# Introduction to Fluid Mechanics

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1 Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009

2 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the solid, cannot sustain a finite deformation under the action of a shear force. Hooke s law σ = Eɛ holds for solids up to the proportion limit of strain. σ: strain and ɛ: normal stress. For most fluids, Newton noted that τ ɛ where τ is a shear force and ɛ is the time rate of change of a fluid element s deformation. Consider a three-dimensional element. Hooke s law of shear: p yx = Gɛ yx Newton s law of viscosity: p yx = 2µ ɛ = µ du dy

3 Classification of Fluid Flows Gases versus Liquid Continuum versus Discrete Fluid Perfect versus Real Fluids Newtonian and Non-Newtonian Fluids Compressible and Incompressible Fluids Steady and Unsteady Fluid Flows One, Two, Three-Dimensional Flows Rotational versus Irrotational Flow

4 Properties of Fluids Mass: M Density: ρ Specific Weight: γ weight per unit volume Specific Gravity: S = ρg the ratio of density to the density of pure water Pressure: p the normal stress Bulk Modulus of Elasticity: K = Vdp dv a measure of the T compressibility of liquids Absolute or Dynamic Viscosity: µ The viscosity of a gas increases with an increase of temperature. The viscosity of a liquid decreases with an increase of temperature. Kinematic Viscosity: ν = µ ρ Surface Tension: σ Capillary Rise or Depression: h

5 Aerohydrostatics For static state, the sum of all external forces acting on the fluid control colume is zero, so is the sum of all momnets of these forces. Consider a static fluid on earth. we will have 1 ρ p = g. Examples: Hydrostatics is the science of the static equilibrium of incompressible fliuds. p 2 p 1 = p = γh Aerostatics differs from hydrostatics in the specific wight γ and/or density is no longer considered constant. Hally s law: p = p0 exp gz RT 0 by assuming the eq of state of air ρ = p RT the perfect gas law and isothermal T = T 0. g Logrithmic Law: p = p 0 1 αz Rα T 0 by assuming the eq of state and T = T 0 αz. The typical value of α is 6.5 C/km

6 Lagrange Description The Lagrangian decription describes the history of the particle exaclty. But it is rarely used in fluid mechanics because of its significant mathematical complexities and experimental limitations. The Lagrangian description is often used to describe the dynamic behaviour of solids. x = xx 0, y 0, z 0, t y = yx 0, y 0, z 0, t z = zx 0, y 0, z 0, t u = x t v = y t w = z t a x = u t = 2 x t 2 a y = v t = 2 y t 2 a w = w t = 2 z t 2

7 Euler Description The Eulerian description is used to describe what is happening at a given spatial location Px, y, z in the flow field at a given instant of time. Substantive Derivative D Dt : the Stoke deriv. u = f 1 x, y, z, t D v = f 2 x, y, z, t Dt t + u x + v y + w z w = f 3 x, y, z, t = t + V The acceration in the Eulerian Description: a x = Du Dt a y = Dv Dt a z = Dw Dt = u t + u u x + v u y + w u z = v t + u v x + v v y + w v z = w t + u w x + v w y + w w z

8 Differential equations of fluid behaviour 6 unkown variable: three scalar velocity components, the temperature, the pressure and the density of the fluid. Here, we use Eulerian description. The equation of state 1 The equation of continuity 1 The equation of conservation of fluid momnetum 3 The equation of conservation of fluid energy 1

9 The conservation of mass The general property of balance: if φ is an intensive continuum qunatitiy of the fluid. D φ dx = φ dx + φv da Dt Ω t Ω Ω Dφ t = φ t + φv The equation of the conservation of mass ρ t + ρv = 0 If the fluid is incompressible ρ =constant, the continuity equation is expressed as V = 0

10 Decomposition of the motion of particlesi Express the velocity u, v, w of a particle at Qx, y, z near Px 0, y 0, z 0 in Taylor s series form: u u u u u 0 x y z x x 0 v = v v v v 0 + x y z y y 0 w w 0 z z 0 w x w y V = V 0 + Ar r 0 + Br r 0 + Ohigh order the anti-symetric part A = 1 2 DV DV T the symetric part B = 1 2 DV + DV T w z 0 +Ohigh order

11 Decomposition of the motion of particlesii The velocity can be expressed as V = V 0 r r 0 ω + r r 0 Ṡ where the angular rotation is ω = 1 2 V the strain rate dyadic is Ṡ = u x u y + v x u z + w x 1 u 2 y + v x 1 2 v y v z + w y 1 u 2 z + w x 1 2 v z + w y w z 0

12 The strain rate dyadic Ṡ Ṡ = t ɛ ij = = u x u y + v x u z + w x ɛ xx ɛ xy ɛ xz ɛ yx ɛ yy ɛ yz ɛ zx ɛ zy ɛ zz u y + v x v y v z + w y 1 u 2 z + w x 1 2 v z + w y w z The strain rate dyadic Ṡ involves the dilatation and shearing strain of th fluid particle at P. The dilatation D is defined as D = ɛ xx + ɛ yy + ɛ zz = V

13 The stress dyadic P Consider the most general form of a linear relation between a stress and a rate of strain. P = aṡ + bi where tensor a contains 36 constants a and tensor b contains 3 constant b. It is called the constitutive equation of fluid dynamics.

14 The stress dyadic P Consider an isotropic fluid no preferred direction, Incompressible fluid: P = 2µṠ pi Compressible fluid: P = 2µṠ p V I where the pressure is p = 1 3 p xx + p yy + p zz.

15 Newton s viscosity potulates Consider the isotropic incompressible fluid. Express the stress tensor p ij as { µ ui p ij = x j + u j x i, j i p + 2µ u i x i, j = i Comparing the stresses with the strain rate tensor S, we see p xy = 2µ ɛ xy p xz = 2µ ɛ xz p yz = 2µ ɛ yz These relations are called Newton s viscosity potulates.

16 Surface forces F s and Vorticity ξ Surface forces: Normal part: Tangential part: Vorticity is defined by F σ = F τ = A A P da, P da, i = j i j ξ = V Note that ξ = 2ω.

17 Cauchy s equation of motion Applying Newton s second law: F s + F b = Ma where F s surface forces and F b body forces If there is only gravitational force acted on the body, we have aρ dx = gρ dx + P da Ω Ω a = g + 1 ρ P Ω This is called the Cauchy s equation of motion. u t + u u x + v u y + w u z v t + u v x + v v y + w v z w t + u w x + v w y + w w z = g x + 1 ρ = g y + 1 ρ = g z + 1 ρ pxx x pxy x pxz x + p yx y + p yy y + p yz y + p zx z + p zy z + p zz z

18 The Navier-Stokes Equations Consider a compressible fluid with the consitutive equation P = 2µṠ p DI Pluging into Cauchy s eq of motion, we obtain V t + V V = g 1 ρ p + ν 2 V + ν D Navier-Stokes eq for incompressible flows: V t + V V = g 1 ρ p + ν 2 V

19 Euler s equation and Stokes flow Navier-Stoke equation for inviscid fluid flow ν = 0: V t + V V = g 1 ρ p It is usually called Euler s equation. For the case of very slow fluid motion, the Navier-Sokes equation becomes p = µ 2 V It is popularly called Stokes flow.

20 The Gromeka-Lamb form of the Navier-Stokes eq V p t + ξ V = g ρ + V 2 ν ξ 2 where ξ = V is a vorticity vector. For a steady and irrotational flow, g = p ρ + V 2 2. For inviscid fluid flow, V t + ξ V = p ρ + V Ω. where g = Ω. For a steady, inviscid and incompressible flow, V ξ = p ρ + V Ω Crocco s or lamb s eq, this gives Bernoulli s equation: p ρ + V Ω = const. For an irrotational flow V = 0, we can set V = φ. Bernoullli s eq: p ρ + V Ω + φ t = ct

21 Conservation of energyi Specific energy: e = i + V gz + e nuclear + e elect + e magn + other In the present discussion, we shall neglect all energies except internal,kinetic and potential. Dρe Dt = ρe t + eρv The first law of thermodynamics: dq dt + dw dt Conservation of heat: dq dt = q Fourier s Law: q = k T The net power: dw dt = dw dt Loss of power due to viscous stress: dw dt v mech + dw dt = 2µ V Ṡ + 2µṠ V The power due to the normal stresses: dw dt = Dρe Dt v + dw dt p p = pv

22 Conservation of energyii dw dt mech = ρe [ + e + p ] ρv k T + 2µV Ṡ t ρ 2µṠ V This equation applies to any Newtonian fluid in a field where the only transfer of heat is by conduction. Examples Steady flow: ρe t flow: µ = 0, we obtain dw dt where specific enthalpy h = i + p ρ The solution is w mech = = 0, No heat transfer: T = 0, Inviscid h mech = + V gz ρv h + V gz. For a fluid at rest or moving with negligible velocity and having no mechanical energy transfer: ρi t = k T In paritcular, if the fluid is a perfect gas, then C v t ρt = k 2 T

23 Dimensional Analysis The Buckingham π theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n k dimensionless variables constructed from the original variables: it is a scheme for nondimensionalization. The Rayleigh Method

24 Dimensionless parameters Dimensionless Navier-Stokes equation: V τ + V V = p k Fr V R L L: a constant characteristic length U: a constant characteristic velocity Reynolds number R L = UL Froude number F r = Mach number M = U c µ U gl Weber number W = U2 Lρ σ Euler number E = ρ C p = p 1/2ρU 2 K and Cauch number C = ρu 2 ρu 2, and the Pressure coefficient

25 Reynolds number R L R L = UL µ Examples where R L is very large or infinite: Turbulent flows Inviscid flows Potential flows Flows far removed from boundary Examples where R L is very small: Creeping flows Laminar flows Stokes flows and lubrication theory Bubble flows Flows very close to a boundary

26 SOme other parameters Reynoolds number can be defined the ratio of the momentum flux to the shearing stress. F r > 1: tranquil flow or rapid flow For large Mach number M 0.3, the effect of compressibility must be considered. 0.3 < M < 1: subsonic flow, M > 1: supersonic flow Mach number can be viewed as the ratio of teh intertial force to the compressibility. Cauchy number is the ratio of the compressibility force to the intertial force. M = 1 C. Large Weber number W indicates surface tension is relatively unimportant, compared to the inertial force.

27 Types of flows Viscous Fluid Flows Laminar Pipe Flow Turbulent Pipe Flow Potential Flow Open-Channel Flow Boundary Layer Flows One-dimensional Compressible Flows

28 References Robert A. Granger, Fluid Mechanics, Dover, 1985.

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