Stokes postulates

Consider first the trivial case of a static fluid. Here there can only be normal forces on a fluid element and they must be in equilibrium. If this were not the case, then the fluid would move and deform. Certainly any valid relationship between stress and strain rate must accommodate the behavior of a static fluid. Hence, for a static fluid the strain-rate tensor must be exactly zero e(/- = 0 and the stress tensor must reduce to 

Under nonflowing, hydrostatic, conditions, the normal stresses are equal to the negative of the pressure. This relationship must hold regardless of the coordinate system used to represent the stress state  

The negative sign is a matter of convention a positive pressure is usually understood to be compressive (i.e., directed inward), whereas a positive normal stress is taken to be tensile (i.e., directed outward). Hence the need for the negative sign. 

It is convenient for subsequent derivations to introduce the notion of deviatoric normal stresses, r/ = m + p, meaning the fluid mechanical normal stress plus the thermodynamic pressure  

The deviatoric stress tensor is related only to fluid motion, since for a fluid at rest the tensor is exactly zero. 

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Viscous Forces In the momentum equation (Navier-Stokes equation), forces F acting on the system result from viscous stresses. It is necessary to relate these stresses to the velocity field and the fluid s viscosity. This relationship follows from the stress and strain-rate tensors, using Stokes postulates. 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

Determine the principal axes for the stress tensor. Why are the principal directions the same for the full stress tensor and the deviatoric stress tensor How does this result relate to the Stokes postulates that are used in the derivation of the Navier-Stokes equations ... 

The 2p(d2u/dz2) term makes one contribution to the Laplacian and one to the set of terms that ultimately makes up a V-V contribution. Recall that when Stokes postulates were used to relate stress and strain, the proportionality constant was 2p, not p. Here we see one of the reasons for that choice. We get terms that split nicely into the Laplacian and... 

In this case the direction of the normal stress and the velocity are the same. Hence the positive sign indicates a negative work rate. Again, as anticipated from the Stokes postulates, P = hz-... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

The basis for connecting the stress and strain-rate tensors was postulated first by G. G. Stokes in 1845 for Newtonian fluids. He presumed that a fluid is a continuous medium and that its properties are independent of direction, meaning they are isotropic. His insightful observations, itemized below, have survived without alteration, and are an essential underpinning of the Navier-Stokes equations ... 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

Edmond Becquerel (1820-1891) was the nineteenth-century scientist who studied the phosphorescence phenomenon most intensely. Continuing Stokes s research, he determined the excitation and emission spectra of diverse phosphors, determined the influence of temperature and other parameters, and measured the time between excitation and emission of phosphorescence and the duration time of this same phenomenon. For this purpose he constructed in 1858 the first phosphoroscope, with which he was capable of measuring lifetimes as short as 10-4 s. It was known that lifetimes considerably varied from one compound to the other, and he demonstrated in this sense that the phosphorescence of Iceland spar stayed visible for some seconds after irradiation, while that of the potassium platinum cyanide ended after 3.10 4 s. In 1861 Becquerel established an exponential law for the decay of phosphorescence, and postulated two different types of decay kinetics, i.e., exponential and hyperbolic, attributing them to monomolecular or bimolecular decay mechanisms. Becquerel criticized the use of the term fluorescence, a term introduced by Stokes, instead of employing the term phosphorescence, already assigned for this use [17, 19, 20], His son, Henri Becquerel (1852-1908), is assigned a special position in history because of his accidental discovery of radioactivity in 1896, when studying the luminescence of some uranium salts [17]. 

Hydrodynamic theory [67], based on Stokes-Einstein equation, postulates that solute is represented by a very large sphere in comparison with the surrounding small liquid phase molecules. Solute mobility, and thus its diffusion coefficient, depends on the frictional drag exerted by liquid phase molecules. For heterogeneous gels (rigid polymeric chains), Cukier [85] suggests... 

Reynolds [127] postulated that the Navier-Stokes equations are still valid for turbulent flows, but recognized that these equations could not be applied directly due to the complexity and irregularity of the fluid dynamic variables. A true description of these flows at all points in time and space was not feasible, and probably not very useful at the time. Instead, Reynolds proposed to develop equations governing the mean quantities that were actually measurable. 

As shown in Figure 8 (41), the tailings solids sediment rapidly over a period of months up to approximately 20% solids (Stokes law sedimentation up to 5% solids, hindered sedimentation after that). The fine tails then consolidate over the next 5 years (or 3-5 years for some ponds reference 44) to approximately 30% solids, at which point consolidation slows dramatically. Scott and Dusseault (40) postulated that this drop in the consolidation rate is due to the high concentration of unrecovered bitumen in the MFT (which leads to a hydraulic conductivity 0.1-0.01 of that of equivalent material with no bitumen) and the low effective density of the MFT solids (5, 41, 45-48). As discussed earlier, the interaction of mineral components has also been postulated to explain the lack of consolidation. This implies a floe structure determined by the clay mineral interactions that are, in turn, a function of the water chemistry. Syncrude MFT (fines content > 30 wt%) have a low permeability of 1-0.01 nm s 1 (46, 47). The dewatering behavior is completely dominated by this low permeability and the long drainage paths that result when the MFT are stored in deep ponds (49). 

Johnson et al. (B12) followed Friedlander s (F2) solution based on the Stokes velocity profiles around solid particles, and numerically calculated the external coefficients for Ar,. < 1. Only a slight difference in the Nusselt number was observed when the velocity profiles of Stokes and Hadamard were postulated. These calculations showed that the transfer coefficient ratio of drops and solids increases from 1 for (Ape), = 1 to 3 for (Ape) = 10 . In the absence of oscillation, similar results may be expected at moderately higher Reynolds numbers (Cl, H3). 

It follows from the linear dependence between the energy yield and the exciting wavelength that the quantum yield of fluorescence as well as the fluorescence spectrum of the dye Rhodamine 6G is independent of the exciting wavelength even in the anti-Stokes region. At least for this dye there does not exist a reduction of the quantum yield in the anti-Stokes region as postulated by Ketskemety and Farkas. 

Two examples of a theoretical approach to the problem of the prediction of diffusion coefficients in fluid media are the equations postulated in 1905 by Einstein and in 1936 by Eyring. The former is based on kinetic theory and a modification of Stokes law for the movement of a particle in a fluid, and is most conveniently expressed in the form... 

These results have given rise to much theoretical and some experimental investigation. Fogg postulated the thermal Wedge hypothesis to explain his results. This induced Cope (11) to undertake a mathematical investigation in which he simultaneously solved the continuity equation, the Navier-Stokes equation and an energy equation in the form shown in equation 2. 

Before leaving this brief discussion of Stokes contribution to the development of the Navier-Stokes equations, we need to point out that his ideas about fluid behavior were more general than the result indicated by Eq. 1-61 would Indicate. In fact, he defined what is now known as a Stokesian fluid (Arts, Sec. 5.21, 1962) in terms of the following four postulates ... 

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