MAT252 cheat sheet
(continuum mechanics)
Tommy Odland
Spectral theorem
Summation bi=Pjaijxj=aij xjdiv(v) = vi,i =∂ vi
∂xi=∂v1
∂x1+∂v2
∂x2+...
Symmetry The matrix Mis symmetric if MT=M. If MT=−M,the matrix
is antisymmetric. Every matrix Mcan be expressed as:
M=A+S=1
2M−MT
| {z }
antisymmetric
+1
2M+MT
| {z }
symmetric
Gram-Schmidt Given vectors {u1,u2, ...., un}, do:
u1:= u1/||u1||
u2:= u2−(u2·u1)u1
u2:= u2/||u2||
u3:= u3−(u3·u1)u1−(u3·u2)u2
u3:= u3/||u3||
un:= un−Pi(un·ui)ui
To create a set in Rn, start with u1and require that uibe orthogonal
to all previous by solving system with (i−1) unknowns.
Navier-Stokes equations in polar coordinates
Gradient The gradient is grad v=∂v
∂xj⊗ej=∂(viei)
∂xj⊗ej
Divergence The divergence is div v=∂v
∂xj·ej=∂(viei)
∂xj·ej.
Digergence of tensor div T=∂T
∂xk·ek=∂
∂xk[Tij (ei⊗ej)] ·ek
Tensor rule (a⊗b)c=a(b·c)
Navier-Stokes ρDu
Dt =∇ · σ+ρf⇔ma=PiFi
Euler ρDu
Dt =∇·−pI(No viscosity)
Material derivative D
Dt ≡∂
∂t +u· ∇
Polar coordinates and solid-body rotation
Solid body rotation The solid body rotation is v(r, θ) = (vr, vθ) = (0, ωr)in polar
coordinates and v(x, y)=(vx, vy) = ω(−y, x)in Cartesian
Lagrangian variables in two dimensions
Strain rate tensor Let vbe a velocity field (deformation mapping), the strain rate
tensor is: dij =1
2(di,j +dj,i)
Change of v
∇v=d
|{z}
strain rate
+w
|{z}
rotation
=df
|{z}
pure strain
+de
|{z}
expansion
+w
|{z}
rotation
where de=1
3tr dand df= (d−de).
Ideal gas computations
Ideal gas law The ideal gas law is P V =nRT , where nis moles
and Ris a gas constant.
First law of thermodynamics The 1st law is dU =dq−dw, where Uis the internal
energy of the fluid, qis heat into fluid and wis work
onto environment.
Entropy Entropy Sis given by dS =dq
T
Exact differential dQ is exact ⇔Qis a state function ⇔HdQ = 0
Second law of thermodynamics
Carnot engine (1) Isothermal expansion, (2) Adiabatic expansion, (3) Isother-
mal compression, (4) Adiabatic expansion. Isothermal means
constant temperature, adiabatic means no heat transfer.
Efficiency of CE Efficiency is given by η=Wout
Qin = 1 −TL
TH
