity gradient linearly through the viscosity. The convective momentum flow tensor pw contains the density and the products of the velocity components. A component of the combined momentum flow tensor of x-momentum across a surface normal to the x-direction is... 

Hydrogen bonding also influences the viscosity of water. Viscosity indicates the resistance to flow, reflecting the cohesion within a fluid as well as transfer of molecular momentum between layers of the fluid. It is thus a... 

The equation for the momentum transport in vectorial form, gives (by particularization) the famous Navier-Stokes equation. This equation is obtained considering the conservation law of the property of movement quantity in the differential form P = mw. At the same time, if we consider the expression of the transport vector Jt = f + w(pw) and that the molecular momentum generation rate is given with the help of one external force F, which is active in the balance point, the par-d(pw)... 

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) ... 

The random eddy motion of groups of particles resembles the random motion of molecules in a gas—colliding with each other after traveling a certain distance and exchanging momentum and licat in the process. Therefore, momentum and beat transport by eddies in turbulent boundary layers is analogous to the molecular momentum and heat diffusion. Then turbulent wall shear stress and turbulent heat transfer can be expressed in an analogous manner as... 

The Schmidt number is defined as the ratio of molecular momentum to mass difiusivity. It is used to characterise fluid flows in which there are simultaneous momentum and mass diffusion convection processes. It is named after Ernst Schmidt and expressed as... 

It is seen in Eq. (17) that the effect of chemical bonding appears as the oscillation term cos (pR) except for the normalization factor. The momentum density vanishes if pR = (2n + l)ir, while it has relative maxima if pR = 2nn. Since the oscillation term is always unity in the momentum direction perpendicular to the bond axis, there is a greater probability of finding a given momentum in the perpendicular direction than in the parallel direction. The resultant molecular momentum distribution is an ellipsoid with its minor axis along the parallel axis, which should be compared to spherical distributions for atoms (see Fig. 16 for an example). 

Prandtl number Fr k oc Ratio of molecular momentum and thermal diffusivities Forced and natural convection... 

Schmidt number Sc V D Ratio of molecular momentum and mass diffusivities Mass transfer... 

Identify the analogies among molecular momentum, heat, and mass transfer for the simple case of fluid flow past a flat plate. 

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... 

Similar agreement with free-molecular momentum calculations for spherical particles has been obtained through study of Brownian diffusion in nonequilibrium gases as described by the Fokker-Planck equation [2.23,24]. 

Prandtl number [ratio of molecular momentum transfer (friction, or viscosity effect) to molecular heat transfer (heat conduction)] Schmidt number [ratio of molecular momentum transfer (friction or viscosity effect) to molecular mass transfer (diffusion effect)]... 

The molecular momentum diffusivity ffp in mVs is a function only of the fluid molecular properties. However, the turbulent momentum eddy diffusivity e, depends on the fluid motion. In Eq. (3.10-29) we related e, to the Prandtl mixing length L as follows ... 

The Schmidt number is the dimensionless ratio of the molecular momentum dilfusi-vity p/p to the molecular mass diffusivity D g. Values of the Schmidt number for gases range from about 0.5 to 2. For liquids Schmidt numbers range from about 100 to over 10 000 for viscous liquids. 

Substituting pilmi = p ilmi + vo in the first term on the right-hand side of (5.84), where p j = pi — mvo is the molecular momentum relative to Vo, and reducing the second term using the definition of vo from Eq. (5.15) gives... 

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... 

The Reynolds number is proportional to the ratio of the inertial and viscous forces, and can also be considered the ratio of the total momentum transfer to the molecular momentum transfer it is generally used to assess hydrodynamic similarity, and to define the critical condition for passage from laminar to turbulent flow. 

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