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.

3 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: 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.

4 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. 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.

5 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).

6 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 .

7 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: 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.

8 With yxas stress in direction x in a plane for constant y This defines the dynamic viscosity (unit: = ) ! 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).

9 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 . 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.

10 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. ; ;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.