FLUID MECHANICS. 1. Division of Fluid Mechanics. Hydrostatics Aerostatics Hydrodynamics Gasdynamics. v velocity p pressure ρ density

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1 FLUID MECHANICS. Diision of Fluid Mechanics elocit p pressure densit Hdrostatics Aerostatics Hdrodnamics asdnamics. Properties of fluids Comparison of solid substances and fluids solid fluid τ F A [Pa] shear stress Solid Fluids (Newtonian) γ (deformation) is proportional to τ shear stress dγ/dt (rate of deformation, strain rate) is proportional to τ shear stress non-newtonian fluids Fluids: no slip condition no change in internal structure at an deformation continuous deformation when shear stress eists no shear stress in fluids at rest Viscosit Velocit distribution: line or surface connecting the tips of elocit ectors the foot-end of which lies on a straight line or on a plane.

2 Turn of the bar M: dγ dγ d d. dγ τ µ µ. Newton's law of iscosit dt d d dt d kgm m kg µ Dnamic iscosit d s m m / s ms [ ] [ τ]. µ ν [ m / s] Kinematic iscosit Compression of water apor heat echanger Tconst If T >> Tkrit : gas O and N T krit 54 [K] and 6 [K] p p RT ideal-gas law where p[pa], [kg/m 3 ], T [K], R R u / M, R u J/kmol/K uniersal gas constant, M kg/kmol molar mass, for air: M9 [kg/kmol], therefore R87J/kg/K. Caitation saturated steam pressure (apor pressure) - temperature. Water 5 0 C, p 700Pa, 00 0 C, p.03*0 5 Pa standard atmospheric pressure

3 3 p Caitation erosion Interactions between molecules (attraction and repulse) repulsion attraction Comparison f liquids and gases liquids gases distance between molecules small d 0 large 0d 0 role of interactions of molecules effect of change of pressure on the olume cause of iscosit relation between iscosit and temperature pressure significant free surface small 000 bar causes 5% decrease in V attraction among molecules T increases µ decreases independent small fill the aailable space large in case of Tconst V proportional to /p momentum echange among molecules T increases µ increases independent Comparison of real and perfect fluids real fluids perfect fluids iscosit iscous iniscid densit compressible incompressible structure molecular continuous

4 4 3. Description of flow field Scalar fields m 3 Densit [ kg / m ] lim V ε 3 V continuum (r,t) (,,,t) V incremental olume ε >> λ (mean free path) Pressure p F / A [N/m], [Pa]. pp(r,t), pp(,,,t) Temperature TT(r,t) Vector fields Velocit ( r,t) Eulerian description of motion Fields (of force) [ g ] N / kg m / s grait field: g. g g k g g 9.8 N/kg field of inertia: accelerating coordinate sstem ( a ai ) g t ai. centrifugal field: rotating coordinate sstem Characteriation of fields Characteriation of scalar fields: g c rω p p p p gradp i j k gradient ector r 4 characteristics of the ector: it is parallel with the most rapid change of p it points towards increasing p its length is proportional to the rate of the change of p it is perpendicular to p constant surfaces Change of a ariable: e.g. increment of pressure p p p p pb pa gradps

5 5 Characteriation of ector fields: ( ) r,t k j i. t),, (, t),,, (, t),,, (,. ector field 3 scalar fields r grad. Diergence: di, [ ] /s m da cos da dq 3 α didv da V A auss-ostrogradskij theorem Rotation, orticit: k j i rot Ω rot. rotda ds A Γ Stokes theorem

6 6 Potential flow gradϕ condition: Γ ds 0, or rot 0 Eample: fields of force for grait force gds 0 work of the field U [m /s ] potential of the field grad U grait field: g g g k Ug gg konst. field of inertia: accelerating coordinate sstem (a ai) g t ai U t a konst. centrifugal field: rotating coordinate sstem g r ω c rω Uc kost. 4. Kinematics Definitions Pathline: loci of points traersed b a particle (photo: time eposure) Streakline: a line whose points are occupied b all particles passing through a specified point of the flow field (snapshot). Plume arising from a chimne, oil mist jet past ehicle model Streamline: ds 0 elocit ector of particles occuping a point of the streamline is tangent to the streamline. Stream surface, stream tube: no flow across the surface. Time dependence of flow: Unstead flow: (r, t) Stead flow: (r) In some cases the time dependence can be eliminated through transformation of coordinate sstem. In stead flows pathlines, streaklines and streamlines coincide, at unstead flows in general not.

7 7 Flow isualiation: quantitatie and/or qualitatie information a) Transparent fluids, light-reflecting particles (tracers) moing with the fluid: particles of the same densit, or small particles (high aerodnamic drag). Oil mist, smoke, hdrogen bubbles in air and in water, paints, plastic spheres in water, etc. PIV (Particle Image Velocmetr), LDA Laser Doppler Anemometr), b) Wool tuft in air flow shows the direction of the flow.,,0,,6,4 0,9 0,8 0

8 8 Irrotational (potential) orte Concept pf two-dimensional (D), plane flows: 0 and 0. Because of continuit consideration at orte flow (r) (r)? Calculation of rot using Stokes theorem: Γ ds rotda 3 d s d s d s d s d s Since ds at nd and 4 th integrals, and at st and 3 rd integral and ds include an angle of 0 0 and 80 0 : d s ( ) ( ) ( ) r dr dϑ r dr rdϑ r Since d ( r dr) ( r) dr dr after substitution d d d s rdϑ dr drdϑ() r drdϑ dr dr dr In plane flow onl (rot) differs from 0. rot d A ( rot) da d rot. dr r ( ) rdϑdr 0 Eample: ω r ( rot ) ω In case of rot 0 d dr K ln ln r ln Konst.. Velocit distribution in an irrotational (potential) r r orte. A

9 9 Motion of a small fluid particle The motion of a FLID particle can be put together from parallel shift, deformation and rotation. In case of potential flow no rotation occurs. 5. Continuit equation [ kg / s] dq m da da cosα integral form of continuit equation: da dv 0 t t A differential form: di( ) 0, if the flow is stead: (r) di ( ) 0 if the fluid is incompressible const. di 0 Application of continuit equation for a stream tube Stead flow, no flow across the surface. V, Integral form of continuit equation for stead flow: da 0. "A" consists of the mantle Ap ( da) and A and A in- and outflow cross sections. da da 0 A A A. Since da da cosα, da cosα da cosα 0 Assumptions: oer A and A ( A) and A A

10 oer A const., oer A const'. 0 A Const., where mean elocit at D changing cross section of a pipeline: A A D 6. Hdrostatics Static fluid: forces acting on the mass (e.g. grait) and forces acting oer the surface (forces caused b pressure and shear stresses) balance each other (no acceleration of fluid). p d d d g d d p() d d p() d 0 p g grad p g fundamental equation of hdrostatics. Assumption: g grad U (potential field of force) grad p grad U pconst. surfaces coincide with U Const. (equipotential surfaces) The surface of a liquid coincides with one of the U Const. equipotential surfaces the surface is perpendicular to the field of force. Assumptions g grad U (potential field of force), const. (incompressible fluid) p p p grad p grad grad U grad U 0 U const. p p U U incomplete Bernoulli equation Pressure distribution in a static and accelerating tank p p p g g gk, where g 9.8N kg. i j k gk dp / d g, áll. p g Const. If 0, then p p0. Const. p 0 p p0 g. In H point p p 0 gh p p U U point on the surface 0), point at the bottom ( H). At coordinate pointing downwards U g, p, 0, p?, H. p 0 p p 0 gh

11 If the tank accelerates upwards, the fluid is static onl in an upwards accelerating coordinate sstem. Here additional (inertial) field of force should be considered: g i a k U i a U U g U i ( g a). After substitution: p ( g a)h p 0 7. Calculation of mean elocit in a pipe of circular cross section? mean elocit In cross section of diameter D the elocit distribution is described b a paraboloid. The difference n r r / R. [ ] of ma and (r) depends on the nth power of r ( ) ( ) 4q D π Mean elocit: [ m /s] 3 where q [ m / s] ma is the flow rate. The flow rate through an annulus of radius r thickness dr, cross section rπdr is dq rπ R n [ ] dr (r)dr q rπ ( r / R) Integration ields: n ma. n. ma 0 q n R π ma, so the mean elocit is: n In case of paraboloid of nd degree (n ) the mean elocit is half of the maimum elocit. 8. Local and conectie change of ariables di t t ( ) 0 grad di( ) 0 In point P the elocit is, the ariation of densit in space is characteried b grad. Unstead flow: / t 0. Variation of densit d in time dt?

12 Two reasons for ariation of : a) Because of time dependence of densit ( / t 0 ), the ariation of densit in point P: d l dt t b) In dt time the fluid particle coers a distance d s dt and gets in P' point, where the densit differs grad ds grad dt from that of in point P. d c d l local ariation of densit (onl in unstead flows) d c conectie ariation of densit is caused b the flow and the spatial ariation of the densit The substantial ariation of the densit is time dt: d dl dc dt grad dt, t d d The ariation in time unit: grad di 0 dt t dt 9. Acceleration of fluid particles The ariation of in unit time. d grad. dt t Acceleration of fluid particle in direction. The first term: local acceleration, the second term: conectie acceleration. d dt t d dt t d dt t r Determining the differential of (r,t): d dt dt. Referring d to unit time, i.e. diiding t r t d r r it b dt:, where dt t r t t Local acceleration is different from 0 if the flow is unstead. The conectie acceleration eists, if the magnitude and/or direction of flow alter in the direction of the motion of the fluid. The formula for acceleration can be transformed: d grad rot. dt t