Nme Due Dte: Problems re collected on Wednesdy. Mth 3150 Problems Hbermn Chpter H1 Submitted work. Plese submit one stpled pckge per problem set. Lbel ech problem with its corresponding problem number, e.g., Problem H3.1-4. Lbels like Problem XC-H1.-4 re extr credit problems. Attch this printed sheet to simplify your work. Lbels Explined. The lbel Problem Hx.y-z mens the problem is for Hbermn 4E or 5E, chpter x, section y, problem z. Lbel Problem H.0-3 is bckground problem 3 in Chpter (tke y = 0 in lbel Problem Hx.y-z ). If not bckground problem, then the problem number should mtch corresponding problem in the textbook. The chpters corresponding to 1,,3,4,10 in Hbermn 4E or 5E pper in the hybrid textbook Edwrds-Penney-Hbermn s chpters 1,13,14,15,16. Required Bckground. Only clculus I nd II plus engineering differentil equtions re ssumed. If you cnnot understnd problem, then red the references nd return to the problem fterwrds. Your job is to understnd the Het Eqution. Chpter H1: 1.1 1. Introduction, Het Eqution Problem H1.0-1. (Prtil Derivtives) Compute the prtil derivtives. x (1 + t + x ) (sinh t sin x) t x (1 + xt + (tx) ) t (et+x sin(t + x)) x (sin xt + t + tx ) t (e3t+x (e t sin x + e x cos t)) Problem H1.0-. (Jcobin) f1 Find the Jcobin mtrix J nd the Jcobin determinnt J for the trnsformtion F(x, y) =, where f f 1 (x, y) = cos(y) sin(3x), f (x, y) = e y sin(y + x). Bckground: J is the ugmented mtrix of column vectors xf(x, y nd yf(x, y). For mtrix A = c d b, the determinnt is defined by A = d bc. Problem H1.0-3. (Directionl Derivtive) Compute the grdient vector grd(f) nd the directionl derivtive of f in the direction of v = ı j t x = y = 1. xf Bckground: grd(f) =, DD(f) = grd(f) v/ v. yf f = x + y + 1 f = x + y f = ye x 1 Problem H1.0-4. (First Order Liner ODE) Solve the following first order ODE problems. y y = 0 y + y = 0 y + y = 4 xy + y = x + 1 y = xy y = y References: Edwrds-Penney, Chpter 1. Appendix A of Asmr, ny edition. The first five re liner, the lst seprble.
Problem H1.0-5. (Wronskin) Compute the Wronskin determinnt of the given functions t x = 0. 1, x 1, e x cos x, sin x e x, e x, e 3x 1, cosh x, sinh x 1, ln 1 + x, x ln 1 + x References: Edwrds-Penney, DE nd LA, Chpter 5. Appendix A of Asmr, ny edition. Bckground: The Wronskin of f, g is the determinnt f f g g. For functions f 1,..., f n, the Wronskin is the determinnt of the n n mtrix formed by tking row 1 to be f 1,..., f n nd then successive rows re the successive derivtives of row 1. Problem H1.0-6. (Newton s Lws) () Stte Newton s second lw F = m in clculus terms for motion of mss m in motion long line with position x(t) t time t. (b) Use the sttement in () to derive the spring-mss undmped free vibrtion model mx (t) + kx(t) = 0. Assume no externl forces nd Hooke s restoring force on the spring of mgnitude kx(t). Symbol x(t) is the signed distnce from the center of mss to equilibrium position x = 0. Symbol m is the mss, symbol k is the Hooke s constnt. References: Edwrds-Penney Chpter 5. The ODE in (b) is clled the hrmonic oscilltor. Serwy nd Vuille, College Physics, 10E. Problem H1.0-7. (PDE Answer Check: Advection Eqution) Show the detils of n nswer check for the solution u(x, t) = e (x t) u(0, t) = e t. of the initil vlue problem u t + u x = 0, Drw four (4) pulse figures for the wveform u, for x = 0, 1,, 3. Ech imge is -dimensionl plot in the tu-plne with t-rnge to 5, clled snpshot or frme (from movie frme). References: The PDE is clled the dvection eqution. The term dvection mens tht prticles re simply crried by the bulk motion of the fluid. It is specil cse of the dvection-dispersion-rection eqution C t C = v x + C DL x q t. The terms on the right re dvection, dispersion nd rection. A historicl reference (1999) for US Geologicl Survey pplictions, which fully explins the symbols in the bove PDE, cn be found Here. A more complete discussion of USGS pplictions is found t http://pubs.er.usgs.gov/publiction/wri99459. Hydrulics in civil engineering uses dvection to model trnsport of chemicl (e.g., pollutnt, lump of ink) in strem of wter flowing through thin tube t constnt speed c. In simple model, the ink moves s lump or pulse, without dispersion. If bottle of ink is dumped in river nd u(x, t) is the downstrem ink concentrtion of the moving lump of ink, then dispersion becomes importnt. Problem H1.-1. (Minus Sign) Explin the minus sign in the conservtion lw u t = φ x. Exmple H1.-1. Explin the minus sign in conservtion lw e t = φ x. Solution. The eqution to be explined is de/dt = (d/dx)φ To mke sense of the eqution we replce derivtives by Newton difference quotients, s follows: e(x, t + h) e(x, t) de/dt h φ(x + k, t) φ(x, t) (d/dx)φ(x, t). k Suppose the therml energy density e(x, t) increses in the time intervl t to t + h, e.g., de/dt > 0. Then the difference quotient for e(x, t) is positive. On the other side of the eqution, the mount of het energy per unit time flowing to
the right hd to decrese, tht is more het hd to be trpped in the rod section in order to increse the het density e(x, t). So the right side of the eqution is which ccounts for the minus sign. Problem H1.-7. (Het Eqution Derivtion) (d/dx)φ = ( 1)(φ(x + k, t) φ(x, t)) = ( )( ) = + Conservtion of therml energy for ny segment of one-dimensionl rod < x < b stisfies the eqution d dt b e(x, t) dx = φ(, t) φ(b, t) + By using the fundmentl theorem of clculus, (d/db) b f(x)dx = f(b), derive the het eqution cρ du dt = d dx References. Hbermn H1. nd Asmr 3.5, Appendix. Problem H1.-8. (Totl Therml Energy) du K 0 + Q. dx If u(x, t) is known, then give n expression for the time-dependent (becuse energy escpes t the ends) totl therml energy L A(x)e(x, t) dx contined in rod x = 0 to x = L. 0 Vlidte the nswer using uniform rod of length L nd cross-sectionl re A, held t stedy-stte temperture u = u 0. Reference: Serwy nd Vuille, College Physics, Chpter 11. Chpter H1: 1.3 1.4 Het Eqution b Q dx Problem H1.3-. (Perfect Therml Contct) Two one-dimensionl lterlly insulted rods of different mterils joined t x = x 0 re sid to be in perfect therml contct if both the temperture u(x, t) nd the het flux φ(x, t) t loction x nd time t vries continuously cross the juncture x = x 0. () The continuous temperture eqution t the juncture x = x 0 is u(x 0, t) = u(x 0 +, t). Illustrte the physicl sitution with digrm. Explin the symbols. (b) Het energy flowing out of one rod mteril t x = x 0 flows into the other with no het loss, which is the het flux eqution φ(x 0 +, t) = φ(x 0, t). Apply Fourier s Lw to re-write this condition s K 0 (x 0 +) u x (x 0+, t) = K 0 (x 0 ) u x (x 0, t). (c) A br of length L is bent into solid ring. It then hs no physicl endpoints for energy to escpe. Assuming perfect therml contct will led to the boundry conditions. Give the ssumptions nd the detils for the formultion of the corresponding periodic boundry vlue problem u t = k u x, u u u(x, 0) = f(x), u( L, t) = u(l, t), ( L, t) = (L, t). x x Don t solve the boundry vlue problem, just explin the equtions. References: Hbermn s text H1., het flux, therml energy nd Fourier s Lw. See lso the 1959 textbook reference in http://en.wikipedi.org/wiki/perfect therml contct. Applictions for therml contct conductnce include het sinks, internl combustion engines, spce vehicles. T There is n eqution for het flux between two solid bodies A nd B: q = 1 T 3 x A/(k A A)+1/(h ca)+ x B /(k B A). Detils re in http://en.wikipedi.org/wiki/therml contct conductnce nd lso in Holmn, J. P. (009) Het Trnsfer, 10E, McGrw-Hill Series in Mechnicl Engineering, ISBN-10: 00735936. 3
Problem H1.4-1. (Equilibrium Temperture in Rod) Determine the equilibrium temperture distribution for one-dimensionl rod cρ u t = K 0 u x +Q with initil temperture u(x, 0) = f(x), constnt therml properties nd the following source nd boundry conditions: () Q = 0, u(0) = 0, u(l) = T (b) Q = 0, u(0) = T, u (L) = α (c) Q/K 0 = x, u(0) = T, u (L) = 0 (d) Q = 0, u (0) [u(0) T ] = 0, u (L) = α Problem H1.4-. (Het Energy in Rod) Determine the het energy generted per unit time inside the entire rod, from the equilibrium temperture distribution for the model cρ u t = K u 0 x + K 0x with boundry conditions zero t x = 0 nd x = L. Problem XC-H1.4-3. (Bi-Metl Rod) Determine the equilibrium temperture distribution for one-dimensionl rod composed of two different mterils in perfect therml contct t x = 1. For x = 0 to x = 1, there is one mteril (cρ = 1, K 0 = 1) with constnt source (Q = 1), wheres for the other x = 1 to x = (cρ =, K 0 = ) there re no sources (Q = 0) with u(1) = 0 nd u() = 0. Reference: Hbermn problem H1.3- nd sections H1., H1.3. Problem H1.4-7. (Equilibrium Temperture) Find the vlues of β for which there exists n n equilibrium temperture distribution for the model u t = u x + 1, u u u(x, 0) = f(x), (0, t) = 1, (L, t) = β. x x References: Hbermn H1.4, equilibrium temperture. The physicl interprettion of the model is lterlly insulted uniform rod with uniform internl heting long its length, with het flux given t x = 0 nd x = L s 1 nd β. Problem XC-H1.4-10. (Totl Therml Energy in Rod) Clculte the totl therml energy t time t in the one-dimensionl rod with model u t = u x + 4, u u u(x, 0) = f(x), (0, t) = 5, (L, t) = 6. x x Chpter 1: H1.5 Het Eqution in Dimensions nd 3 Problem H1.5-. (Fourier s Het Conduction Lw, Convection nd Advection) The trnsfer of het from torched end of welding rod to the fr end is modeled by Fourier s Lw of het conduction, φ = K 0 grd(u). Wter boiling in pst cooking pot illustrtes convection, the movement of molecule ggregtes in response to het. Silt in river illustrtes dvection, the movement in fluid of suspended prticles or dissolved mteril. Suppose R is region in spce contining mteril. A velocity field V defines molecule movement: molecule ggregtes in R move in direction V with speed V. Advection is ssumed zero, due to no prticultes. Vribles c, ρ, u, φ re functions of loction (x, y, z) in region R, with u, φ lso dependent on time t. Symbol c is the specific het, the het energy needed to dd one degree of temperture to unit mss of the mteril. Symbol ρ is mteril mss per unit volume. Symbol u is temperture. Vector symbol φ is het flux, which stisfies Fourier s Lw φ = K 0 grd(u) (therml conductivity)(temperture difference). Explin in fewest words or equtions the following sttements. () Therml energy e 1 (x, y, z, t) is given by e 1 = cρu. Then het flux from convection is e 1 V. 4
(b) Het flux from conduction is given by Fourier s Lw, K 0 grd(u). (c) The totl het flux from conduction nd convection is φ = K 0 grd(u) + cρu V. (d) Assume symbols c, ρ, K 0 re constnts nd Q = 0. Substitute the het flux into the conservtion lw cρ u t = Div( φ) + Q to obtin for k = K 0 /(cρ) the prtil differentil eqution u t = k (u) V grd(u) u Div( V ). References: Hbermn H1.5. In the cse of n incompressible flow, the eqution Div( V ) = 0 is vlid, then (d) reduces to the eqution u t = k (u) V grd(u), which is the clssicl eqution for het conduction with convection, ignoring dvection. Compre this eqution to the chemicl dvection-dispersion-rection eqution C t Problem XC-H1.5-14. (Isotherms nd Insulted Boundry) C = v x + D C L x q t. Isotherms re lines of constnt temperture. Show tht isotherms re perpendiculr to ny prt of the boundry tht is insulted. Problem XC-H1.5-15. (Het Eqution 3D Derivtion) Derive the het eqution in three dimensions ssuming constnt therml properties nd no sources. 5