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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
Flat earth
g= (0, 0, -g0) | Φ=g0z
26
Hydrostatic eqn with constant density
Nabla(Φ*)=0 Φ*=Φ+p/ρ0
27
Hydrostatic eqn with barotropiv fluid
Nabla(Φ*)=0 Φ*=Φ+w(p) w(p)=integral(p,p0) dp/ρ(p)
28
Specific heat at constant pressure
Polytropic FP=γ/(γ-1) * R p/ρ = RT
29
Bulk modulus
Large: small changes in volume
30
Bulk modulus, isothermal
K = p (ideal gas)
31
Bulk modulus, isentropic
K = γp (polytropic)
32
Bulk modulus, polytropic water
K = K0 + γ(p-p0)
33
Adiabatic
No heat transfer with environment
34
Isentropic
Adiabatic and reversible (delta(S)=0)
35
Homentropic
Constant entropy
36
Homentropic ideal gas
pρ^-γ=p0/ρ0^-γ Polytropic with C=p0/ρ0^γ
37
Direct product
ab^T=(axbx axby axbz aybx ayby aybz azbx azby azbz)
38
Volume product
a x b • c = det(cx cy cz ax ay az bx by bz) Volume of parallelepiped spanned by a, b, c
39
Tensor product (indicial)
(ab^T)ij = aibj
40
Cross product (indicial notation)
(a x b)i = εijk*aj*bk
41
Volume product (indicial notation)
a x b • c = εijk*ai*bj*ck
42
Rotation matrices
Orthogonal (B^-1 = B^T) | (BB^T)ij = δij
43
Reflections
Convert right handed coord sys to left handed
44
Vector transformation
u’ = Au
45
Matrix transformation
T’ = ATA^T
46
Scalar field
S’=S | Rank 0 tensor
47
Vector field
Ui’ = aij*Uj Rank 1 tensor
48
Tensor field
T’=ATA^T Tij’ = aik*ajl*Tkl Rank 2 tensor
49
Contractions (indicial notation)
If two indices of a rank r tensor are equal (summed over), the result is an r-2 rank tensor
50
Tensor products
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
Quotient rule (indicial notation)
Tij satisfies Ai=Tij*Bj in all coord systems and Ai and Bj are vectors Then Tij is a rank 2 tensor
52
Stress field
- Local and can be normal or tangential - units of pressure - sigma = F/A - sigma is shear stress (either internal or external)
53
External stress
Stress on interface between body and environment
54
Internal stress
Act at imaginary surfaces inside the body
55
Tensile strength
Maximum tension a material can sustain before it breaks
56
Yield stress
Stress beyond which solids don’t bounce back
57
Cauchy’s stress tensor
- a stress tensor field (rank 2) - σij: i shows force, j shows surface - dFi = sigma ij*dSj - ti = σij*nj (stress vector)
58
Stress tensor: diagonal components
Normal stresses
59
Stress tensor: off-diagonal components
Shear stresses
60
Stress vectors (not t vector, but with sigma)
Sigma x = dFx/dSx (similar for y and z) Sigma x = (sigma xx, sigma yx, sigma zx)
61
Mechanical pressure
P = -1/3 Tr(sigma) Rank 0 tensor
62
Cauchy’s equilibrium equation
fi + dσij/dxj = 0 | f + divergence(sigma T) = 0
63
Constitutive equation
Defines the behaviour of a continuum
64
Normal and shear stresses (formulas with stress vector)
``` tnormal = (t*n)n tshear = t - tnormal t = σ*n = tnormal + tshear ``` n is unit outward normal
65
Symmetric stress tensor
Gives no torque (bc moment of force is 0)
66
Is the stress tensor is symmetric, what do we know?
- eigenvalues are real
67
Stress tensor eigenvalues
Principle stresses
68
Stress tensor eigenvectors
Principle directions of stress In eigenvector basis, stress tensor is diagonal (no shear stresses)
69
Stress vector boundary conditions
Must be continuous across a surface
70
Eulerian displacement
u(x) = x - X(x)
71
Lagrangian displacement
u(X) = x(X) - X
72
Linear displacement
x = aX + b a not singular
73
Linearize needle
δa = a - a0 such that δaj = aj*dui/dxj Alt: δa = (nabla(u))^T*a ``` δ|a| = |a| - |a0| δ(|a|^2) = 2|a|*δ|a| ```
74
Displacement gradient
Nabla(u) = dui/dxj Rank 2 tensor field
75
Slowly varying
|ui| << L L is length of domain
76
Cauchy’s strain tensor
uij = 1/2 * (duj/dxi + dui/dxj) Symmetric u~0 gives no deformations Contains all local geometry info
77
Eigenvalues and eigenvectors for cauchy’s strain tensor
Eigenvectors: principle axes of strain Eigenvalues: how much material is being deformed along principle axes
78
Diagonal elements of uij
Fractional rate of change in that index’s direction
79
Off-diagonal elements of uij
Fractional rate of change in angle between the two needles a and b Projection of uij onto vectors a and b: uab=a*u*b
80
Curve element
δ(dli) = δlj*dui/dxj
81
Volume element
δ(dV) = (divergence(u))dV Divergence if u sets spread in volume dρ/ρ = - divergence(u)
82
Young’s modulus
Displacement parallel to force E = kL/A, k as in hooke’s law Units of pressure
83
Poisson’s ratio
Dimensionless and positive (except for crystals) Displacement orthogonal to force
84
Isotropic materials
Uniform / homogenous in all directions No preferred direction of matter
85
Lamé coefficients
Units of pressure μ is shear modulus (if it’s 0, no shear stress)
86
Mechanical pressure and bulk modulus
K = E/3(1+ν) dp = -1/3 σii = -(λ+2/3 μ)uii change in mechanical pressure
87
Anisotropic materials
Elasticity tensor: σij = Eijkl*ukl | Rank 4 tensor
88
Norm of a matrix
|A| = sqrt(sum(i,j) aij^2)

AMATH 361 (2 decks)

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