Lecture 2

Question 1

Systems of units

Answer 1

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.

Question 2

Scaling

Answer 2

Question 3

Nondimensionalization

Answer 3

Question 4

Fluid mechanics

Answer 4

Question 5

The sress tensor

Answer 5

Question 6

Viscosity

Answer 6

Question 7

. The Reynolds number(part 1)

Answer 7

Question 8

. The Reynolds number(part 2)

Answer 8

Question 9

The Navier-Stokes equations

Answer 9

Question 10

Porous medium eqaution

Answer 10

Question 11

Prolongation of vector fields

Answer 11

Question 12

Transformations of function

Answer 12

Question 13

Transformations of the plane

Answer 13

Question 14

The Lie bracket

Answer 14

Question 15

Lie groups and Lie algebras

Answer 15

Question 16

Continuous symmetries of differential equations

Answer 16

Question 17

Translational invariance

Answer 17

Question 18

Similarity solution

Answer 18

Question 19

Scaling invariance

Answer 19

Question 20

pedestrian derivation

Answer 20

Question 21

point source solution

Answer 21

Question 22

The porous medium equation(part 1)

Answer 22

Question 23

The heat equation

Answer 23

Question 24

Self-similarity

Answer 24

Question 25

Validity of the five-thirds law

Answer 25

Question 26

The Kolmogorov length scale

Answer 26

Question 27

The five-thirds law

Answer 27

Question 28

Correlation functions and the energy spectrum

Answer 28

Question 29

Homogeneous, isotropic turbulence

Answer 29

Question 30

Kolmogorov’s 1941 theory of turbulence

Answer 30

Question 31

Stokes formula for the drag on a sphere

Answer 31

Question 32

Stokes equations

Answer 32

Question 33

Euler equations

Answer 33

Question 34

The benefits and drawbacks of dimensional arguments

Answer 34