0-1 AME 526: Homework 12, Due April 20, 2016 Problem 1 For a half-cylinder of length l and radius R, solve r 2 T .r; ; z/ D 0; (25 points) 0 r R; 0 ; 0 z l; (1) with boundary conditions @T k D q0 ; @z zD0 T .r; ; l/ D 0I (2) T .r; ; z/ D 0I (3) T .r; 0; z/ D 0; and T .R; ; z/ D 0; T .r ! 0; ; z/ < 1 (4) Problem 2 For a semi-innite cylinder of length of radius R, solve r 2 T .r; ; z/ D 0; 0 r R; 0 2 ; (30 points) 0 z 1; (5) The boundary z D 0 is subjected to a uniform heat ux q0 over half of that end. The lateral surface is kept at T D 0. .T D 0 .. . .. ............................ .. .......................................... .. ............... ....... .. ..... .. . ............... ................ ...... ....................... .. .. ............. ..... z ! 1 .......... . .. ..... .. .. ... ..................... .. .... . . . . .. ........ ..................... .. .. T D 0 .. .. q0......................................................... . .... .. .. ..... . .......................... .. ... .. . . .. ... ........................................... .. .. ........ .. . . .. ..... . ..................... .................... ....... ... . .......... . . .... .. ............................. ...................... . ....... .. . .. .... .. .................. . .... ...................... . ............................................................ ......... . ... insulated zD0 Non-Homogeneous PDEs Problem 3 The problem here has a forcing function representing a constant pressure gradient divided by the kinematic viscosity. This quantity is named as P0 . (20 points) .............................. ...... ........ Solve the steady-state velocity prole u.r; / for ..... .... . ... D 0 u D 0 ... uid ow in circular pipe with a longitudinal n by .. .. . .' r D R; u D 0 . . . . solving Poisson's equation, . . . .. .. . s d ... .. d . 2 .... .. @ u 1 @u 1 @2 u d ..... ... C C 2 2 D P0 (6) ....... ...... d D 2 ............................... uD0 @r 2 r @r r @ Also calculate the total volumetric ow rate through the pipe. 0-2 Follow the procedure: 1. In this case, D 0 and D 2 are real boundaries. Identify the three homogeneous boundary conditions on rectangularized .r; / coordinates. At r D 0, we do not have a specic condition but we need not get too concerned. With continuity around the n, it can be taken to be zero. 2. With homogeneous conditions all around, either r or can be taken to be a variable for eigenfunction expansion. Let us choose because it will be easier. 3. Multiply the differential equation by r 2 , and isolate the @2 u=@ 2 term. Set up the operator 2 as an equation for n . / so that the second derivative operation on it yields n n . /. 4. Solve the equation, satisfy the homogeneous conditions at D 0 and D 2 , obtain values, and establish specic expressions for n . / n 5. Expand the right-hand side of the differential equation in terms of the eigenfunctions n . /, i.e., 1 X P0 D pn n . /; nD1 and determine pn by orthogonality of the eigenfunctions, using the specic expressions for n . /. 6. Next, express u.r; / as an eigenfunction expansion in using the same set of eigenfunctions established in Item 4 and used in Item 5, i.e., u.r; / D 1 X Un .r/n . /; nD1 7. Substitute both expansions [for u.r; / and P0 ] into the differential equation (6). Keep the pn notation for the RHS. 8. Equate both sides term-by-term, and obtain an O.D.E. for Un .r/. Multiply the equation by r 2 and note that it is a non-homogeneous Cauchy-Euler equation. 9. Obtain the general solution to this non-homogeneous equation (homogeneous + particular). 10. Eliminate any terms that go to innity as r ! 0. Satisfy the remaining homogeneous condition at r D R. This, being homogeneous, can be satised term-by-term. 11. Assemble the solution for u.r; /. Put back the expression for pn derived in Item 5. Integrate u.r; / across the pipe cross-section to get the owrate. You're done. 0-3 Problem 4 D0 ................................... r D c; ....... ..... The elastic stress on a shaft of uniform cross..... . D 0; D0 ....... .. ... ........ ............ ..... section undergoing torsion can also be described .. ..... .. .. . . . . . . .. by the two-dimensional Poisson's equation, . . . . .. .. .. .. r .. r D 2 ; .. 2 D0 ... ...... . r .r; / D G0 ; .. ... ..................... .... ... ..... . where .r; / is known as the torsion function, .... r r D R; ....... r D0 ................................. and G0 is a constant related to the shear modulus of the solid and the angle of twist of the shaft. For the present problem we have a hollow shaft of inner radius c and outer radius R and a longitudinal slit throughout its length. Follow the same procedure as the previous problem but take into consideration that r D 0 is not included in the problem, and apply the boundary condition at r D c. The torque required can be calculated by integrating .r; / over the cross section (this is not required for this HW). (25 points)