Lecture 17 Fluid Dynamics Handouts

Srinivasan u 0 t continuity equation 10 5 continuity

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Unformatted text preview: + ∂x ∂y ∂z EE C245 ∇U = 9 U. Srinivasan © Continuity Equation • Convert surface integral to volume integral using Divergence Theorem EE C245 • For differential control volume ∂ρ ∫∫∫V + ∇ ⋅ ( ρ U ) dV = 0 ∂t U. Srinivasan © ∂ρ + ∇ ⋅ (ρ U) = 0 ∂t Continuity Equation 10 5 Continuity Equation • Material derivative measures time rate of change of a property for observer moving with fluid ∂ρ + (U ⋅ ∇ )ρ + ρ∇ ⋅ U = 0 ∂t Dρ ∂ρ = + ( U ⋅ ∇ )ρ Dt ∂t ∂ ∂ ∂ ∂ D∂ + U y + Uz = + U ⋅ ∇ = + Ux ∂z ∂t ∂x ∂y Dt ∂t For incompressible fluid EE C245 Dρ + ρ∇ ⋅ U = 0 Dt Dρ + ρ∇ ⋅ U = 0 Dt U. Srinivasan © 11 Lecture Outline • Today’s Lecture Basic Fluidic Concepts Conservation of Mass → Continuity Equation Newton’s Second Law → Navier-Stokes Equation Incompressible Laminar Flow in Two Cases Squeeze-Film Damping in MEMS EE C245 • • • • • U. Srinivasan © 12 6 Newton’s Second Law for Fluidics • Newton’s 2nd Law: • Time rate of change of momentum of a system equal to net force acting on system Sum of forces acting on control volume EE C245 ∑F = Rate of momentum efflux = from control volume dp d = ∫∫∫V Uρ dV dt dt Net momentum accumulation rate Rate of accumulation of momentum in control volume + + ∫∫S Uρ (U ⋅ n ) dS Net momentum efflux rate 13 U. Srinivasan © Momentum Conservation • Sum of forces acting on fluid ∑F = pressure and shear stress forces gravity force EE C245 • Momentum conservation, integral form ∫∫S (− Pn + τ )dS + ∫∫∫V ρgdV = U. Srinivasan © d ∫∫∫ ρUdV + ∫∫S ρU(U ⋅ n ) dS dt V 14 7 Navier-Stokes Equation • Convert surface integrals to volume integrals η τ dS = ∫∫∫ η∇ 2 U + ∇(∇ ⋅ U ) dV ∫∫ S V 3 ∫∫ − Pn dS = ∫∫∫ − ∇P dV S V ∫∫ ρU( U ⋅ n ) dS = ∫∫∫ ρU( ∇ ⋅ U ) dV V EE C245 S 15 U. Srinivasan © Navier-Stokes Differential Form η 2 ∫∫∫ − ∇P + ρg + η∇ U + ∇(∇ ⋅ U ) dV = V 3 ∂U dV + ρU(∇ ⋅ U ) dV ∫∫∫ ρ ∫∫∫ V V ∂t EE C245 DU dV ∫∫∫ ρ V Dt ρ DU η = −∇P + ρg + η∇ 2 U + ∇(∇ ⋅ U ) Dt 3 U. Srinivasan © 16 8 Incompressible Laminar Flow • Incompressible fluid ρ ∇⋅U = 0 DU η = −...
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