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AMATH 361 > Pre-midterm > Flashcards

Pre-midterm Flashcards

1
Q

Molecular scale

A

Length of a cube with the volume of a molecule

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2
Q

Density field

A

ρ(x)= lim(l->Lmicro) Ml(x)/Vl

l is small but large enough for V to contain many molecules

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3
Q

Relative size of density fluctuations

A

Δρ/ρ = 1/sqrt(N)

More molecules = smaller changes in density due to molecular motion

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4
Q

Micro scale

A

Length at which density fluctuations are less than ε

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5
Q

Macro scale

A

Length over which density changes due to fluid

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6
Q

Continuum approximation

A

Valid for L&raquo_space; Lmacro

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7
Q

Conservation laws (def)

A

Related cinematic and dynamic quantities

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8
Q

Conservation laws (examples)

A 
dP/dt = F
P = momentum (vector), F=force
dL/dt = M
L= angular momentum, M=moment of force

dK/dt = P
K=kinetic energy, P=power (scalar)

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9
Q

Centre of mass

A

xm = 1/m * sum(N, i=1)mixi

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10
Q

Moment of force

A

M = sum(N, i=1) xi x(cross) fi

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11
Q

Angular momentum

A

L = sum(N, i=1) xi x(cross) mi*dxi/dt

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12
Q

Power

A

P=sum(N, i=1) fi*dxi/dt

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13
Q

Body force

A

External force that acts over the entire volume of the object

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14
Q

Contact force

A

Force that acts at contact of surfaces of objects

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15
Q

Tension

A

Normal component of force in same direction as unit outward normal

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16
Q

Shear

A

Tangential component of force

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17
Q

Pressure

A

Normal component of force in opposite direction of unit outward normal

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18
Q

Stress vector t

A

dF = t(x, t, n)dS
F=force, x=position, t=time, n=normal, S=surface
Part of “stress principle of Euler and Cauchy”
Fluid at rest cannot have shear stresses (t parallel to n)

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19
Q

Pascal’s law

A

The pressure in a fluid is the same in all directions

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20
Q

Hydrostatic equilibrium

A

Global: triple integral over V (ρg-nabla(p)) = 0

Local: nabla(p) = ρg

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21
Q

Barotropic

A

Independent of temperature

Density and pressure are functions of each other

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22
Q

Isothermal

A

Constant temperature

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23
Q

Polytropic and ideal gas density

A

ρ=(RT/C)^(1/γ-1)

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24
Q

Pressure potential

A

Gravity is conservative: g=-nabla(Φ)

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25
Q

Flat earth

A

g= (0, 0, -g0)

Φ=g0z

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26
Q

Hydrostatic eqn with constant density

A

Nabla(Φ*)=0

Φ*=Φ+p/ρ0

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27
Q

Hydrostatic eqn with barotropiv fluid

A

Nabla(Φ*)=0

Φ*=Φ+w(p)
w(p)=integral(p,p0) dp/ρ(p)

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28
Q

Specific heat at constant pressure

A

Polytropic

FP=γ/(γ-1) * R

p/ρ = RT

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29
Q

Bulk modulus

A

Large: small changes in volume

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30
Q

Bulk modulus, isothermal

A

K = p (ideal gas)

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31
Q

Bulk modulus, isentropic

A

K = γp (polytropic)

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32
Q

Bulk modulus, polytropic water

A

K = K0 + γ(p-p0)

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33
Q

Adiabatic

A

No heat transfer with environment

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34
Q

Isentropic

A

Adiabatic and reversible (delta(S)=0)

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35
Q

Homentropic

A

Constant entropy

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36
Q

Homentropic ideal gas

A

pρ^-γ=p0/ρ0^-γ

Polytropic with C=p0/ρ0^γ

37
Q

Direct product

A

ab^T=(axbx axby axbz
aybx ayby aybz
azbx azby azbz)

38
Q

Volume product

A

a x b • c = det(cx cy cz
ax ay az
bx by bz)

Volume of parallelepiped spanned by a, b, c

39
Q

Tensor product (indicial)

A

(ab^T)ij = aibj

40
Q

Cross product (indicial notation)

A

(a x b)i = εijkajbk

41
Q

Volume product (indicial notation)

A

a x b • c = εijkaibj*ck

42
Q

Rotation matrices

A

Orthogonal (B^-1 = B^T)

(BB^T)ij = δij

43
Q

Reflections

A

Convert right handed coord sys to left handed

44
Q

Vector transformation

A

u’ = Au

45
Q

Matrix transformation

A

T’ = ATA^T

46
Q

Scalar field

A

S’=S

Rank 0 tensor

47
Q

Vector field

A

Ui’ = aij*Uj

Rank 1 tensor

48
Q

Tensor field

A

T’=ATA^T
Tij’ = aikajlTkl

Rank 2 tensor

49
Q

Contractions (indicial notation)

A

If two indices of a rank r tensor are equal (summed over), the result is an r-2 rank tensor

50
Q

Tensor products

A

Product of rank r and rank S tensor is rank r+S tensor

Zero tensor: all components 0 in one coord sys, then they’re 0 in all rotations

51
Q

Quotient rule (indicial notation)

A

Tij satisfies Ai=Tij*Bj in all coord systems and Ai and Bj are vectors

Then Tij is a rank 2 tensor

52
Q

Stress field

A
  • Local and can be normal or tangential
  • units of pressure
  • sigma = F/A
  • sigma is shear stress (either internal or external)
53
Q

External stress

A

Stress on interface between body and environment

54
Q

Internal stress

A

Act at imaginary surfaces inside the body

55
Q

Tensile strength

A

Maximum tension a material can sustain before it breaks

56
Q

Yield stress

A

Stress beyond which solids don’t bounce back

57
Q

Cauchy’s stress tensor

A
  • a stress tensor field (rank 2)
  • σij: i shows force, j shows surface
  • dFi = sigma ij*dSj
  • ti = σij*nj (stress vector)
58
Q

Stress tensor: diagonal components

A

Normal stresses

59
Q

Stress tensor: off-diagonal components

A

Shear stresses

60
Q

Stress vectors (not t vector, but with sigma)

A

Sigma x = dFx/dSx (similar for y and z)

Sigma x = (sigma xx, sigma yx, sigma zx)

61
Q

Mechanical pressure

A

P = -1/3 Tr(sigma)

Rank 0 tensor

62
Q

Cauchy’s equilibrium equation

A

fi + dσij/dxj = 0

f + divergence(sigma T) = 0

63
Q

Constitutive equation

A

Defines the behaviour of a continuum

64
Q

Normal and shear stresses (formulas with stress vector)

A 
tnormal = (t*n)n
tshear = t - tnormal
t = σ*n = tnormal + tshear

n is unit outward normal

65
Q

Symmetric stress tensor

A

Gives no torque (bc moment of force is 0)

66
Q

Is the stress tensor is symmetric, what do we know?

A
  • eigenvalues are real
67
Q

Stress tensor eigenvalues

A

Principle stresses

68
Q

Stress tensor eigenvectors

A

Principle directions of stress

In eigenvector basis, stress tensor is diagonal (no shear stresses)

69
Q

Stress vector boundary conditions

A

Must be continuous across a surface

70
Q

Eulerian displacement

A

u(x) = x - X(x)

71
Q

Lagrangian displacement

A

u(X) = x(X) - X

72
Q

Linear displacement

A

x = aX + b

a not singular

73
Q

Linearize needle

A

δa = a - a0 such that δaj = ajdui/dxj
Alt: δa = (nabla(u))^Ta

δ|a| = |a| - |a0|
δ(|a|^2) = 2|a|*δ|a|
74
Q

Displacement gradient

A

Nabla(u) = dui/dxj

Rank 2 tensor field

75
Q

Slowly varying

A

|ui| &laquo_space;L

L is length of domain

76
Q

Cauchy’s strain tensor

A

uij = 1/2 * (duj/dxi + dui/dxj)

Symmetric
u~0 gives no deformations
Contains all local geometry info

77
Q

Eigenvalues and eigenvectors for cauchy’s strain tensor

A

Eigenvectors: principle axes of strain
Eigenvalues: how much material is being deformed along principle axes

78
Q

Diagonal elements of uij

A

Fractional rate of change in that index’s direction

79
Q

Off-diagonal elements of uij

A

Fractional rate of change in angle between the two needles a and b

Projection of uij onto vectors a and b: uab=aub

80
Q

Curve element

A

δ(dli) = δlj*dui/dxj

81
Q

Volume element

A

δ(dV) = (divergence(u))dV

Divergence if u sets spread in volume

dρ/ρ = - divergence(u)

82
Q

Young’s modulus

A

Displacement parallel to force
E = kL/A, k as in hooke’s law

Units of pressure

83
Q

Poisson’s ratio

A

Dimensionless and positive (except for crystals)

Displacement orthogonal to force

84
Q

Isotropic materials

A

Uniform / homogenous in all directions

No preferred direction of matter

85
Q

Lamé coefficients

A

Units of pressure

μ is shear modulus (if it’s 0, no shear stress)

86
Q

Mechanical pressure and bulk modulus

A

K = E/3(1+ν)

dp = -1/3 σii = -(λ+2/3 μ)uii change in mechanical pressure

87
Q

Anisotropic materials

A

Elasticity tensor: σij = Eijkl*ukl

Rank 4 tensor

88
Q

Norm of a matrix

A

|A| = sqrt(sum(i,j) aij^2)

AMATH 361 (2 decks)

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