Course summary 18.354 Dimensional analysis Keplers problem L = r - PowerPoint PPT Presentation

Course summary 18.354

Dimensional analysis

Kepler’s problem L = r × md r dt ,

Random walks & diffusion ∂ x = D ∂ 2 n ∂ n ∂ t = − ∂ J x ∂ x 2 , Mark Haw David Walker

(In)stability analysis & pattern formation ∂ t ψ = � U � ( ψ ) + γ 0 ⇥ 2 ψ � γ 2 ( ⇥ 2 ) 2 ψ U ( ⇤ ) = a 2 ⇤ 2 + b 3 ⇤ 3 + c 4 ⇤ 4 ,

Calculus of variations � I [ Y ] 1 = lim ✏ { I [ f ( x ) + ✏� ( x − y )] − I [ f ( x )] } � Y ✏ ! 0  @ f Z x 2 � @ Y � ( x − y ) + @ f @ Y 0 � 0 ( x − y ) = dx x 1  @ f Z x 2 @ Y − d @ f � = � ( x − y ) dx. dx @ Y 0 x 1 @ Y − d @ f @ f 0 = dx @ Y 0

Surface tension

Elasticity

Hydrodynamics ∂ρ Z ∂ρ Z Z ∂ t dV = � ρ u · n dS = � r · ( ρ u ) dV. ∂ t + r · ( ρ u ) = 0 . V S V ρ D u Z Z D u Dt = �r p Dt dV = ( �r p + ρ g ) dV + g . ρ V ( t ) V ( t )

Low Re 10 ㎛

Singular perturbations ✏ d 2 u dx 2 + du dx = 1 .

Conformal mappings dw dz = ∂φ ∂ x + i ∂ψ ∂ x = u − iv. Ze − i α + R 2 ✓ ◆ � i Γ Z e i α W ( Z ) = u 0 2 π ln Z.

Rotating flows � 1 ∂ u ρ r p Ω + ν r 2 u � 2 Ω ⇥ u , ∂ t + u · r u + Ω ⇥ ( Ω ⇥ r ) = = 0 . r · u Taylor columns, etc

Solitons KdV equation

Topological defects

Active matter