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Massachusetts Institute of Technology Mathematical Methods for Materials Scientists and Engineers 3.016 Fall 2005 W. Craig Carter Department of Materials Science and Engineering Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139 Problem Set 4: Due Fri. Nov. 4, Before 5PM: email to 1 the TA.� � � � Individual Exercise I4-1 rKreyszig Mathematica  Computer Guide: problem 6.14, page 78 Individual Exercise I4-2 rKreyszig Mathematica  Computer Guide: problem 6.16, page 78 Individual Exercise I4-3 rKreyszig Mathematica  Computer Guide: problem 7.12, page 87 Individual Exercise I4-4 rKreyszig Mathematica  Computer Guide: problem 8.10, page 96 Individual Exercise I4-5 rKreyszig Mathematica  Computer Guide: problem 8.22, page 96 Group Exercise G4-1 The shape of the catenary x + B y(x) = A cosh A is very important. The catenary is the shape of a flexible chain at equilibrium and the rotation of the catenary around y = 0 creates a surface of revolution called the catenoid. In the absence of gravity, a soap film suspended between two rings with radii R1 and R2, axes lying along y = 0, and separated by distance L has a catenoid shape. Consider a soap film suspended between two identical concentric rings o f radius R and separated by distance L. Let the soap film have surface tension γ. Surface tension has units energy/area. 1. Find a parametric representatio n of the catenoid. 2. The mean curvuture of a surface is the sum of two curvatures. These two curvatures are obtained by slicing the surface with two orthogonal planes—creating two curves—and then using the formula for curvature for a curve. One of the curvatures is simply 1/y(x); the second can be obtained by using the result in Kreyszig page 443. Calculate the total mean curvature κ(x) of the catenary and plot it. 3. Write a function that calculates the constants A and B given R and L. What are the conditions that there is one solution, two solutions, no solutions? 4. Write a function that calculates the t otal surface energy, E(R, L), of a soap film. The equation for the area of a surface of revolution is: x2 dy A[y(x)] = 2 π y 1 + ( )2dx dx x1 Plot the normalized energy surface(s) E(R, L)/(γRL). 2� � � � � � Group Exercise G4-2 The diffusion equation ∂c = D∇2 c ∂t describes how the concentration field c (~r, t) changes with time proportio nal to spatial second derivatives. A solution to the diffusion equation requires that initial conditions and boundary conditions be specified. Boundary conditions specify how c(~r, t) behaves at particular points in space for all t imes. Initial conditions specify how c(~r, t) behaves throughout all space at a particular time. For some boundary conditions (BCs) and initial conditions (ICs), it is possible to write a solution to the diffusion equation in terms of an int egra l. For solutions in the infinite domain, the following BCs and ICs are a pair of such conditions, c(x = ±∞, y = ±∞, z = ±∞, t) = 0 (1) ac c0 if|x| ≤ 2 and |y| ≤ b and |z| ≤ 2 2 (2) c(x, y, z, t = 0) = 0 otherwise where a, b, and c are finite (i.e., the initial conditions have uniform concentration, c0, inside a rectangular box and zero outside. 1. Show that a b c 2 2 2 c0dζdηdχ −(x−χ)2+(y−η)2+(z−ζ)2 c(x, y, z, t) = e (3) 4Dt −a −b −c (4πDt)3/2 2 2 2 always satisfies the diffusion equation (independent of BCs and ICs). 2. Show that Eq. 3 always satisfies the boundary conditions, independent of the ICs. 3. Find the closed form of c(x, y, z, t) that satisfies both Eq. 1 and 2. 4. Show by a graphical means that c(x, y, z, t) plausibly approaches the ICs (Eq. 2) as t → 0. 5. Show that the total number of at oms is conserved for c(x, y, z, t). Group Exercise G4-3 The potential energy of two small magnetic dipoles µ~1 and µ~2 located at points r~1 and r~2 are given by µo µ~1 · µ~2 3[µ~1 · (r~1 − r~2)][µ~2 · (r~1 − r~2)]U(r~1, ~r2) = − 4π |r~1 − r~2|3 |r~1 − r~2|5 Suppose the first magnetic dipole is located at the origin and po ints towards the z-direction. 1. Illustrate the potential energy of the two-dipole system as a function of the second magnet’s position r~2 if it is also directed towards the z-direction. 3~2. Illustrate the potential energy of the two-dipole system if t he second magnet is fixed at the location r~2 but is rotated by θ about t he normal to the plane conta ining both magnets and the z-axis. 3. Illustrate the potential energy of the two-dipole system as a function both the second mag-net’s position r~2 and its rotation θ about the normal to the plane cont aining both magnets and the z-axis. 4. Suppose the second magnet is moved along a trajectory, (x, y, z) = r0(cos(2πt), sin(2πt), 0), and the magnet is always directed towards the trajectory’s tangent. Calculate and illustrate the potent ia l energy and the rate of work done on the system as a function of time. 5. Extra Credit: Suppose the two magnets are immersed in a viscous fluid and the first magnet is fixed as above. The rate of rotation is given by (approximately) dθ τ = dt 4πηR2L where R and L are the radius and length of the cylindrical magnet and η is the viscosity in the fluid medium. τ is the torque applied to the magnet. The velocity is given by (very approximately) d~r F = dt 6πηR where ~F is the force applied to the mag net. Graphically illustrate the position of the rod as a function of time, if the rod is initially at rest at t = 0 and located at ~r = r0 for the following initial inclination angles: θ = (0◦ , 1◦ , 45◦ , 89◦ , 90◦ , 91◦ , 135◦ , 179◦ , 180◦ , 181◦ , 225◦ , 269◦ , 270◦ , 271◦ , 315◦ , 359◦)


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