Abstract
The equations of motion for viscous fluids are obtained from the mass and momentum equations by parametrization of the viscous stress tensor as an isotropic function of the strain rate tensor \({\varvec{D}}\) with scalar coefficients, which are themselves functions of the invariants of the latter (and possibly additional scalar field quantities such as e.g., the temperature and salinity). Newtonian fluid materials emerge by linearization of this parametrization in \({\varvec{D}}\). They are materially described by a shear viscosity and for compressible liquids (gases) an additional bulk viscosity. The emerging equations appear as Navier-Stokes equations. For water, experimentally supported expressions for these parameters are given in Chap. 1. The simplest nonlinear forms of the viscosity functions generate the material descriptions of dilatant and pseudoplastic liquids, which appear in rheology as power law fluids for which plane wall shear flows are studied. Applications address viscometry, hover craft and viscous Hagen-Poiseuille flows, the Reynolds-Sommerfeld slide bearing model, shallow three-dimensional free surface flows applied to the flows in large ice sheets such as Greenland and Antarctica, significant for estimations of sea level rise due to anthropogenic production of greenhouse gases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In that equation the surface term was only written for an isotropic (hydrostatic) stress state.
- 2.
Any rank-2 function, continuously differentiable in a domain centered at the origin where \({\varvec{D}}={\varvec{0}}\), can be expressed as a power series of \({\varvec{D}}\), which through the Caley-Hamilton theorem, can be replaced by a second order polynomial in \({\varvec{D}}\). This is demonstrated in books on Linear Algebra.
- 3.
For a biography of M. Reiner and circumstances why the Reiner-Rivilin fluid should be named Reiner-Riwlin fluid see Fig. 7.2 .
- 4.
For a short biography of Navier see Fig. 7.3 .
- 5.
For a short biography of Stokes see Fig. 7.4 .
- 6.
- 7.
See however also [28] and http://www.deutsche-biographie.de/ppn11638039X.html.
- 8.
Incidentally, this work has been Albert Einstein ’s doctoral dissertation at the University of Zurich. His goal in this paper was the determination of the size of the sugar molecules in water solution by measuring the viscosities of the bearer fluid, \(\eta _0\), and of the mixture of water plus sugar, \(\eta \). If one then also knows the mass fractions of water and sugar, then one may with the simultaneous assumption of the spherical shape of the sugar molecules compute the volume fraction n of the sugar in the mixture from which an estimate for the diameter of the spheres ensues. This paper of Einstein on hydrodynamics has been published after his pioneering publications on special relativity, Brown ian motion and photoelectric effect (for which he captured the 1921 Nobel prize) in the year 1905. The likely reason was that Einstein ’s first version of his doctoral dissertation was not accepted for reasons of its brevity (\(\sim \)30 pages). Einstein then prolonged it by a single additional sentence at the end. Whether the thesis was then accepted, because Einstein had gained fame in the meantime, is unknown [6–8]. Interesting is also, that a computational error remained undiscovered; the amendment was published by Einstein himself in 1911.
- 9.
In the literature, pseudoplastic behavior is also known as shear thinning behavior, because the effective viscosity decreases with increasing shear stress. This effect occurs because e.g. in polymers the structure of the entangling of the molecules changes under deformation. For this reason pseudoplastic materials are in German also called structurally viscous materials.
- 10.
-
Thomas Graham (1805–1869); British chemist from Glasgow, who worked on the diffusion of gases, and rheology of colloids.
-
Wilhelm Ostwald (1853–1932) was a Baltic-German physical chemist who is known in rheology through his rheometer. In 1909 he won the Nobel prize in chemistry.
-
Arnand de Waele (1887–1966) was a British chemist noted for his contributions to rheology.
-
John Glen (*1927) is a British ice physicist. His power law for the creep deformation of ice appeared in 1953 [10 ] .
-
- 11.
Power law structure for the creep law is not the only condition for making the creep function an infinite viscosity law. Neither is a polynomial law with linear behavior close to small stresses or stretching the only way of regularizing the creep law . If one uses \(\sinh \) laws then the \(\sinh ^{(n-1)/2}\)-law exhibits finite viscosity behavior for \(n = 1\) and infinite viscosity characteristics for \(n \ne 1\). Incidentally, material scientists attribute the non-triviality of k to “diffusion creep”.
- 12.
For a short biography of Coulomb see Fig. 13.8 in Vol. 2, Chap. 13.
- 13.
Interesting accounts on history, design and uses of hovercraft can be found in the internet.
- 14.
For the definitions of the integrals \(J_{j}, j = 1,2,...,5\) and their values see Table 7.2.
- 15.
Arnold Sommerfeld (1868–1951), Professor of theoretical physics for decades at the University in Munich and Osborn Reynolds (1842–1912), Physics Professor at Cambridge University UK, developed this theory of lubrication of gears . Theirs and other pioneers’ work on the theory of hydrodynamic lubrication are collected in “Ostwalds Klassiker der exakten Wissenschaften, Vol. 218: Theorie der hydrodynamischen Schmierung” with fundamental contributions by N. Petrow , O. Reynolds , A. Sommerfeld and A.G.M. Michel . The second treatise of this Volume of Ostwald ’s classics has been edited by S. Zima and is published by Verlag Harri Deutsch (227 pp.) in the year 2000. ISBN 978-3-8171- 3218-8. For a brief biographical sketch of Sommerfeld, see Vol. 2, Chap. 14, Fig. 14.5.
- 16.
For a short biography of Froude see Fig. 7.25 .
- 17.
E.g. \(T_1\) the temperature; \(T_{j}, j \ge 2\) for polycrystalline ice a parameter identifying impurities, such as dust, or a bubble parameter in a foamy material, etc.
- 18.
The computations for the right panel of Fig. 7.27 were performed by R. Greve with the software SICOPOLIS, available through http://www.sicopolis.net.
- 19.
We propose here the denotation Shallow Flow Approximation, SFA, since it applies in many practical examples of geophysical and environmental fluid mechanics, e.g., in creeping flow of earth masses on mountain slopes, in certain lava flows, in deformations of salt domes, etc. The SFA is similar to the shallow water approximation (SWA), but it is not the same. We also wish to mention that there exist several variants of the SFA which differ from one another whether the flow is primarily horizontal or downhill, see the following chapter.
- 20.
This equation follows from the balance law of energy (first law of thermodynamics) and will be derived from first principles in Chap. 17 (Vol. 2), see equation (17.85).
- 21.
- 22.
At the present stage of knowledge, where thermodynamics of phase changes have not yet been presented, \(a_{\perp }^{b}\) cannot be given in detail yet, see e.g. Hutter (1984) [16].
- 23.
\(\int \limits _{b}^{h}1\int \limits _{b}^{z} f(\bar{z})\mathrm {d}\bar{z} \mathrm {d}z= \left[ z\int \limits _{b}^{z}f(\bar{z})\mathrm {d}\bar{z}\right] _{b} ^{h} - \int \limits _{b}^{h}zf(z)\mathrm {d}z = \int \limits _{b}^{h}\underbrace{(h - b)}_{H(z)}f(z)\mathrm {d}z\).
- 24.
For a short biography of Clapeyron see Fig. 7.30 .
- 25.
One of the most important reasons of such computations is the estimation of how much ice of the Earth’s ice sheets (Greenland and Antarctica and others, including glaciers) will melt under the anthropogenically caused increase of the Greenhouse gases in the Earth’s atmosphere in the coming decades and centuries and how much this melting contributes to a possible Earth’s surface temperature increase and sea level rise.
- 26.
For a short biography of Frobenius see Fig. 7.31 .
- 27.
The reader should be aware of the fact that the result (7.204) holds, provided the no-slip boundary condition applies. When basal slip is included the singularity may depend on either sliding or differential creep or both. The behavior depends upon the form of the sliding law and the flow law of the fluid and how they compete.
References
ASME-Steam Tables: Thermodynamics properties of steam. Am. Soc. Mech. Eng. New York (1940)
Baral, D.: Asymptotic theories of large scale motion, temperature and moisture distributions in land based polythermal ice shields and in floating ice shelves—A critical review and new developments, Ph.D. thesis, Darmstadt University of Technology, Darmstadt, Germany (1999)
Baral, D.R., Hutter, K., Greve, R.: Asymtotic theories of large scale motion, temperature and moisture distributions in land based polythermal ice sheets. A critical review and new developments. Appl. Mech. Rev. 54(3), 215–256 (2001)
Batchelor, G.K., Green, J.T.: The hydrodynamic interaction of two small freely moving spheres in a linear flow field. J. Fluid Mech. 56, 375 (1972)
Dorsey, N.E.: Properties of Ordinary Water Substance. Reinhold Publishing Corporation, New York (1940)
Einstein, A.: Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik 322(6), 132–148 (1905)
Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik. 322(8), 549–560 (1905)
Einstein, A.: Zur Elektrodynamik bewegter Körper. Annalen der Physik und Chemie 17, 891–921 (1905)
Einstein, A.: Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik und Chemie, Bd. 29, 289 (1906), with correction in Bd. 34, 591 (1911)
Glen, J.W.: The creep of polycrystalline ice. Proc. R. Soc. Lond. A228, 519–538 (1953)
Hagen, G.H.L.: Ueber die Bewegung des Wassers in engen cylindrischen Röhren. Annalen der Physik 46, 423–442 (1839)
Hagenbach-Bischoff, J.E.: Über die Bestimmung der Zähigkeit einer Flüssigkeit durch Ausflu aus Röhren. Verhh. d. Naturforsch. Ges. in Basel 2 (1860)
Hager, W.H., Castro-Orgaz, O.: William Froude and the Froude number. ASCE Forum Paper. (submitted)
Hager, W.H.: Hydraulicians in Europe, 1800–2000. IHAR-monograph (2003)
Hutter, K.: Theoretical Glaciology. Reidel Dortrecht (1983)
Hutter, K.: Ice and snow mechanics, a challenge to theoretical and applied mechanics. In: Proceedings XVI-ICTAM Congress. Copenhagen, North Holland Publishing Company (1984)
Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)
Jeffrey. G.B.: The motion of elliptical particles immersed in a viscous fluid. Proc. R. Soc. Lond. 102A, 161 (1923)
Johnson, R.E., McMeeking, R.M.: Near surface flow in glaciers obeying Glen’s flow law. Q. J. Mech. Appl. Math. 37, 273–291 (1984)
Kirchner, N., Hutter, K., Jakobsson, M., Gyllencreutz, R.: Capabilities and limitations of numerical ice sheet models: a discussion for Earth scientists and modelers. Quat. Sci. Rev. 30(25–26), 3691–3704 (2011)
Mallock, A.: Experiments on fluid viscosity. Philos. Trans. R. Soc. Lond. 187, 41–56 (1895)
Newton, I.: Principia. Original title: Philosophiae Naturalis Principia (Latin, 1687). Published in English (1728)
Norton, F.H.: The Creep of Steel at High Temperatures. McGraw-Hill, New York (1929)
Piau, J.M., Bremond, M., Couette, J.M., Piau, M.: Maurice Couette, one of the founders of rheology. Rheolog. Acta 33, 357–368 (1994)
Poiseuille, J.L.: Recherces experimentales sur le movement des liquids dans les tubes de trés petits diamétres. Compes Rendue, 11, 961 (1840), 12 (1841)
Reiner, M.: Selected Papers on Rheology. Amsterdam (1975)
Scott Blair, G.W.: Markus Reiner 1886–1976: Obituary. J. Texture Stud. 7, 151–152 (1976)
Sutera, S.P.: The history of Poiseuille’s law. Annu. Rev. Fluid Mech. 25, 1–19 (1993)
Taylor, G.I.: The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. 138A, 41 (1932)
Wilshinsky, A., Chugunov, V.A.: Modeling ice mechanics by perturbation methods. J. Glaciol. 43(144), 352–358 (1997)
Wilshinsky, A., Chugunov, V.A.: Modeling ice flow in various glacier zones. GAMM. J. Appl. Math. Mech. 65(3), 479–493 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hutter, K., Wang, Y. (2016). Viscous Fluids. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33633-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-33633-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33632-9
Online ISBN: 978-3-319-33633-6
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)