Hydrodynamics. l. mahadevan harvard university
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1 Thermodynamics / statistical mechanics : Hydrodynamics N>>1; large number of interacting particles! Hydrodynamics : L >> l; T >> t; long wavelength, slow time - average over (some) microscopic length and time scales... continuum field theories! microscopic length : l? (particle size, mean-free path, pore size,... ) microscopic time : t? (particle relaxation times, hopping times,... ) Continuum theory? 1. Microscopic picture + systematic averaging... "rigorous"! 2. Equations of state, constitutive equations... "empirical"! 3. Symmetry, invariance... "intuitive"! l. mahadevan harvard university Balance laws: mass, momentum (linear, angular), energy, entropy...
2 Variables - independent? (r,t)=(x, y, z, t) space-time - dependent? (r,t), u(r,t), (r,t),... density (scalar), velocity (vector), stress (tensor),... displacement... elasticity; orientation... liquid crystals; polarity... magnetic field etc. Frames: 1. Lagrangian - material frame : follow parcels of material... finite bodies. 2. Eulerian - spatial (inertial) frame : lab-based... infinite bodies. order parameter Laws + boundary conditions: Qualitative behavior of solutions: Lecture 1. Balance laws, constitutive equations and boundary conditions. Symmetry, invariance and how to derive equations for simple and complex fluids. Compressibility, Viscosity, Inertia, Capillarity. Dimensional analysis and scaling laws. Analogies to other field theories (electrostatics, elasticity, transport). Lecture 2. Stability and instability. Rayleigh instabilities. Pattern formation. Turbulence. Kolmogorov scaling. Low dimensional flows. Landau-Levich problem. Washburn problem. Lecture 3. Porous media - convection, soft hydraulics. Complex fluids - suspensions, polymers. Locomotion in fluids.
3 Laws: differential form (integral form - useful in the presence of discontinuities - e.g. shocks) Conservation Eulerian Incompressible fluid or + r ( u) =0 D (r,t) D Dt Dt Material derivative =0 r u =0 =(@ t + u r) = r u where dr dt = u velocity Conservation of linear momentum Du Dt = f + r accln force Cauchy stress theorem Newton II for deformable continua ( per unit volume) stress ~ force/area (interactions!) ratio of two vectors?? T (n) da = i T (e i) da i da i =(n e i )da Stress tensor completely defines force/area locally! T (n) = T n all images from wikipedia
4 Stress tensor (Newtonian fluid) - isotropic, homogeneous = µ(ru + ru T 2 )+[ p +( 3 µ)(r u)]i 2 constant fluid! (anisotropic) shear (isotropic) "pressure" = thermodynamic + dynamic pressure dilatation + u ru) = rp + + equation of state µr2 u +( + µ/3)r(r u)+f p = p(, T ) acceleration shear viscosity body force Conservation of angular momentum? = T Conservation of entropy? heat conduction + u rs) =appler2 T µ[ru + 2 rut 3 rui]2 + (r u) 2 + f u entropy production viscous dissipation external work Initial conditions: u(r, 0) T (r, 0) unknowns u,p,t Boundary conditions = U b or = p b = s b or T = q b open boundaries
5 Solid (rigid) boundaries =0 no slip Free boundaries A very slippery condition! Fails at - low density - porous interface - hydrophobic surface - contact line (triple phase boundary) generalizing Laplace's law p = apple n = applen traction is continuous apple = 1 R R 2 2 X mean curvature n surface tension Physical parameters: dynamic viscosity? surface tension? µ fluid (intrinsic) cone-plate rheometer falling ball viscometer L U extrinsic pendant drop sessile drop air Pa.s water Pa.s honey 1.0 Pa.s water alcohol 0.07 Pa.m 0.02 Pa.m
6 Incompressible viscous + u ru) = r u =0 rp + µr2 u + + u rt )=Dr2 T + µ 2 C p [ru + ru T ] 2 + f u C p Navier-Stokes eqns. Incompressible, inviscid fluid (and f = + u r u =0 rp + u rs = Ds Dt =0 Euler eqns. Steady incompressible, inviscid = 0; u 2 /2+p = const Bernoulli eqn. (energy cons.) Incompressible, inertia-less flow 2d flow? axisymmetric flow? rp = µr 2 u + f r u =0 Stokes eqns. Linear!
7 Laws: symmetry + invariance? a hydrodynamic theory of self-propelled objects? birds, bees, + r ( + 1u ru + 2 (r u)u + 3 r( u 2 )= u u 2 u rp + D 1 r 2 u + D 2 r(r u)+d 3 (u r) 2 u + f not Galilean invariant 1. Truncation? 2nd order in space + cubic in velocity Coefficients? empirical / experimental... - Unusual phases/ instabilities - Long range fluctuation effects Compare with balance laws : symmetry/invariance used in constitutive equation. What next? - Nonlinear PDE... all terms are not equally relevant in all situations. - Approximations: analysis (perturbation/ asymptotics), computation, SCALING
8 Dimensional analysis and the Pi theorem In any equation, all terms must have the same dimensions! n k number of physical variables (dimensional) number of independent physical units p = n k dimensionless numbers E. Buckingham (1914) examples: + u ru) = L fluid parameters U extrinsic rp + µr2 u n = 4; k = 3; p =1 force/volume dimensionless variables ˆt = tu/l; û = u/u; ˆp = pl/µu + û ˆrû = µ UL ( ˆrˆp + ˆr2û ) = µ/ kinematic viscosity (momentum diffusivity) Re = UL = U 2 µu/l Reynolds # = inertial pressure viscous stress Re =0?; Re = 1? Stokes Euler
9 Fun with scaling! 1. Frequency of a bobbing buoy? 2. Oscillations of a drop?! = f(, w,r,g)! = f(,, R) 3. Diffusion of a polymer? 4. Tsunami warning time? T = f(l, g, H, ) D = f(k B T,µ,R) 5. Yield of an atomic explosion? 6. Drag on a body? T s F = f(r,, U, µ) E = f(r, t, )
10 Small does not mean negligible! F = U 2 g( UL ) C d = F U 2 = g(re) Re!1? Example µẋ + kx = f; x(0) = 0 exact solution... exponentials! i.e. ẋ + x = f/k; x(0) = 0 natural time scale = µ/k 2 regimes t ; x(t) =ft/k = ft/µ t ; x(t) =f/k initial layer... memory of init. condn. - Singular ODE - Divide and conquer! Home work? Langevin equation ẍ +ẋ = (t); x(0) = 0, ẋ(0) = 0 < (t) >= 0, < (t) (t 0 ) >= G (t t 0 ) In space - boundary layers... Prandtl (1905), but also Laplace, Stokes, Rayleigh, Lamb, Stokes' problems: : fluid driven by transient wall motion boundary data: I u(x, y =0,t)=UH(t); y II u(x, y =0,t)=Usin!t x u =(u(x, y, t),v(x, y, t)) - Scaling approach - Analytical approach
11 I Suddenly moving boundary... no intrinsic/extrinsic length scale! zone of influence: y ( t) 1/2 momentum diffusion II Oscillating boundary.. extrinsic frequency scale! zone of influence: l S ( /!) 1/2 Stokes length r u =0 X t u yy u I II similarity solution (heat equation) separation of variables Home work! Prandtl boundary layer Steady flow past a semi-infinite flat plate - effect of wall is limited to a (small) neighborhood of the wall... at low viscosity (high Re!) - zone of influence cannot depend on the length of the plate (infinite!)... must be self-similar
12 r u =0! u/x v/ u ru = rp + µr 2 u boundary layer assumption x! v u u 2 /x µu/ 2 ( p/x) ( x U )1/2 x Re 1/2 x where Re x = Ux so that Re x 1! x Wall shear stress µ@ y u µu/ ( µu 3 x )1/2 decreases with distance along plate! Total drag force / width Z l 0 dx ( µu 3 l) 1/2 skin drag Comparison F p = C d U 2 l pressure drag / width F s = CµU Stokes drag / length Careful analysis: Stream function approach (see Wikipedia or Batchelor, Landau/Lifshitz) Q. Are these steady state solutions stable? Q. Instability and transition (to turbulence)? NEXT TIME: Hydrodynamic Instability, Turbulence (briefly) + Free surface flows
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