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# ME 3350 Week 7 Notes ME 3350 - 01

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This 3 page Class Notes was uploaded by Jonah Leary on Saturday October 15, 2016. The Class Notes belongs to ME 3350 - 01 at Wright State University taught by Philippe Sucosky in Fall 2016. Since its upload, it has received 3 views. For similar materials see Fluid Dynamics in Mechanical and Materials Engineering at Wright State University.

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Date Created: 10/15/16

ME 3350 Notes – Week 7 2 – Material vs. Spatial Derivatives Associated with these descriptions are two time derivatives: δ δ D 1. δt|particleidentity|r= Dt 2. 0 δ = δ = δ δtlocationidentityt|r δt D Df = lim∆ f Express Dt in terms of spatial coordinates: Dt ∆t→ 0∆t|particle ∆ f = t+∆t ft= (x+∆x ,y+∆ y,z+∆ z,t+∆t − f (x,y,z ,t) Taylor Series Expansion: δf δf δf δf fx+∆x ,y+∆y,z+∆z ,t+∆t =f x,y,z ,)+ ∆x+ ∆ y+ ∆ z+ ∆ t+… δx δy δz δt ∆ f = t+∆t ft=δf ∆x+ δf ∆ y+δf∆ z+δf ∆t+… δx δy δz δt ∆ f= δf ∆ x +δf ∆y + δf ∆z +δf ∆t δx ∆t ( δy ∆t δz ∆t δt Df ∆ f δf δf δf δf Dt= ∆t→ 0t = δxu+ δyv+ δzw+ δt where u, v, and w are x-, y-, and z- components, respectively, of velocity δ ^ δ ^ δ ^ V=ui+v j+w k^ ∇= δxi+δy j+δz k Df δf δ δ δ δf = +(u +v +w )f= + V∙∇ )f Rate of change of f Dt δt δx δy δz δt following motion = Local rate of change of f + convective rate of change of f Application: Acceleration field: DV δV δV δV δV δV a= = +V ∙∇)V= + ( +v +w ) dt δt δt δx δy δz Newton’s 2 ndLaw expressed in spatial coordinates is non-linear PDE Example: Plane-stagnation point flow u=Kx v=−Ky K is a constant Streamline equation: Everywhere tangent to velocity field ds×V=0 dx u wdy−vdz 0 dy × v = udz−wdx = 0 vdx−udy=0 ||| | In 2D, dz w vdx−udy 0 dy v −Ky −y −dy dx dx= u Kx = x separate variables: y = x C integrate: −ln ( )ln( )+C 1ln y( )ln x( ) →y1e −l( )=−C 2 x dx u=Kx= dt Pathline equation: Trajectory of particle {v=−Ky= dy dt dx Kt =Kdt→ln x)=Kt+C →1x (t=C 2 x Kt dy −Kt x ( )C 2 y =−Kdt→ln (y=−Kt+C →y3 ()=C 4 { −Kt y(t=C 4 Kt x C2C 4 C e = C →y= x = x 2 Streakline equation: Trajectory of particles that pass through a point u=Kx {v=−Ky t=t (x ,y ) Draw at time f the streakline whose origin is at 0 0 at time x0 KT C2= KT (0<T<t ) x0=C 2 → e x ( ) x0 ekt f : {y =C e −KT y0 e KT 0 3 {C3= KT e y0 −Kt y( )= −KTe e Equations provide location (x,y) at time t of particle that was at (x0,y0) at time T. x0 ktf y0 −Kft Location of particle at time t f x( f= KTe y (f) −KT e e e Location at tfof all particles released from (x0,y0) is obtained by considering all values of T between 0 and t f x(T)=x e Kfe−KT y(T)=y e −KfeKT 0 0 Kf K f e =T x 0 →y=y e −Ktfx0e = x0y0 x 0 x x u=Kx Acceleration field: {v=−Ky a= DV = δV + V ∙∇ V= δ (Kxi−Ky j^)+ Kx δ −Ky δ Kxi−Ky j ^=0+Kx δ (Kxi−Ky j^)−Ky δ (Kxi−Ky j^)→a=K (xi−y j) dt δt δt ( δx δy ) δx δy 4 – Differential Form of the Conservation of Mass Principle Consider arbitrary differentially small CV (∆V) : δ −δ δρ δρ 1 δt ∫ ρdV+ ∮V ∙dA=0→ ∮ ρV∙dA= δt ∫ ρdV=− δt|atsome pt(x,y,z ∈CV ∆V=− δt |x,y,z ,t) ∆V ∮ ρV ∙dA CV CS CS CV CS −δρ 1 shrink as ∆V→0 : δt = ∆ V →0V ∮ ρV ∙dA Divergence CS lim 1 f ∙dA=∇∙ f →− δρ =∇∙ (ρV )→ δρ +∇∙ ρV )=0 Theorem: ∆V → 0 CS δt δt continuity equation (most general form) In Cartesian coordinates: δρ + δ (ρu + δ ρv )+ δ (ρw =0 δt δx δy δz δρ 1 δ 1 δ δ In cylindrical coordinates: δt+ r δr(ρrv r) r δθ (v θ) δz(ρv z0

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