Geodynamics - Fluids 2

Lecture Notes 22 Fluids (continued) Material Derivative Laws of physics – conservation of mass, conservation of energy, etc. Express in reference frame of material, e.g. rod T=0 T = T0 x L Figure 22.1 Steady state: T = T0 x / L; Lagrangian frame: ρ c p ∂T =0 ∂t ∂T = −k∇ 2T + A ∂t Eulerian frame – material is moving. There would be a ∂T for the above rod moving ∂t through. Marching band example. Need to account for “non physical” change due to motion. Above example: ∂T ∂T ∂T where −v is advection term. = −v ∂x ∂x ∂t 1 Material derivative: D ∂ = + v ⋅∇ Dt ∂t Heat conduction DT ∂T = + v∇T = − k ∇ 2T + H ∂t Dt Conservation of Mass – Continuity Equation Consider motion in x2 direction: X3 ρv2 dx3 ] dx ρv2 + ∂ (ρv2) dx2 ∂x2 1 dx2 X2 X1 Figure 22.2 Sides: mass in - mass out = - ∂ ∂ (ρ v2 )dx2 dx1dx3 = (ρ v2 )dV ∂x 2 ∂x 2 Front, back: mass in - mass out = - ∂ (ρ v1 )dV ∂x1 Top, bottom: mass in - mass out = - ∂ (ρ v3 )dV ∂x 3 2 For all 3 directions: - ∂ ∂ ∂ρ ∂ (ρ v1 )(ρ v2 )( ρ v3 ) = ∂x 2 ∂x 3 ∂t ∂x1 ∂ρ ∂ + (ρ vi ) = 0 ∂t ∂x i ∂ρ ∂ρ ∂v + vi +ρ i =0 ∂t ∂x i ∂x i Dρ ∂v + ρ i = 0 (Law of conservation of mass) Dt ∂x i For an incompressible fluid with constant properties −∇p + µ∇ 2 v + ρ x = ρ Dv Dt or, with ν ≡ µ / ρ (dynamic viscosity) 1 ∂p Dv ∂2 vi − +ν + xi = i ρ ∂xi ∂x j ∂x j Dt (Navier-Stokes equation) “Plane strain” V0 BOAT X2 ν (water) X1 Figure 22.3 3 t = 0, v = 0, v( x1 = 0) = (0, v0 , 0) only have v2 ≠ 0 ∞ in x2 direction ⇒ ∂ =0 ∂x2 Subtract out hydrostatic ∂v2 ∂2 v = ν 22 ∂t ∂x1 The solution becomes v = v0 (1 − erf where erf (y) = 2 π y x1 2 νt ) ∫ e ξ dξ . − 2 0 Velocity propagates downward a characteristic depth, x1 = 2 ν t . Example: canoe 5 meters long, v0 = 5 m/sec ⇒ t : 1 sec water ν : 10 −2 cm2/sec ⇒ x1 : 2 10−2 = 2 mm A canoe will drag along about 2 mm water. 4