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Journal articles on the topic 'STOKES LAW'

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Published: 4 June 2021

Last updated: 17 June 2025

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1

Cartwright, Julyan H. E. "Stokes' law, viscometry, and the Stokes falling sphere clock." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2179 (2020): 20200214. http://dx.doi.org/10.1098/rsta.2020.0214.

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Clocks run through the history of physics. Galileo conceived of using the pendulum as a timing device on watching a hanging lamp swing in Pisa cathedral; Huygens invented the pendulum clock; and Einstein thought about clock synchronization in his Gedankenexperiment that led to relativity. Stokes derived his law in the course of investigations to determine the effect of a fluid medium on the swing of a pendulum. I sketch the work that has come out of this, Stokes drag, one of his most famous results. And to celebrate the 200th anniversary of George Gabriel Stokes’ birth I propose using the time of fall of a sphere through a fluid for a sculptural clock—a public kinetic artwork that will tell the time. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

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2

Auerbach, David. "Some limits to Stokes’ law." American Journal of Physics 56, no. 9 (1988): 850–51. http://dx.doi.org/10.1119/1.15442.

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3

Nasyrov, V. V., and M. G. Nasyrova. "About the Stokes law applicability." Mathematical Structures and Modeling, no. 2 (54) (October 5, 2020): 40–48. http://dx.doi.org/10.24147/2222-8772.2020.2.40-48.

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We find the correction coefficient for the Stokes law that permit to use this formula in case of a spherical body in a tube with the glycerol. An interpolation formula for the correction coefficient for a motion with low-Reynolds-number is obtained.

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4

Nabarro, F. R. N. "Cottrell-stokes law and activation theory." Acta Metallurgica et Materialia 38, no. 2 (1990): 161–64. http://dx.doi.org/10.1016/0956-7151(90)90044-h.

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5

Wojnar, Ryszard. "Heuristic derivation of Brinkman's seepage equation." Technical Sciences 4, no. 20 (2017): 359–74. http://dx.doi.org/10.31648/ts.5433.

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Brinkman’s law is describing the seepage of viscous fluid through a porous medium and is more acurate than the classical Darcy’s law. Namely, Brinkman’s law permits to conform the flow through a porous medium to the free Stokes’ flow. However, Brinkman’s law, similarly as Schro¨dinger’s equation was only devined. Fluid in its motion through a porous solid is interacting at every point with the walls of pores, but the interactions of the fluid particles inside pores are different than the interactions at the walls, and are described by Stokes’ equation. Here, we arrive at Brinkman’s law from Stokes’ flow equation making use of successive iterations, in type of Born’s approximation method, and using Darcy’s law as a zero-th approximation.

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6

Schiller, Robert. "The Stokes-Einstein law by macroscopic arguments." International Journal of Radiation Applications and Instrumentation. Part C. Radiation Physics and Chemistry 37, no. 3 (1991): 549–50. http://dx.doi.org/10.1016/1359-0197(91)90033-x.

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7

Djoko, J. K., J. Koko, M. Mbehou, and Toni Sayah. "Stokes and Navier-Stokes equations under power law slip boundary condition: Numerical analysis." Computers & Mathematics with Applications 128 (December 2022): 198–213. http://dx.doi.org/10.1016/j.camwa.2022.10.016.

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8

Yang, Hailing, and Yi Xia. "Hydrodynamic instability of nanofluids in round jet for small Stokes number." Modern Physics Letters B 33, no. 33 (2019): 1950419. http://dx.doi.org/10.1142/s0217984919504190.

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The flow instability of particle-laden jet has been widely studied for large Stokes numbers. However, there is little attention on the case with small Stoke number, which often occurs in practical applications with nanoparticle-laden fluid. In this paper, the instability of nanofluids in round jet is studied numerically for [Formula: see text]. The results show that the law of nanofluids instability is quite similar to regular particle instability for axisymmetric azimuthal mode [Formula: see text]. However, for asymmetric azimuthal mode [Formula: see text], the regular pattern of instability is quite complex and different compared to common particle instability. The variations of wave amplification with wave number for different jet parameter [Formula: see text], Reynolds number Re, particle mass loading [Formula: see text], Knudsen number Kn, Stokes number St and the azimuthal modes [Formula: see text] are given. The flow usually gets more unstable as Knudsen number Kn increases, but the varying law gets inverse at high Reynolds number and at [Formula: see text]. The flow gets more unstable as Stokes number St increases at [Formula: see text] but gets more stable at [Formula: see text]. The decreases in wave number stimulate the flow instability at [Formula: see text] which shows distinct difference for the case at [Formula: see text]. Some unusual results of the effect of B, Re, Z on the flow instability are also discussed.

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9

Barton, I. E. "Exponential-Lagrangian Tracking Schemes Applied to Stokes Law." Journal of Fluids Engineering 118, no. 1 (1996): 85–89. http://dx.doi.org/10.1115/1.2817520.

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The exponential-Lagrangian tracking scheme applied to Stokes Law is developed by introducing a predictor-corrector formulation. The new predictor-corrector schemes are more accurate than the original scheme and are estimated to give a better performance taking into account the increased computational effort. The schemes are tested on two simple problems and the results are compared with the analytical solutions.

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10

Greenwood, Margaret Stautberg, Frances Fazio, Marie Russotto, and Aaron Wilkosz. "Using the Atwood machine to study Stokes’ law." American Journal of Physics 54, no. 10 (1986): 904–6. http://dx.doi.org/10.1119/1.14786.

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11

Straub, Dieter, and Michael Lauster. "Angular momentum conservation law and Navier-Stokes theory." International Journal of Non-Linear Mechanics 29, no. 6 (1994): 823–33. http://dx.doi.org/10.1016/0020-7462(94)90055-8.

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12

Bochniak, W. "The cottrell-stokes law for F.C.C. single crystals." Acta Metallurgica et Materialia 41, no. 11 (1993): 3133–40. http://dx.doi.org/10.1016/0956-7151(93)90043-r.

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13

Hong, Sun Ig, and Campbell Laird. "Deviations from Cottrell-Stokes law in cyclic deformation." Scripta Metallurgica et Materialia 26, no. 7 (1992): 1113–18. http://dx.doi.org/10.1016/0956-716x(92)90239-b.

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14

Hu, Yuxi, and Reinhard Racke. "Compressible Navier–Stokes Equations with Revised Maxwell’s Law." Journal of Mathematical Fluid Mechanics 19, no. 1 (2016): 77–90. http://dx.doi.org/10.1007/s00021-016-0266-5.

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15

Marušić-Paloka, E. "On the Stokes Paradox for Power-Law Fluids." ZAMM 81, no. 1 (2001): 31–36. http://dx.doi.org/10.1002/1521-4001(200101)81:1<31::aid-zamm31>3.0.co;2-g.

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16

Hu, Yuxi, and Reinhard Racke. "Compressible Navier–Stokes Equations with hyperbolic heat conduction." Journal of Hyperbolic Differential Equations 13, no. 02 (2016): 233–47. http://dx.doi.org/10.1142/s0219891616500077.

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We investigate the system of compressible Navier–Stokes equations with hyperbolic heat conduction, i.e. replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time [Formula: see text], global smooth solution exists for small initial data. Moreover, as [Formula: see text] goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier–Stokes equations.

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17

Lushnikov, Pavel M. "Structure and location of branch point singularities for Stokes waves on deep water." Journal of Fluid Mechanics 800 (July 12, 2016): 557–94. http://dx.doi.org/10.1017/jfm.2016.405.

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The Stokes wave is a finite-amplitude periodic gravity wave propagating with constant velocity in an inviscid fluid. The complex analytical structure of the Stokes wave is analysed using a conformal mapping of a free fluid surface of the Stokes wave onto the real axis with the fluid domain mapped onto the lower complex half-plane. There is one square root branch point per spatial period of the Stokes wave located in the upper complex half-plane at a distance $v_{c}$ from the real axis. The increase of Stokes wave height results in $v_{c}$ approaching zero with the limiting Stokes wave formation at $v_{c}=0$. The limiting Stokes wave has a $2/3$ power-law singularity forming a $2{\rm\pi}/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but non-zero $v_{c}$, making the limit of zero $v_{c}$ highly non-trivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of the Riemann surface to find coupled square root singularities at distances $\pm v_{c}$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of the Stokes wave into all of these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that a non-limiting Stokes wave at the leading order consists of an infinite number of nested square root singularities which also implies the existence in the third and higher sheets of additional square root singularities away from the real and imaginary axes. These nested square roots form a $2/3$ power-law singularity of the limiting Stokes wave as $v_{c}$ vanishes.

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18

Wright, Steve. "Time‐dependent Stokes flow through a randomly perforated porous medium." Asymptotic Analysis 23, no. 3-4 (2000): 257–72. https://doi.org/10.3233/asy-2000-398.

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An incompressible fluid is assumed to satisfy the time‐dependent Stokes equations in a porous medium. The porous medium is modeled by a bounded domain in $R^n$ that is perforated for each ε > 0 by ε‐dilations of a subset of $R^n$ arising from a family of stochastic processes which generalize the homogeneous random fields. The solution of the Stokes equations on these perforated domains is homogenized as ε → 0 by means of stochastic two‐scale convergence in the mean, and the homogenized limit is shown to satisfy a two‐pressure Stokes system containing both deterministic and stochastic derivatives and a Darcy‐type law with memory which generalizes the Darcy law obtained for fluid flow in periodically perforated porous media.

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19

Rana, Anirudh Singh, Vinay Kumar Gupta, and Henning Struchtrup. "Coupled constitutive relations: a second law based higher-order closure for hydrodynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2218 (2018): 20180323. http://dx.doi.org/10.1098/rspa.2018.0323.

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In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

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20

Guo, Xiaoyi. "New Exact Solutions for Stokes First Problem of a Generalized Jeffreys Fluid in a Porous Half Space." Applied Mechanics and Materials 477-478 (December 2013): 246–53. http://dx.doi.org/10.4028/www.scientific.net/amm.477-478.246.

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The fractional calculus approach has been taken into account in the Darcys law and the constitutive relationship of fluid model. Based on a modified Darcys law for a viscoelastic fluid, Stokes first problem is considered for a generalized Jeffreys fluid in a porous half space. By using the Fourier sine transform and the Laplace transform, two forms of exact solutions of Stokes first problem for a generalized Jeffreys fluid in the porous half space are obtained in term of generalized Mittag-Leffler function, and one of them is presented as the sum of the similar Newtonian solution and the corresponding non-Newtonian contributions. As the limiting cases, solutions of the Stokes first problem for the generalized second fluid, the fractional Maxwell fluid and the Newtonian fluid in the porous half space are also obtained.

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21

Gorski, Patrick R., and Stanley I. Dodson. "Free-swimming Daphnia pulex can avoid following Stokes' law." Limnology and Oceanography 41, no. 8 (1996): 1815–21. http://dx.doi.org/10.4319/lo.1996.41.8.1815.

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22

Bar-Ziv, Ezra, Bin Zhao, Elaad Mograbi, David Katoshevski, and Gennady Ziskind. "Experimental validation of the Stokes law at nonisothermal conditions." Physics of Fluids 14, no. 6 (2002): 2015–18. http://dx.doi.org/10.1063/1.1476305.

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23

Pau, Paul Chi Fu, J. O. Berg, and W. G. McMillan. "Application of Stokes' law to ions in aqueous solution." Journal of Physical Chemistry 94, no. 6 (1990): 2671–79. http://dx.doi.org/10.1021/j100369a080.

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24

Carlson, Edward H. "A microscopic picture of Reynolds number and Stokes’ law." American Journal of Physics 56, no. 11 (1988): 1045–46. http://dx.doi.org/10.1119/1.15341.

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25

Pritchard, David, Catriona R. McArdle, and Stephen K. Wilson. "The Stokes boundary layer for a power-law fluid." Journal of Non-Newtonian Fluid Mechanics 166, no. 12-13 (2011): 745–53. http://dx.doi.org/10.1016/j.jnnfm.2011.04.011.

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26

Tanner, R. I. "Stokes paradox for power-law flow around a cylinder." Journal of Non-Newtonian Fluid Mechanics 50, no. 2-3 (1993): 217–24. http://dx.doi.org/10.1016/0377-0257(93)80032-7.

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27

Borggaard, Jeff, Traian Iliescu, and John Paul Roop. "An improved penalty method for power-law Stokes problems." Journal of Computational and Applied Mathematics 223, no. 2 (2009): 646–58. http://dx.doi.org/10.1016/j.cam.2008.02.002.

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28

Boukrouche, Mahdi, Imane Boussetouan, and Laetitia Paoli. "Unsteady 3D-Navier–Stokes system with Tresca’s friction law." Quarterly of Applied Mathematics 78, no. 3 (2019): 525–43. http://dx.doi.org/10.1090/qam/1563.

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29

Voronel, A., E. Veliyulin, V. Sh Machavariani, A. Kisliuk, and D. Quitmann. "Fractional Stokes-Einstein Law for Ionic Transport in Liquids." Physical Review Letters 80, no. 12 (1998): 2630–33. http://dx.doi.org/10.1103/physrevlett.80.2630.

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30

Rehmeier, Marco, and Andre Schenke. "Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations." Nonlinear Analysis 227 (February 2023): 113179. http://dx.doi.org/10.1016/j.na.2022.113179.

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31

Darvell, B. W., and N. B. Wong. "Viscosity of dental waxes by use of Stokes' Law." Dental Materials 5, no. 3 (1989): 176–80. http://dx.doi.org/10.1016/0109-5641(89)90009-2.

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32

OGOLO, NAOMI A., MIKE O. ONYEKONWU, and ABBEY T. MICHAEL. "Problems of Stokes’ law application in determining the settling velocity of clays." Journal of Engineering Sciences and Innovation 9, no. 3 (2024): 277–86. https://doi.org/10.56958/jesi.2024.9.3.277.

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Clay sedimentation is important for various applications including settling of drilling mud during downtime in drilling operations, and it is essential to model such processes. A widely accepted theory that explains the kinetics of dispersed particles under gravitational pull in a quiescent medium is Stokes’ law. This law is commendable and reliable for modelling the settling velocity of particles but has been reported to be inadequate for modelling the settling velocity of clays. This has prompted a re-examination of the law with regard to assumptions made in deriving it, and it was found that several factors associated with clay sedimentation were not considered in the law. These factors include particle shape, size, interactions, flocculation, salinity, pH value, concentration and hindrance velocity of settling. Hence, this paper discusses these factors and how they defy assumptions made in Stokes’ law, thereby rendering the law unsuitable for modeling clay settlement in liquids.

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33

Palaniappan, D., S. D. Nigam, and T. Amaranath. "Shear-free boundary in Stokes flow." International Journal of Mathematics and Mathematical Sciences 19, no. 1 (1996): 145–50. http://dx.doi.org/10.1155/s016117129600021x.

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A theorem of Harper for axially symmetric flow past a sphere which is a stream surface, and is also shear-free, is extended to flow past a doubly-body𝔅consisting of two unequal, orthogonally intersecting spheres. Several illustrative examples are given. An analogue of Faxen's law for a double-body is observed.

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34

Neumann, Wladimir, Doris Breuer, and Tilman Spohn. "Water-Rock Differentiation of Icy Bodies by Darcy law, Stokes law, and Two-Phase Flow." Proceedings of the International Astronomical Union 11, A29A (2015): 261–66. http://dx.doi.org/10.1017/s174392131600301x.

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AbstractThe early Solar system produced a variety of bodies with different properties. Among the small bodies, objects that contain notable amounts of water ice are of particular interest. Water-rock separation on such worlds is probable and has been confirmed in some cases. We couple accretion and water-rock separation in a numerical model. The model is applicable to Ceres, icy satellites, and Kuiper belt objects, and is suited to assess the thermal metamorphism of the interior and the present-day internal structures. The relative amount of ice determines the differentiation regime according to porous flow or Stokes flow. Porous flow considers differentiation in a rock matrix with a small degree of ice melting and is typically modelled either with the Darcy law or two-phase flow. We find that for small icy bodies two-phase flow differs from the Darcy law. Velocities derived from two-phase flow are at least one order of magnitude smaller than Darcy velocities. The latter do not account for the matrix resistance against the deformation and overestimate the separation velocity. In the Stokes regime that should be used for large ice fractions, differentiation is at least four orders of magnitude faster than porous flow with the parameters used here.

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35

Ouya, Jules, and Arouna Ouedraogo. "Rigorous justification of hydrostatic approximation of compressible fluid flow equations." Gulf Journal of Mathematics 18, no. 1 (2024): 72–94. https://doi.org/10.56947/gjom.v18i1.2239.

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In this paper, we obtain the 3D compressible primitive equations approximation with gravity by taking the small aspect ratio limit to the Navier-Stokes equations. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations with a pressure law of the form p(ρ) = ρ2 to the compressible primitive equations.

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36

Saeger, R. B., L. E. Scriven, and H. T. Davis. "Transport processes in periodic porous media." Journal of Fluid Mechanics 299 (September 25, 1995): 1–15. http://dx.doi.org/10.1017/s0022112095003399.

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The Stokes equation system and Ohm's law were solved numerically for fluid in periodic bicontinuous porous media of simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) symmetry. The Stokes equation system was also solved for fluid in porous media of SC arrays of disjoint spheres. The equations were solved by Galerkin's method with finite element basis functions and with elliptic grid generation. The Darcy permeability k computed for flow through SC arrays of spheres is in excellent agreement with predictions made by other authors. Prominent recirculation patterns are found for Stokes flow in bicontinuous porous media. The results of the analysis of Stokes flow and Ohmic conduction through bicontinuous porous media were used to test the permeability scaling law proposed by Johnson, Koplik & Schwartz (1986), which introduces a length parameter Λ to relate Darcy permeability k and the formation factor F. As reported in our earlier work on the SC bicontinuous porous media, the scaling law holds approximately for the BCC and FCC families except when the porespace becomes nearly spherical pores connected by small orifice-like passages. We also found that, except when the porespace was connected by the small orifice-like passages, the permeability versus porosity curve of the bicontinuous media agrees very well with that of arrays of disjoint and fused spheres of the same crystallographic symmetry.

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37

Wei, Yizheng, and Chao Sun. "The Depth Distribution Law of the Polarization of the Vector Acoustic Field in the Ocean Waveguide." Journal of Marine Science and Engineering 12, no. 8 (2024): 1325. http://dx.doi.org/10.3390/jmse12081325.

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The polarization of the acoustic field in the ocean waveguide environment is a unique property that can provide new ideas for locating and detecting the underwater target, so it is interesting to study the polarization. This paper extends the Stokes parameters to a broadband form, and uses the non-stationary phase approximation method to simplify the expressions, reducing the complexity of theoretical derivation. A physical phenomenon is observed where polarization exhibits significant variations concerning the sea surface, seafloor, source depth, and the source symmetrical depth. Simulation results demonstrate that the simplified equations using the non-stationary phase approximation are effective. Additionally, by normalizing the broadband Stokes parameters, the effects of horizontal range on the depth distribution law of polarization can be eliminated. Subsequently, using the normalized broadband Stokes parameters, the influence of environmental and source parameters on the depth distribution law of polarization is analyzed. The effectiveness of the non-stationary phase approximation and the range-independence property of the normalized broadband Stokes parameters are verified by processing RHUM-RUM experimental data. Based on the conclusions of this paper, it is expected that the polarization can be used for target depth estimation.

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38

Smoleń, Jakub, Piotr Olesik, Jakub Jała, Hanna Myalska-Głowacka, Marcin Godzierz, and Mateusz Kozioł. "Application of Mathematical and Experimental Approach in Description of Sedimentation of Powder Fillers in Epoxy Resin." Materials 14, no. 24 (2021): 7520. http://dx.doi.org/10.3390/ma14247520.

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In this paper, sedimentation inhibition attempts were examined using colloidal silica in a mathematical and experimental approach. Experimental results were validated by a two-step verification process. It was demonstrated that application of quantitative metallography and hardness measurements in three different regions of samples allows us to describe the sedimentation process using modified Stokes law. Moreover, proper application of Stokes law allows one to determine the optimal colloidal silica amount, considering characteristics of applied filler (alumina or graphite). The results of mathematical calculations have been confirmed experimentally—the experimental results show good agreement with the calculated data.

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39

Chapuis, Robert P., François Duhaime, and Simon Weber. "Simplifying the calculation of equivalent diameter in sedimentation tests." Canadian Geotechnical Journal 52, no. 8 (2015): 1186–89. http://dx.doi.org/10.1139/cgj-2013-0467.

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In sedimentation tests, the equivalent diameter of particles, D, is calculated using an equation derived from Stokes’ law and a factor K interpolated from a table listing values of suspension temperature and the specific gravity of solids. This paper explains how to start with Stokes’ law and obtain the equation used in standards. Then it provides two equations for K, both of which are accurate for the usual temperature range for hydrometer tests, and for any specific gravity. The two equations can be used in spreadsheets to automatically calculate D, an easier process than obtaining or interpolating K from a table.

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40

BOURGEAT, ALAIN, EDUARD MARUŠIĆ-PALOKA, and ANDRO MIKELIĆ. "WEAK NONLINEAR CORRECTIONS FOR DARCY’S LAW." Mathematical Models and Methods in Applied Sciences 06, no. 08 (1996): 1143–55. http://dx.doi.org/10.1142/s021820259600047x.

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We consider the Navier-Stokes system in a periodic porous medium Ωε where ε is the characteristic pore size. The viscosity is of order εβ with 0≤β<3/2, sufficiently close to the critical exponent β=3/2. An asymptotic expansion for the velocity and the pressure, in terms of the local Reynolds number Reε=ε3−2βis set and a second-order error estimate is proved.

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41

Shevchenko, Heorhii, Valentyna Cholyshkina, Vladyslav Kurilov, Halyna Lipska, and Oleksandr Havrosh. "Patterns of constrained particle settling in water mineral suspensions of different densities." Geo-Technical Mechanics, no. 169 (2024): 140–52. https://doi.org/10.15407/geotm2024.169.140.

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The settling velocity of particles in mineral suspensions is a crucial parameter for calculating the design of various hydraulic devices and equipment used for mineral pulp benefication. In studies of gravity separation of heterogeneous particles by settling, the determination of mass settling velocity, the influence of suspension density on the process, and the applicability of classical hydrodynamics laws remain the least explored aspects. Often, free settling conditions are used for calculating hydraulic separation processes, but this introduces significant error in the velocity magnitude, as, in practice, the process occurs under constrained conditions. The purpose of this work was to analyze the patterns of constrained settling using the example of coal particle settling in fly ash suspensions from thermal power plants. The article employs an original method for calculating the characteristics of suspensions and the velocity of constrained settling depending on the density. Experimental data on the mass settling velocity of natural fly ash are presented, which indicate the order of velocities and give grounds for the velocity calculation. Given the fine particle size of the ash, the main focus was on the settling of fine coal in the ash. The analysis covered a database in which the characteristics of suspensions and the velocity of constrained coal settling were determined by varying the density of the ash suspension from 1.05 g/cm³ to 1.3 g/cm³ and the size of the settling coal from 0.01 mm to 4 mm. The database was analyzed using the Reynolds number and the applicability of Stokes' law and Lyashenko's law. It was found that the more dilute is the suspension, the smaller is the particle size that follows Stokes' law, and the smaller is the range of particle sizes that Stokes' law covers, and vice versa. For fine coal fractions of 0.001–0.1 mm, the numerical coefficient in Stokes' law decreases according to an inverse power law depending on the pulp density. The ratio of free to constrained settling velocities decreases according to a power law, similar to Lyashenko's law for porosity. The conducted research expands scientific understanding of the processes of constrained settling, facilitates engineering calculations when designing hydraulic devices, and optimizes their operational modes.

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42

Sinha, Nityanand, Andres E. Tejada-Martínez, Cigdem Akan, and Chester E. Grosch. "Toward a K-Profile Parameterization of Langmuir Turbulence in Shallow Coastal Shelves." Journal of Physical Oceanography 45, no. 12 (2015): 2869–95. http://dx.doi.org/10.1175/jpo-d-14-0158.1.

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AbstractInteraction between the wind-driven shear current and the Stokes drift velocity induced by surface gravity waves gives rise to Langmuir turbulence in the upper ocean. Langmuir turbulence consists of Langmuir circulation (LC) characterized by a wide range of scales. In unstratified shallow water, the largest scales of Langmuir turbulence engulf the entire water column and thus are referred to as full-depth LC. Large-eddy simulations (LESs) of Langmuir turbulence with full-depth LC in a wind-driven shear current have revealed that vertical mixing due to LC erodes the bottom log-law velocity profile, inducing a profile resembling a wake law. Furthermore, in the interior of the water column, two sources of Reynolds shear stress, turbulent (nonlocal) transport and local Stokes drift shear production, can combine to lead to negative mean velocity shear. Meanwhile, near the surface, Stokes drift shear serves to intensify small-scale eddies leading to enhanced vertical mixing and disruption of the surface log law. A K-profile parameterization (KPP) of the Reynolds shear stress comprising local and nonlocal components is introduced, capturing these basic mechanisms by which Langmuir turbulence in unstratified shallow water impacts the vertical mixing of momentum. Single-water-column, Reynolds-averaged Navier–Stokes simulations with the new parameterization are presented, showing good agreement with LES in terms of mean velocity. Results show that coefficients in the KPP may be parameterized based on attributes of the full-depth LC.

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43

Alam, M., S. Saha, and R. Gupta. "Unified theory for a sheared gas–solid suspension: from rapid granular suspension to its small-Stokes-number limit." Journal of Fluid Mechanics 870 (May 15, 2019): 1175–93. http://dx.doi.org/10.1017/jfm.2019.304.

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A non-perturbative nonlinear theory for moderately dense gas–solid suspensions is outlined within the framework of the Boltzmann–Enskog equation by extending the work of Saha & Alam (J. Fluid Mech., vol. 833, 2017, pp. 206–246). A linear Stokes’ drag law is adopted for gas–particle interactions, and the viscous dissipation due to hydrodynamic interactions is incorporated in the second-moment equation via a density-corrected Stokes number. For the homogeneous shear flow, the present theory provides a unified treatment of dilute to dense suspensions of highly inelastic particles, encompassing the high-Stokes-number rapid granular regime ($St\rightarrow \infty$) and its small-Stokes-number counterpart, with quantitative agreement for all transport coefficients. It is shown that the predictions of the shear viscosity and normal-stress differences based on existing theories deteriorate markedly with increasing density as well as with decreasing Stokes number and restitution coefficient.

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McKenna, Brian. "Outsourcing stokes financial crime threat." Computer Fraud & Security 2004, no. 12 (2004): 1–2. http://dx.doi.org/10.1016/s1361-3723(05)70177-9.

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Aguayo, Jorge, and Hugo Carrillo Lincopi. "Analysis of Obstacles Immersed in Viscous Fluids Using Brinkman's Law for Steady Stokes and Navier--Stokes Equations." SIAM Journal on Applied Mathematics 82, no. 4 (2022): 1369–86. http://dx.doi.org/10.1137/20m138569x.

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Szücs, Mátyás, and Róbert Kovács. "Gradient-dependent transport coefficients in the Navier-Stokes-Fourier system." Theoretical and Applied Mechanics, no. 00 (2022): 9. http://dx.doi.org/10.2298/tam221005009s.

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In the engineering praxis, Newton?s law of viscosity and Fourier?s heat conduction law are applied to describe thermomechanical processes of fluids. Despite several successful applications, there are some obscure and unexplored details, which are partly answered in this paper using the methodology of irreversible thermodynamics. Liu?s procedure is applied to derive the entropy production rate density, in which positive definiteness is ensured via linear Onsagerian equations; these equations are exactly Newton?s law of viscosity and Fourier?s heat conduction law. The calculations point out that, theoretically, the transport coefficients (thermal conductivity and viscosity) can also depend on the gradient of the state variables in addition to the wellknown dependence of the state variables. This gradient dependency of the transport coefficients can have a significant impact on the modeling of such phenomena as welding, piston effect or shock waves.

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Bresch, Didier, Pierre—Emmanuel Jabin, and Fei Wang. "Global Existence of Weak Solutions for Compresssible Navier—Stokes—Fourier Equations with the Truncated Virial Pressure Law." Communications in Applied and Industrial Mathematics 14, no. 1 (2023): 17–49. http://dx.doi.org/10.2478/caim-2023-0002.

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Abstract This paper concerns the existence of global weak solutions á la Leray for compressible Navier–Stokes–Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

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Batsanov, Stepan S., Dmitry A. Dan’kin, Sergey M. Gavrilkin, Anna I. Druzhinina, and Andrei S. Batsanov. "Structural changes in colloid solutions of nanodiamond." New Journal of Chemistry 44, no. 4 (2020): 1640–47. http://dx.doi.org/10.1039/c9nj05191k.

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Robbins, M. L., R. Varadaraj, J. Bock, and S. J. Pace. "EFFECT OF STOKES’ LAW SETTLING ON MEASURING OIL DISPERSION EFFECTIVENESS." International Oil Spill Conference Proceedings 1995, no. 1 (1995): 191–96. http://dx.doi.org/10.7901/2169-3358-1995-1-191.

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ABSTRACT Industry laboratory tests to measure dispersion effectiveness for oil spills on water measure only the volume percentage of oil dispersed and not the dispersed particle size. The effect of particle size on settling behavior is particularly pronounced in tests that use long settling times to superimpose a dispersion stability criterion on the effectiveness rating. The authors have studied the effect of settling time on the volume cumulative particle size distribution measured by the Coulter Multisizer II. Using Stokes’ law settling to analyze the results, we have demonstrated the effects of settling flask geometry and sample volume on measured effectiveness. These arbitrary test variables control the settling path height and vary markedly from test to test. The intrinsic variables that control settling vs time—initial particle size distribution, aqueous viscosity, and aqueous and oil densities—are functions of aqueous, oil, and dispersant compositions; temperature; and dispersion energy. The author's analysis shows that the effect of settling variables is to cut off the initial cumulative particle size distribution above a certain particle size, thereby fixing measured effectiveness. Stokes’ law provides a measure of this cutoff size. Experimental data have been developed to support this theoretical analysis. This analysis points to the variables that must be considered with different laboratory tests to rank dispersants when settling is part of the test procedure. Even with a single test, ranking may change with settling time given an initially large fraction of large particles and a sufficiently large difference between the densities of water and oil.

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Kim, Jae-Myoung. "3D Navier-Stokes equations of power law type with damping." Archiv der Mathematik 118, no. 3 (2022): 323–35. http://dx.doi.org/10.1007/s00013-021-01684-z.

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