LECTURES SCHEDULE FOR 18.311; Spring semester 2008. Lect. # & date. ========================================================================== General mechanics of class. #1 Tue-Feb-05 ========================================================================== Outline and physical phenomena to be covered: --- Hydraulic Jumps [kitchen; river floods; flash floods; dams; etc]. --- Shock waves [sonic boom, explosions, super-novas & crab nebula]. --- Traffic flow waves. Others (not mentioned in this lecture) --- Solitary waves [say, in lakes]. --- Diffusion. ========================================================================== % For the material below see the conservation law notes on the WEB page. % ========================================================================== Discrete to continuum limits: --- Densities and fluxes. --- Issues relating to the meaning of the "continuum" limit. Examples: Car densities and flux. Interstellar media (where do super-nova shocks travel on?). Pressure in a gas. Other fluid properties. Other examples, not mentioned in the lecture: Grade "curve" (idealization versus reality: histograms). River density and flux. Forestry; molds; bacteria. How conservation principles lead to pde's; illustrate with Traffic Flow. --- Integral and differential form of a conservation law. --- Closure issues & "constitutive equations." #02 Thu-Feb-07 Quasi-equillibrium approximations and time-scale limitations. The case of traffic flow and the traffic flow curve. --- Another example: river flow equations. Summarize general machinery; conservation laws to pde models. Further examples: --- Derive Euler Equations of Gas Dynamics in 1-D. Closure, quasi-equillibrium and thermodynamics. Polytropic gas. --- Heat flow along an insulated wire. #03 Tue-Feb-12 Closure, Fick's law, and intuitive justification using stat. mech. interpretation of heat and temperaure. --- Higher order (TRANSPORT) effects to quasi-equillibrium. Important when gradients are not so small. Connection with randomness at microscopic level. Examples: Traffic flow and burger-s like equations Heat conductivity and viscosity. A first peak at the Navier-Stokes equations. --- Longitudinal vibrations of an elastic rod & the linear wave equation. x = Lagrangian coordinate (particle position at equillibrium). \rho = density (mass per unit length) of rod u = u(x, t) displacement. Thus \rho*u_t = Longitudinal linear momentum density. T = T(x, t) tension ( T(x0, t) is force by x > x0 on x < x0). Thus -T = momentum flux Elasticity: T is a function of the strain = f(u_x - 1) Explain why u_x-1 is strain. Derive equation & consider Hooke's law case when T = \kappa*(u_x-1). Alternative derivation using F = m*a on rod differential elements --- Small transve vibrations of a string under constant tension T, with motion restricted to a plane (again: the linear wave equation). x = coordinate along string u = u(x, t) transversal displacement of string from equillibrium. \rho = density of string (assume constant). Thus \rho*u_t = transversal linear momentum density. T*u_x = Transversal component of tension = transversal momentum flux. Use general conservation law machinery now: --- Obtain wave equation: u_{tt} - c^2 u_{xx} = 0; c^2 = T/rho. --- Note c is a velocity (next lecture will show of what). NOTE: the students should check elementary physics book [say, as used in 8.01/02] where the equation is derived using force {F = m*a] balance on string differential elements. Conservation is a more powerful method, generalizable to many other contexts. Conservation law methods. Generalizations: --- Higher order (TRANSPORT) effects to quasi-equillibrium (see above). --- Adding sources and sinks. Examples: cars flowing in/out of highway through commuter township. water flowing into river from small affluents. --- More than one dimension. #04 Thu-Feb-14 Examples: Euler equations of fluid dynamics, plus body forces. Compressible and incompressible cases. Slow granular flow in a silo. ========================================================================== % The material below is in the books by Wan, Haberman, Whitham, etc. % ========================================================================== Solution of 1-st order (scalar) quasilinear equations by characteristics. Examples: traffic flow and river waves. ............... about 4 lectures. Traffic Flow equations. Traffic density, flow and car velocity. Linearize equation near a constant density and solve u_t+c_0*u_x = 0. Generalize to solution of u_t+c_0*u_x = a*u Solution for general kinematic wave u_t + c(u)*u_x = 0. #05 Thu-Feb-21 Characteristic form. Characteristic speed. Note that, for traffic flow u > c and for river flows u < c. Definition of characteristics and solution by characteristics. Start by first looking at examples of linear problems with variable coefficients, where all the calculations can be done exactly x*u_x + y*u_y = y Write characteristics in parametric form. Draw the characteristics. Then solve u(x, 1) = g(x) for -inf < x < inf Where is solution defined? u_x + x^2*u_y = y Write characteristics in parametric form. Draw the characteristics. Solve u(x, 0) = g(x) for x > 0. Where is solution defined? Back to traffic flow: Examples: --- Red light turns red. Gap in characteristic field. #06 Tue-Feb-26 Argue nice dependence on perturbations to data. Fill gap by taking limit of smeared discontinuity. Obtain expansion fan solution c = x/t. Example with Q quadratic. --- Green light turns red. Split into two initial-boundary value problems. Do case of "light" traffic first. Ahead of light: characteristics cross. Resolve issue by introduction of the last car to make the light. Discontinuity in solution where characteristics are chopped. Speed of this discontinuity is obvious. Behind the light: again, characteristics cross. Argue that in real life drivers wait till the last moment to break. Another discontinuity needed [location of thin layer where cars break]. How does one compute it's velocity? Get law speed = [Q]/[\rho] by arguing "flow of cars into discont. = flow of cars out" Note flow = rho*(u-s). Note limit as [rho] vanishes is characteristic speed. Again: characteristics chopped at discontinuity and crossing avoided. Consider Traffic Flow with smooth initial wave profile rho = f(x). Show that characteristics cross almost always. What does this mean? Examine evolution of wave profile, as given by the characteristic solution. Graphical interpretation: --- Move each point on graph at velocity c(rho). Evolution as sliding of horizontal slabs at different velocities (guarantees #07 Thu-Feb-28 conservation ... connection with Lebesgue integral). --- Show wave steepening and wave breakdown. Multiple values. --- Mathematical model breakdown. Quasi-equillibrium assumption fails and PDE model breaks down. --- Back to "physics". Need to augment model with shocks. IMPORTANT: not always true. Shocks introduce new physics! Explain what is needed to get them: flow against gradient and locality. Shocks/hydraulic jumps: --- Rankine Hugoniot jump conditions for shocks and integral form of the conservation laws. Derive using conservation in shock frame, where flux is (u-s)*rho. Hence ((u-s)*rho)- = ((u-s)*rho)+. --- Shocks as curves along which characteristics end, so crossing does no occur. NOT NEEDED OTHERWISE! --- Allowed and not-allowed shocks. Entropy condition: c- > s > c+. The arrow of time and causality: characteristics die at shock, hence INFORMATION IS LOST. Irreversible evolution, FORWARD IN TIME ONLY. --- Assume piecewise smooth solution with shocks. At a shock problem then splits into three subproblems: - Solution ahead of the shock. Smooth and satisfying characteristic form. Determined at each point by a single characteristic connecting to data. - Solution behind the shock: same deal as solution ahead of shock. - Shock position determined by solving o.d.e. given by jump cond. Graphic depiction of shock conditions in Q-rho Flow-Density plane. --- Need consistency between entropy condition and jump condition. Jump condition must yield shock velocity satisfying c- > s > c+. Can be made to work for Q concave or convex. --- Traffic flow: backward facing shocks, moving slower than the cars. Car enter shock region from behind. (Q concave). --- River flow: forward facing shocks, moving faster than the water flow. Water enters hydraulic jumps from ahead. Note: all this assumes standard drivers and standard driving conditions. Using "special" drivers you can arrange to have discontinuities that do not satisfy the entropy condition: e.g.: Example (traffic flow): consider the situation at the start of a car race, with all the racing cars neatly organized in a pack behind a lead car. Hence a discontinuity in rho is produced where rho goes down and violates entropy. This is "car ballet", not traffic flow. Example (traffic flow): driver that goes slower than road conditions allow, creates long line of cars behind [common in mountain roads] --- again, requires special driver. Example (river flow): push water from behind with a paddle (this is the equivalent of the second example above in traffic flow). #08 Tue-Mar-04 Examples of traffic flow problems. Take special case with Q = 4*rho*(1-rho) and solve/show --- Shock speed average of wave speeds on each side. --- Red light turns green, with finite line of cars stopped behind the light: rho = 0 for 0 < x; rho = 1 for -2 < x < 0; rho = 1/2 for x < -2; Solve using characteristics. Then plug in expansion fans and shocks. --- In prior situation, let light turn red at some later time tau > 1. Continue solution. Identify shocks that correspond to car paths (last car through light) and check shock speed = u for them. --- Turn light back to green at t = 2*tau. Continue solution. Another example, signaling problem: --- Solve u_t + u*u_x = 0 on x > 0 with u(0, t) = 1 for t < 0; u(0, t) = 1/2 for t > 0; #09 Thu-Mar-06 Shock structure due to "diffusion" type effects. --- Diffusion effects in traffic flow: "look ahead" by drivers. --- The role diffusion plays in stopping steepening and wave breaking. --- Shock as thin layer where diffusion and nonlinearity balance. --- Shock structure: Make argument that shock zone in space time is very thin, so shock looks like a plane wave. --- Traveling wave for augmented equation. Recover jump and entropy conditions. ========================================================================== Gas Dynamics: --- Write isentropic equations in 1-D, where p = p(rho). Example: p = kappa*rho^gamma --- For smooth solutions, manipulate equations into the form rho_t + u*rho_x + rho*u_x = 0 u_t + (a^2/rho)*rho_x + u*u_x = 0 where a^2 = dp/drho [calculate case ideal gas]. --- Notice form Y_t + A(Y)*Y_x, similar to the scalar case, but with the wave velocity replaced by a matrix. Look for o.d.e. forms [i.e. characteristics] by doing linear combinations of the equations. Show it works if using an eigenvector L*A = c*L Apply to Gas Dynamics example above. Find the eigenvalues of the matrix (namely c = u +/- a), and the corresponding eigenvectors. Then get characteristic form: +/- (a/rho)*(drho/dt) + du/dt = 0 along dx/dt = u +/- a. Introduce h = h(rho) by property dh/drho = a/rho. Show for ideal gas h = 2*a/(gamma-1). Then d/dt (u +/- h) = 0 along dx/dt = u +/- a. i.e (u +/- h) is constant along characteristics. Show how this, in principle, determines the solution. At each point in space time two characteristics [C+ and C-], each carrying information from a different part of the initial data, which combined gives the solution at the point. But now the characteristics are neither straight, nor can we solve for them explicitly, because they interact with each other. ==========================================================================#10 Tue-Mar-11 ========================================================================== % %% EOF