Advanced Fluid Mechanics Chapter 1 Introduction 1.1 Classification of a Fluid (A fluid can only substain tangential force when it moves) 1.) By viscous effect: inviscid & Viscous Fluid. 2.) By compressible: incompressible & Compressible Fluid. 3.) By Mack No: Subsonic, transonic, Supersonic, and hypersonic flow. 4.) By eddy effect: Laminar, Transition and Turbulent Flow. The objective of this course is to examine the effect of tangential (shearing) stresses on a fluid. Remark: For a ideal (or inviscid) flow, there is only normal force but tangential force between two contacting layers. 1.2 Simple Notation of Viscosity U F (tangential force required to move upper plate at velocity of U ) h y Fluid (e.g. water) x u(y) = y/h U From observation, the tangential force per unit area required is proportional to U/h, or du/dy. Therefore U τ ≡ shear stress = tangential force per unit area (F/A) ∝ h or U ∂ u τ =µ = µ 〝Newton’s Law of function〞 (1.1) h ∂ y µ: Constant of proportionality The first coefficient of viscosity Remark: E.g. (1.1) provides the definition of the viscosity and is a method for measuring the viscosity of the fluid. Chapter1- 1 Advanced Fluid Mechanics In generally, if ε represent the strain rate, then XY τ = f(ε ) (1.2) xy xy plastic τ yielding fluids Dilatent fluid Non- Newtonian Pesudoplastic fluid fluid Newtonian fluid Yield stress ε Newtonian fluid: linear relation between τ and ε Pesudoplastic fluid: the slope of the curve decrease as ε increase (shear-thinning) of the shear-thinning effect is very strong. The fluid is called plastic fluid. Dilatent fluid: the slope of the curve increases as ε increases (shear-thicking). Yielding fluid: A material, part solid and part fluid can substain certain stresses before it starts to deform. Note Newton 1 Pa (Pascal) ≡ (Pascal, a French philosopher and Mathematist) m2 (a unit of pressure ) m kg⋅ s2 kg g [µ] = [pa · sec] (= ⋅s = = 10 ) m2 m⋅s cm⋅s The metric unit of viscosity is called the poise (p) in honor of J.L.M. Poiseuille (1840), who conducted pioneering experiment on viscous flow in tubes. 1g 1 P ≡ ( )( ) = 0.1 pa⋅sec cm s Chapter1- 2 Advanced Fluid Mechanics The unit of viscosity:      F  [ ]  τ   L 2   F  µ = = = T ← (Old -English Unit: F-L-T) αu   L  L2   ∂y    T   L  or ML  [µ] =  T2 ⋅T =  M  ← (international system SI unit: M-L-T)    L2  LT   N Denote: ≡ Pa, then M2 µ =1.01×103Pa⋅sec water,20°c (liquid): T → µ µ = 283Pa ⋅sec water,100°c µ =17.9Pa⋅sec air,20°c (gas): T → µ µ = 22.9Pa ⋅sec air,100°c For dilute gas: n µ  T  ≈     (Power- law) µ T   ° ° 3 µ  T  2 T + S ≈   °   (Sutherland’s law) µ T T + S   ° ° Whereµ, T and S depends on the nature of the gases. 0 0 µ Kinematics Viscosity υ≡ ρ Chapter1- 3 Advanced Fluid Mechanics Exp: (Effect of Viscosity on fluid) u U ∞ Flow past a cylinder θ w R Foe a ideal flow: R2  u(r,θ)=U cosθ −1 ∞  r2  1 R2  v(r,θ)=U sinθ +1 ∞  r2  At r = R, u=0, v = 2U∞ sinθ -3 The Bernoulli e.g. along the surface is: 1 1 ρU2 + P = ρv2 + p (Incompressible flow) 2 ∞ ∞ 2 P− P v2 1 C = ∞ =1− =1− sin2θ p 1 U2 4 ρU2 ∞ 2 ∞ D’Albert paradox: No Drag. For a real flow: (viscous effect in) ρVD Re = µ (前後幾乎對稱) Re=0.16 (fig 6) (前後不對稱) Re=1.54 (fig 24) Chapter1- 4 Advanced Fluid Mechanics Separation occur (pair of recirculating eddies) Re=9.6 (fig 40) (6 < Re <40) d ∼ Re R=26 (fig. 26) d L ∼ Re L supercritical θ > 90° Subcritical θ < 90° sep Chapter1- 5 Advanced Fluid Mechanics The pressure distribution then becomes: θ C p 1 θ Supercritical (separation) -1 Subcritical (separation) -2 Theoretical (invuscid) -3 (White. P.9. Fig. 1-5) Chapter1- 6 Advanced Fluid Mechanics Remark: Newtonian Fluid Non-Newton Fluid For a Newtonian fluid: ↔ ↔ ↔ τ = − µ ε τ: stress tension ↔ ε: rate of strain tension µ = a constant for a given temp, pressure and composition Lf µ is not a constant for a given temp, pressure and composition, then the fluid is called Non-Newtonian fluid. The Non-Newtonian fluid can be classified into several kinds depending on how we model the viscosity. For example: （I）Generalized Newtonian fluid ↔ ↔ ↔ τ = −ηε η= a function of the scalar invariants of ε (i) The Carreau-Yusuda Model η−η (n−1) ↔ ∞ =[1+(λε)a] a ε: magnitude of the ε η −η 0 ∞ (ii) power-Law model η = mεn−1 n<1: pseudo plastic (shear thinning) n=1: Newtonian fluid n>1: dilatant (shear thickening) （II）Linear Viscoelastic Fluid → polymeric fluids （III）Non-linear Viscoelastic Fluid → The fluid has both 〝viscous〞 and 〝elastic〞 properties. By 〝elasticity〞one usually means the ability of a material to return to some unique, original shape on the other hand, by a 〝fluid〞, one means a material that will take the shape of any container in which it is left, and thus does not possess a unique, original shape. Therefore the viscoelastic fluid is often returned as 〝memory fluid〞. Chapter1- 7 Advanced Fluid Mechanics FIGURE 2.2- 1 Tube flow and “shear thinning.” In each part, the Newtonian behavior is shown on the left ○N ; the behavior of a polymer on the right ○P . (a) A tiny sphere falls at the same through each; (b) the polymer out faster than Newtonian fluid. Chapter1- 8 Advanced Fluid Mechanics ○ FIGURE2.3-1. fixed cylinder with rotating rod N . The Newtonian liquid, glycerin, shows a vortex; ○P the polymer solution, polyacrylamide in glycerin, climbs the rod. The rod is rotated much faster in the glycerin than in the polyacrylamide solution. At comparable low rates of rotation of the shaft, the polymer will climb whereas the free surface of the Newtonian liquid will remain flat. [Photographs courtesy of Dr. F. Nazem, Rheology Research Center, University of Wisconsin- Madison.] Chapter1- 9 Advanced Fluid Mechanics P =P P*= (P+ τ ) - (P+ τ ) > 0 1 2 yy 1 yy 2 FIGURE 2.3-4 A fluid is flowing from left to right between two parallel plates across a deep transverse slot. “Pressure”are measured by flush-mounted transducer “1.”and recessed transducer “2.”○N For the Newtonian fluid P =P . ○P For 1 2 polymer fluids (P+ τ ) > (P+ τ ) . The arrows tangent to the streamline indicate yy 1 yy 2 how the extra tension along a streamline tends to “lift ”the fluid out of the holes, so that the recessed transducer gives a reading that is lower than that of the flush mounted transducer. Chapter1- 10