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Numerical modeling of rock deformation: Dimensional analysis

Numerical modeling of rock deformation: Dimensional analysis. Jonas Ruh jonas.ruh@erdw.ethz.ch NO E 69 HS2011, Thursday 10-12, NO D11. Introduction. Wiki: “[…] a tool to find or check relations among physical quantities by using their dimensions.”

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Numerical modeling of rock deformation: Dimensional analysis

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  1. Numerical modeling of rock deformation:Dimensional analysis Jonas Ruh jonas.ruh@erdw.ethz.ch NO E 69 HS2011, Thursday 10-12, NO D11 Numerical modeling of rock deformation: Dimensional analysis.

  2. Introduction Wiki: “[…] a tool to find or check relations among physical quantities by using their dimensions.” Internet: “[…] any number or expression can be multiplied by one without changing its value […]” Why doing dimensional analysis? • Converting units • Checking equations • First order estimate of results • Reducing number of parameters • Determine controlling parameter • Determine relative importance of parameters Numerical modeling of rock deformation: Dimensional analysis.

  3. Basic idea “Physical laws do not depend upon arbitrariness in the choice of the basic units of measurement” Mass * Acceleration = Force Newton’s law is true for: [kg] [m/s2] [N] But also for: [slugs] [ft/s2] [pounds] Numerical modeling of rock deformation: Dimensional analysis.

  4. We can use four fundamental units: Length (L), Time (T), Mass (M) and Temperature (q) (Electric charge, etc.) Always know the units of the parameters you use! Units Numerical modeling of rock deformation: Dimensional analysis.

  5. Units Correct conversion between different units: We want to convert inches to feet. How many feet are 60 inches? We know: 12 inches = 1 foot Therefore: 60 inches = 5 feet Solving it with dimensional analysis: We know: Therefore: Or: Numerical modeling of rock deformation: Dimensional analysis.

  6. Units Correct conversion between different units: We want to convert mi/hr to m/s. How many m/s are 55 mi/hr? We know: How to implement the different conversion fractions to get m/s? XX = 24.4 m/s Numerical modeling of rock deformation: Dimensional analysis.

  7. Checking the units The viscosity [Pa s] of rocks is a function of several parameters. What are the units of R, V and A? Units • E and V must have the same units, because they are added. • => • n must be dimensionless, because it is an exponent. • The ratio E/(nRT) is dimensionless since the exponential function is dimensionless. • => E: activation energy; V: activation volume Numerical modeling of rock deformation: Dimensional analysis.

  8. Units Checking the units Defining the unit of A? Numerical modeling of rock deformation: Dimensional analysis.

  9. Dimensionless Functions The following functions and quantities have no dimension: • Sine, cosine, etc. Functions • Exponential functions • Logarithms • Angles Numerical modeling of rock deformation: Dimensional analysis.

  10. Atomic explosion The British physicist G.I.Taylor estimated the energy of the first atomic explosion in 1945 based on a series of pictures that where published in a popular magazine. At that time the energy of the atomic explosion was considered top secret and Taylor’s estimate caused “much embarrassment” in American government circles, because the series of pictures was not classified. Numerical modeling of rock deformation: Dimensional analysis.

  11. Atomic explosion [R] has the unit of L (length): Solution: a = 1/5, b = 2/5, c = -1/5 Dimensional analysis is a powerful tool! Numerical modeling of rock deformation: Dimensional analysis.

  12. The quantity P is dimensionless and does not change when the fundamental units of measurements are changed Period of oscillation: The pendulum 1 What is the period of the pendulum? a On what can the period of oscillation depend? l • Length of pendulum • Mass of bob • Gravitational acceleration Length: L (e.g., meter) Time: T (e.g., second) Mass: M (e.g., kilogram) m Numerical modeling of rock deformation: Dimensional analysis.

  13. Generate units for period out of the three independent units The pendulum 2 Dimensionless form 3 independent units l m On what can the period of oscillation depend? Numerical modeling of rock deformation: Dimensional analysis.

  14. Analytical solution: Folding wavelength m2 P m1 H l What is the characteristic wavelength of viscous folds? Numerical modeling of rock deformation: Dimensional analysis.

  15. Analytical solution: Folding wavelength Note: The relation between the dominant wavelength and the viscosity ratio can be determined with only one set of experiments only varying the viscosity ratio. Dummies would perform four sets of experiments varying individually the parameters H, P, m1 and m2. Numerical modeling of rock deformation: Dimensional analysis.

  16. 1 parameter: S Controlling parameters Folding of a ductile layer resting on an infinite, viscous matrix affected by gravity 7 parameters: n, m, e, H, Dr, g, h Schmalholz et al., JGR, 2002 Numerical modeling of rock deformation: Dimensional analysis.

  17. Literature • Scaling, self-similarity and intermediate asymtotics. Barenblatt, G.I. • Dimensional analysis. Barenblatt, G.I. • Dimensinal analysis and the theory of models. Langhaar, H.L. Numerical modeling of rock deformation: Dimensional analysis.

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