6. Fluid mechanics: fluid statics; fluid dynamics

1 1/966. Fluid mechanics : Fluid statics; Fluid dynamics (internal flows, external flows)Ron Zevenhoven bo Akademi UniversityThermal and Flow Engineering/ V rme- och str mningstekniktel. 3223 ; grunder ( PTG ) Introduction to Process bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland2 Fluid statics bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland3/96 Fluid statics, static pressure/1 Two types of forcesact on a Fluid volumeelement: surface (pressure) forcesand body(gravitational) forces: see Figure Pressure(a scalar!) is defined as surface force / area, for examplepb= Fb/ (d w) = p @ z = z1 Picture: KJ05 Fluid volume h d wwith density and mass m = h d w z = z1 In engineering applications, a Fluid (sv: Fluid )is a liquid or a gas The behaviour of stationary fluidsis described by Fluid statics A liquidin a container forms a layer with a distinct surface, and exerts forces on the walls supporting it, while a gaswill fill the whole container.

2 Bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland4/96 Fluid statics, static pressure /2 For the horizontal forces Fn+ Fs= 0 or -py h w + py h w = 0 py= 0 Similarly Fw+ Fe= 0 gives px= 0, There are three vertical forces: -Ft h d - m g + Fb h d = 0 (gravity g) The pressure differencebetweenz = z1and z = z1+ h follows from -Ft- h d w g = - Fb, with -Fb/ (d w) = -pz@ z = z1; andFt/ (d w) = -pz@ z = z1+h; givespz(z1)= pz(z1+h) + h g If z = z1+h is at the Fluid surface exposed to atmosperic pressure p0thenpz(z1)= p0+ h gPicture: KJ05 Fluid volume h d wwith density and mass m = h d w z = z1 Picture: ~sitko/CollegePhysicsIII/9-Solids&Fluids /Solids& bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland5/96U-tube manometer The U-tube manometeris based on the relation between depth and pressure in static fluids, with one end open to the atmosphere at patm For the Figure, with gravity g and densities gand lfor gas and liquid: pC= g h1 g + pBpD= l h2 g + pC= l h2 g + g h1 g + pBand also, from the other sidepD= l (h3+h2) g + pF= l (h3+h2) g + patmwhich gives, with pB= pA l h2 g + g h1 g + pA= l (h3+h2) g + patmpA patm= l h3 g - g h1 g and noting that l g: pA patm= l h3 gPicture.

3 KJ05 Note that the U-tube manometermeasurespressure differences6/96 BarometerPicture: KJ05the density of liquid Hg is kg/m3at 20 Cafter Torricelli:1 torr = 1 mm Hg pressure 1 atm = 760 torr at 0 C A device for measuring atmosphericpressure (which cannot be done using an U-tube manometer) is referred to as barometer A closed tube filled with mercury (Hg) is quickly put upside-down in an opencontainer filled with Hg Gravity causes the Hg level in the tube to fall, but no air can enter the tube. The small gas volume trapped is Hg vapour at equilibrium with liquid Hg. For the tube pvapor,Hg+ Hg hHg g = patm At 20 C, pvapor,Hg= Pa patm, thus patm Hg hHg g 7/96 Example: a manometer Two piston-cylinder assemblies are connected by a tube filled with mercury (Hg) at 20 C (density 13546 kg/m3) The diameter of each piston is m, the mass of each piston is kg.

4 Mass m1= kg Use the data to calculate mass mPicture: KJ05 bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland8/96 Buoyancy /1 Buoyancy(sv: flytkraft, fi: nostovoima)or buoyant forceacts on all objects immersed or submerged (sv: s nkad)in a Fluid It is an overall upwards forceas the result of the fact that pressure p in a static Fluid increases with depthPicture: : ~crorres/Archimedes/ : KJ05surface For an immersed object, horizontalforces cancel each other, and the twovertical forces are gravity and buoyancy. The forces on the surface of the objectare the same as when that surfacewould be filled with the Fluid Thus, the buoyant force on a masswith volume V is equal (but opposite in sign) to the weight of the Fluid in the volume V, and acts on the same centre of gravity(CG): FB= - mfluid g = - Fluid V g9/96 Buoyancy/2 Picture: KJ05 Picture: ~ /3 For any object the buoyancy force it experiences may be less than, equal to or larger than its weight If FB> weight, the object will rise / floatIf FB< weight, the object will sinkIf FB= weight, the will float in suspension For example, for the two fluids geometry FB= ( 1 V1+ 2 V2) gin equilibrium with Fgravity= m0 g = 0 Vtot gfor object mass m0(kg).

5 0 Vtot= 1 V1 + 2 V2and Vtot= V1+V2 For example, for cases with water + air FB= ( a Va+ w Vw) g w Vw g( a >> w) 0 Vtot= w Vw, or : 0 / w= Vw/ VtotPictures: KJ0511/96 Example: buoyancy The tip of a certain iceberg (which is the volume of the iceberg above the water surface) is Vtip= 79 m3, in seawater of with density sea= 1027 kg/m3. Calculate the submerged ( water) volume of the iceberg. For ice the density is ice= 920 : : KJ05 Surface tension A liquid at a material interface, usually liquid-gas, exertsa forceFintper unit length L alongthe surface. It is the result of molecularattraction at a liquid surfacebeing different from that in the liquid the surface acts like a stretched membrane Surface tension( or , unit: N/m) quantifies this force:Fint= L Result phenomena: Contact angle Capillary action (rise or drop) Bubbles, droplets12/96 ambient water-air: = N/m bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland13 Fluid dynamics :viscosity, laminar, turbulent flow,boundary layer bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland14/96 Fluids will (try to) resist a change in shape, as will occur in Fluid flow situations wheredifferent Fluid elements have different velocities Note the definitionof a Fluid : a Fluid is a substance that deformscontinuously under the application of a shear stress (sv.)

6 Skjuvsp nning) Consider Fluid flow between plates: The no-slip conditionsays that at the wall the velocity of the Fluid is the same as the wall velocity *), for a fixed wallvfluid= 0 at the wall Between the plates a velocity profileexists: it can be decribed as vx= vx(y) Shear stresses, Fluid , arise due to velocitydifferences between different Fluid elements Internal friction in Fluid flow /1*) this applies alwaysexcept for very low pressure gases, for example in the upper atmospherexyPicture T06 bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland15/96 Internal friction in Fluid flow /2 For a Fluid between plates with width W (m), distance d (m) the shear force F = (Fx,Fy,Fz) = (Fx,0,0) (unit: N)to pull the Fluid at velocity v = (vx,vy,vz) = (vx,0,0) gives a shear stress yx(unit: N/m2)in the Fluid at y = d that is equal to:with yxas stress in direction x in a plane for constant y This defines the dynamic viscosity (unit: = ) !

7 Note: yxat y = y0is the shear stress of Fluid elements with y < y0on the Fluid elements with y > y0. As a result Fx> 0 if dvx/dy < 0 !Picture: ~sitko/CollegePhysicsIII/9-Solids&Fluids /Solids& , wall fluidy v dydv yxLWFsurfaceFxxdyfluidwall,xwallfluid,x Lvx= 0 @ y = 0 SIGN: bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland16/96 Internal friction in Fluid flow /3 The linear relation between yxand dvx/dy is referred to as Newton sLawwhich holds for so-calledNewtonian fluids For non-Newtonian fluids, other relations between shear force and velocity gradient hold, for example Bingham fluids (toothpaste, clay)or pseudo-plastic (Ostwald) fluids (blood, yoghurt). For those,viscosity is a function of the velocity gradient: yx= (dvx/dy) dvx/dyPicture: BMH99 Note:The flow of a Fluid between plates, or in a tube or on a surface doesn tnecessarily requiremoving walls: usually the drivingforce is gravity, or a static pressure difference bo Akademi University | Thermal and Flow Engineering | 20500 Turku | FinlandNewtonianvs non-Newtonianfluids17/96 Viscosity Viscosity(sv: viskositet) is a measure of a Fluid 's resistance to flow; it describes the internal friction of a moving Fluid .

8 More specifically, it defines the rate of momentum transfer in a Fluid as a result of a velocitygradient. Dynamic viscosity (unit: ) is related to a kinematic viscosity, (unit: m2/s) via Fluid density (kg/m3) as: = / Picture T06 Picture: KJ0518/96 Internal friction in Fluid flow /5 Concentration, c, temperature, T, and energy, E, are scalars, and their gradient is a vectorsuch as dT/dx or T = ( T/ x, T/ y, T/ z), etc. Velocity is a vector v, for example v = (vx, vy, vz) and it s gradient is a (second order) tensorwith elements such as dvx/dy (gradient of vxin y-direction) zvzvzvyvyvyvxvxvxvvzyxzyxzyx)(.zvyvxvv :notezyx Gradients of a scalar propertygive a vector (or 1storder tensor);gradients of a vector property give a 2ndorder tensor, friction in Fluid flow /6 v results in 3 compressive stresses(sv: trycksp nningar) xx, yyand zzand 6 shear stresses(sv: skjuvsp nningar) xy, xz, yz, zx, yxand zy: etc.

9 ; ;dyvddydvdyvddydvzzyzxxyx Picture: SSJ84 yxis in x-direction in plane of constant y 20/96 Viscous work The shear stresses can be expressed as tensor , resulting in a viscous shear force on a certain area A that is equal to Fvisc= A, with A = An with normal vector n If the velocity v at surface A the rate of viscous workdone by the Fluid at surfaceA equals Wvisc= Fvisc v = A v , which for a certain volume element of controlvolume (inside which v and can vary) with total outside surface A gives the rate of work done: Note: at the wall v = 0 so no work is done; also at points where velocity and shear are perpendicular v = 0 and no work is AviscAd)v (W ===Picture friction work is dissipated as HEATV ector/tensor calculationslike this are beyond this course bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland21/96 Example: shear stress concentric cylinders /1 Oil with viscosity = Pa sfills a mm gap between twocylinders of which the inner onerotates whilst the outer one is fixed.

10 The diameter of the inner cylinder is 8 cm, the length is 20 cm. Question: How much power is required to rotate the inner cylinder at 300 rpm?Picture: KJ05 Question : shear stress concentric cylinders /2 Picture: KJ05 Question *) The space between the two cylinders is very small and may be treated as a flat plate For circular tube flow, the laminar turbulent flow transition occurs at Reynolds number Re2100 - 2300, with the dimensionless numberdefined as Re = <v> d/ for = Fluid s density (kg/m3), <v> = Fluid saveragevelocity (m/s), d = tube diameter (m) and = Fluid s dynamic viscosity (Pa s)23/96 Laminar turbulent Fluid flowPictures: T06 Osborne Reynolds s dye-streakexperiment (1883) for measuring laminar turbulent flow transitionlaminar: Re < 2100laminar turbulentturbulent: Re > 4000 bo Akademi University | Thermal and Flow Engineering | 20500 Turku | Finland24/96 Example.