Lecture 2 Flashcards
Systems of units
The numerical value of any quantity in a mathematical model is measured with respect to a system of units (for example, meters in a mechanical model, or dollars in a financial model). The units used to measure a quantity are arbitrary, and a change in the system of units (for example, from meters to feet) cannot change the model. A crucial property of a quantitative system of units is that the value of a dimensional quantity may be measured as some multiple of a basic unit. Thus, a change in the system of units leads to a rescaling of the quantities it measures, and the ratio of two quantities with the same units does not depend on the particular choice of the system. The independence of a model from the system of units used to measure the quantities that appear in it therefore corresponds to a scale-invariance of the model.
Scaling
Nondimensionalization
Fluid mechanics
The sress tensor
Viscosity
. The Reynolds number(part 1)
. The Reynolds number(part 2)
The Navier-Stokes equations
Porous medium eqaution
Prolongation of vector fields
Transformations of function
Transformations of the plane
The Lie bracket
Lie groups and Lie algebras
Continuous symmetries of differential equations
Translational invariance
Similarity solution
Scaling invariance
pedestrian derivation
point source solution
The porous medium equation(part 1)
The heat equation
Self-similarity
Validity of the five-thirds law
The Kolmogorov length scale
The five-thirds law
Correlation functions and the energy spectrum
Homogeneous, isotropic turbulence
Kolmogorov’s 1941 theory of turbulence
Stokes formula for the drag on a sphere
Stokes equations
Euler equations
The benefits and drawbacks of dimensional arguments
