Lecture 5. Differential Analysis of Fluid Flow Navier-Stockes equation

Transcription

1 Lectre 5 Differential Analsis of Flid Flo Naier-Stockes eqation

2 Differential analsis of Flid Flo The aim: to rodce differential eqation describing the motion of flid in detail

3 Flid Element Kinematics An flid element motion can be reresented as consisting of translation, linear deformation, rotation and anglar deformation

4 Velocit and acceleration field Material deriatie Dt D t t r V V V V V, a V t t Dt D Acceleration Velocit field ˆ ˆ ˆ V i j k

5 Linear motion and deformation Let s consider stretching of a flid element nder elocit gradient in one direction Volmetric dilatation rate: δδtδδ 1 d δ V lim δv dt δt δδδδt 1 d δ V V δ V dt

6 Anglar motion and deformation Flid elements located in a moing flid moe ith the flid and generall ndergo a change in shae anglar deformation. A small rectanglar flid element is located in the sace beteen concentric clinders. The inner all is fied. As the oter all moes, the flid element ndergoes an anglar deformation. The rate at hich the corner angles change rate of anglar deformation is related to the shear stress casing the deformation

7 Anglar motion and deformation ω ω OA OB δα / δδt 1 t t lim δt δ δ δ Rotation is defined as the aerage of those elocities: ω 1

8 Anglar motion and deformation ω ω ω iˆ ˆj kˆ 1 1 ω V 1 ˆ 1 ˆ 1 ˆ i j k Vorticit is defined as tice the rotation ector ζ ω V If rotation and orticit is ero flo is called irrotational

9 Anglar motion and deformation Rate of shearing strain or rate of anglar deformation can be defined as sm of flid element rotations: γ

10 Conseration of mass DM Ss As e fond before: dv da Dt t CV Vn CV t dv δδδ Flo of mass in -direction: t CV δ δ δ V t t

11 Conseration of mass Incomressible flo V Flo in clindrical coordinates 1 rr 1 θ t r r r θ Incomressible flo in clindrical coordinates 1 rr 1 θ r r r θ

12 Stream fnction D incomressible flo We can define a scalar fnction sch that ψ ψ Stream fnction Lines along hich stream is const are stream lines: d d Indeed: ψ ψ dψ d d d d

13 Stream fnction Flo beteen streamlines dq d d ψ ψ dq d d dψ q ψ ψ 1

14 Descrition of forces Bod forces distribted throgh the element, e.g. Grait δ F δ mg Normal stress Forces Srface forces reslt of interaction ith the srronding elements: e.g. Stress Shearing stresses Linear forces: Srface tension

15 Stress acting on a flidic element normal stress σ n lim δ A δ F n δ A shearing stresses δ F τ lim τ δ A lim δ F δ A 1 1 δa δa

16 Stresses: doble sbscrit notation normal stress: σ τ shearing stress: τ normal to the lane direction of stress sign conention: ositie stress is directed in ositie ais directions if srface normal is ointing in the ositie direction

17 Vectors and Tensors To define stress at a oint e need to define stress ector for all 3 erendiclar lanes assing throgh the oint σ τ τ τ τ σ τ τ τ σ

18 Force on a flid element To find force in each direction e need to sm all forces normal and shearing acting in the same direction σ τ τ δ F δδδ

19 t g τ τ σ F ma δ δ Acceleration material deriatie t g τ τ σ t g τ τ σ Differential eqation of motion, F ma F ma m F ma δ δ δ δ δ δ δ δ δ δ

20 Viscos Flo σ σ σ d µ d d µ d d µ d τ τ τ τ τ τ d µ d d µ d d µ d d d d d d d for iscos flo normal stresses are not necessar the same in all directions

21 Naier-Stokes Eqations g t g t g t µ µ µ 4 eqations for 4 nknons,,, Analtical soltion are knon for onl fe cases

22 Stead Laminar Flo beteen arallel lates g ; µ g t g t g t µ µ µ

23 g ; µ 1 1 ; 1 1 f g C C C d d µ µ Bondar condition no sli h h 1 h µ Velocit rofile Flo rate h d h d q h h h h µ µ What is maimm elocit ma and aerage elocit?

24 No-sli bondar condition Bondar conditions are needed to sole the differential eqations goerning flid motion. One condition is that an iscos flid sticks to an solid srface that it toches. Clearl a er iscos flid sticks to a solid srface as illstrated b lling a knife ot of a jar of hone. The hone can be remoed from the jar becase it sticks to the knife. This no-sli bondar condition is eqall alid for small iscosit flids. Water floing ast the same knife also sticks to it. This is shon b the fact that the de on the knife srface remains there as the ater flos ast the knife.

25 Coette flo 1 1 ; 1 1 f g C C C d d µ µ Bondar condition no sli U b ; Please find elocit rofile and flo rate

26 Bondar condition no sli U b ; 1 b b U µ Velocit rofile 1 b b U b b U µ Dimensionless arameter P

27 Hagen-Poiseille flo Poiseille la: Q π R 8 R 8 4 µ l µ l ;

28 The elocit distribtion is arabolic for stead, laminar flo in circlar tbes. A filament of de is laced across a circlar tbe containing a er iscos liqid hich is initiall at rest. With the oening of a ale at the bottom of the tbe the liqid starts to flo, and the arabolic elocit distribtion is reealed. Althogh the flo is actall nstead, it is qasi-stead since it is onl slol changing. Ths, at an instant in time the elocit distribtion corresonds to the characteristic stead-flo arabolic distribtion. Laminar flo

29 Problems 6. A certain flo field is gien b eqation: V 3 1 i 6j Determine eression for local and conectie comonents of the acceleration in and directions 6.8 An incomressible iscos flid is laced beteen to large arallel lates. The bottom late is fied and the to moes ith the elocit U. Determine: olmetric dilation rate; rotation ector; orticit; rate of anglar deformation. U b

30 Problems 6. The stream fnction for an incomressible flo sketch the streamline assing throgh the origin; determine of flo across the strait ath AB 6.74 Oil SAE3 at 15.6C steadil flos beteen fied horiontal arallel lates. The ressre dro er nit length is kpa/m and the distance beteen the lates is 4mm, the flo is laminar. Determine the olme rate of flo er nit idth; magnitde and direction of the shearing stress on the bottom late; elocit along the centerline of the channel