Abstract
The fundamental concepts necessary to understand soft impact will be presented in this and the next chapter. First, definitions of unit, dimension, stress, and strain are introduced. Then, the basic ideas of scaling and dimensional analysis are briefly explained on the basis of fundamental continuum mechanics. After reviewing the elementary theory of the fluid drag force, a list of meaningful dimensionless numbers is provided. Finally, the concept of the similarity law, which is important in the design and analysis of the experimental system, is described. In the next chapter, constitutive laws of soft matter particularly for granular matter are intensively discussed. Because this and the next chapters concern fundamentals, those who already have a good understanding of continuum mechanics and granular matter do not need to read these chapters. Note that, however, many equations derived in these chapters will be used in the subsequent chapters.
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.
- 2.
This assumption is the extremely important to reach the reasonable scaling. If the energy released by explosion is mainly transmitted to the blast wave kinetic energy, these energies should be of the same order of magnitude. Therefore, Π 1 should have an order of magnitude of 100.
- 3.
This notation is used in the remainder of this book. The symbol \(\simeq \) represents an approximate equality, and the symbol \(\propto \) denotes the proportional relation of different dimensional quantities.
- 4.
The derivation of this form is not so simple. Some tips for the vector analysis in spherical coordinate and a fundamental knowledge of fluid mechanics are necessary for the calculation. While the details of the derivation are skipped here, an outline is briefly provided below. For the Stokes dynamics , the solution called Stokeslet which satisfies \(\nabla p =\eta \nabla ^{2}\boldsymbol{u}\) and Eq. (2.26) is useful. In spherical coordinates (r, \(\theta\), ϕ), the stream function \(\varphi\) of the Stokeslet obeys the equation, \(\mathcal{E}^{2}\varphi = 0\), where \(\mathcal{E} = (\partial ^{2}/\partial r^{2}) + (\sin \theta /r^{2})(\partial /\partial \theta )[(1/\sin \theta )(\partial /\partial \theta )]\). Here \(\theta\) corresponds to the zenith angle from the flow (z) axis. The stream function \(\varphi\) is related to the velocity components u r and \(u_{\theta }\) as \(u_{r} = (1/r^{2}\sin \theta )(\partial \varphi /\partial \theta )\) and \(u_{\theta } = (-1/r\sin \theta )(\partial \varphi /\partial r)\). Considering the boundary conditions \(\varphi (r = R_{i}) = 0\), \((\partial \varphi /\partial r)(r = R_{i}) = 0\) (no slip on the surface of the spherical object in radius R i ) and \(\varphi (r \rightarrow \infty ) = (1/2)vr^{2}\sin ^{2}\theta\) (uniform flow of velocity v along z axis), the solution for \(\mathcal{E}^{2}\varphi = 0\) is obtained by assuming the variable separation form \(\varphi = f(r)\sin ^{2}\theta\) as \(\varphi = (1/2 - 3R_{i}/4r + R_{i}^{3}/4r^{3})vr^{2}\sin ^{2}\theta\). Then, the velocity components are written as \(u_{r} = (1 - 3R_{i}/2r + R_{i}^{3}/2r^{3})v\cos \theta\) and \(u_{\theta } = -(1 - 3R_{i}/4r - R_{i}^{3}/4r^{3})v\sin \theta\). The shear-originated drag force F s is computed by \(F_{s} =\int _{s}\eta \dot{\gamma }_{r\theta }\sin \theta ds\), where s is the small surface unit of the object and \(\dot{\gamma }_{r\theta } = r(\partial /\partial r)(u_{\theta }/r) + (1/r)(\partial u_{r}/\partial \theta )\) is the shear strain rate. The normal drag force F n can be computed by \(\int _{s}2\eta (\partial u_{r}/\partial r)\cos \theta ds\) at r = R i . Finally, \(F_{\eta }\) can be computed from \(F_{\eta } = F_{s} + F_{n}\) and D i = 2R i .
- 5.
Using usual notations of representative quantities (v = U, D i = l, and ρ t = ρ), Eq. (2.71) becomes \(C_{D} \sim F_{D}/\rho U^{2}l^{2}\) by omitting a numerical factor.
- 6.
All other drag force forms used in this section (\(F_{\eta }\), F i , and F E ) are also written using their absolute values. Hence the motion’s direction is considered as a positive direction (Fig. 2.7).
- 7.
This form is understandable by considering a centrifugal acceleration.
- 8.
This velocity can roughly be estimated by the impulse balance, \(m_{i}gT_{\mathrm{step}} = 0.5\rho _{t}v^{2}\pi R_{i}^{2}l_{\mathrm{leg}}/v\).
- 9.
This equation dimensionally represents a simple relation, \(m_{m}\langle u_{t}\rangle ^{2} \sim k_{B}T\). More precisely, this relation can be computed by \(\langle u_{t}\rangle = (m_{m}/2\pi k_{B}T)^{2/3}\int _{0}^{\infty }u_{t}\exp (-m_{m}u_{t}^{2}/2k_{B}T)4\pi u_{t}^{2}du_{t}\).
- 10.
F r is sometimes defined as \(F_{r} = U/\sqrt{gl}\).
- 11.
This form comes from the stress balance, \(\rho g\lambda _{c} =\gamma _{c}/\lambda _{c}\).
- 12.
This relation can also be derived by the simple linear approximation of the energy balance per infinitesimal radius variation of the bubble dR b , \(\varDelta p4\pi R_{b}^{2}dR_{b} \simeq 4\pi [(R_{b} + dR_{b})^{2} - R_{b}^{2}]\gamma _{c}\).
- 13.
References
J.C. Maxwell, Proc. Lond. Math. Soc. 3, 257 (1871)
G.I. Barenblatt, Scaling (Cambridge University Press, Cambridge, 2003)
Y. Fung, A First Course in Continuum Mechanics, 3rd edn. (Prentice Hall, New Jersey, 1993)
P.K. Kundu, I.M. Cohen, D.R. Dowling, Fluid Mechanics, 5th edn. (Academic, Waltham, 2011)
P. Oswald, Rheophysics: The Deformation and Flow of Matter (Cambridge University Press, Cambridge, 2009)
K.K. Rao, P.R. Nott, An Introduction to Granular Flow (Cambridge University Press, Cambridge, 2008)
E. Buckingham, Phys. Rev. 4, 345 (1914)
E. Buckingham, Nature 96, 396 (1915)
D. Bolster, R.E. Hershberger, R.J. Donnelly, Phys. Today 64, 42 (2011)
J.C. Gibbings, Dimensional Analysis (Springer, London, 2011)
A.A. Sonin, PNAS 101, 8525 (2004)
G. Taylor, Proc. R. Soc. A 201, 159 (1950)
G. Taylor, Proc. R. Soc. A 201, 175 (1950)
K.L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985)
K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. A 324, 301 (1971)
M.K. Chaudhury, G.M. Whitesides, Langmuir 7, 1013 (1991)
P. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979)
P. de Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and Wetting Phenomena – Drops, Bubbles, Pearls, Waves (Springer, New York, 2004)
L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon, London, 1960)
A. May, J.C. Woodhull, J. Appl. Phys. 19, 1109 (1948)
D. Gilbarg, R.A. Anderson, J. Appl. Phys. 19, 127 (1948)
T.T. Truscott, A.H. Techet, J. Fluid Mech. 625, 135 (2009)
J.W. Glasheen, T.A. McMahon, Phys. Fluids 8, 2078 (1996)
J.W. Glasheen, T.A. McMahon, Nature 380, 340 (1996)
A. May, J.C. Woodhull, J. Appl. Phys. 21, 1285 (1950)
W.G.V. Rosser, An Introduction to Statistical Physics (Wiley, New York, 1982)
P.S. Epstein, Phys. Rev. 23, 710 (1924)
W.A. Allen, E.B. Mayfield, H.L. Morrison, J. Appl. Phys. 28, 370 (1957)
W.A. Allen, E.B. Mayfield, H.L. Morrison, J. Appl. Phys. 28, 1331 (1957)
R.A. Bagnold, Proc. R. Soc. A 225, 49 (1954)
C.S. Campbell, M.L. Hunt, R. Zenit, C.E. Brennen, J. Fluid Mech. 452, 1 (2002)
GDR MiDi, Eur. Phys. J. E 14, 341 (2004)
F. da Cruz, S. Emam, M. Prochnow, J.N. Roux, F.m.c. Chevoir, Phys. Rev. E 72, 021309 (2005)
S.B. Savage, K. Hutter, J. Fluid Mech. 199, 177 (1989)
C. Ancey, P. Coussot, P. Evesque, J. Rheol. 43, 1673 (1999)
M.K. Hubbert, Bull. Geol. Soc. Am. 48, 1459 (1937)
L.L. Nettleton, Bull. Geol. Soc. Am. 18, 1175 (1934)
D.C. Barton, Bull. Geol. Soc. Am. 17, 1025 (1933)
M.K. Hubbert, Bull. Geol. Soc. Am. 62, 356 (1951)
P. Cobbold, L. Castro, Tectonophys. 301, 1 (1999)
Y. Yamada, Y. Yamashita, Y. Yamamoto, Tectonophys. 484, 156 (2010)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Japan
About this chapter
Cite this chapter
Katsuragi, H. (2016). Scaling and Dimensional Analysis. In: Physics of Soft Impact and Cratering. Lecture Notes in Physics, vol 910. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55648-0_2
Download citation
DOI: https://doi.org/10.1007/978-4-431-55648-0_2
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55647-3
Online ISBN: 978-4-431-55648-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)