Laplace equation solutions February 11 2009 Overview Laplace Equation Solutions Review last class Laplace equation solutions for homogenous boundary conditions on three boundaries Larry Caretto Mechanical Engineering 501B Solutions of Laplace s equation for more than one nonzero boundaries Seminar in Engineering Analysis February 11 2009 Superposition solutions Superposition for gradient and other boundary conditions Cylindrical coordinates 2 Review Laplace s Equation Similar to Diffusion Equation Used to express equilibrium fields of engineering variable like temperature species concentration electrostatic potential and ideal fluid flow Written in general coordinates as 2u 0 2u Cartesian Cylindrica l Sphere 2u 2u 2u 2u x 2 y 2 z 2 2u 1 u 1 2u 2u r r r r r 2 2 z 2 1 2 u 1 2u 1 2u cot u 2 r 2 2 2 r r r r sin 2 r 2 2 r Diffusion Laplace 1 u 2u t x 2 0 x L t 0 u 0 t 0 u 2u y 2 x 2 0 x L 0 y H u 0 y 0 u L t 0 u r t u0 r u L y 0 u x H u0 r open boundary in t u x 0 0 2 3 Separation of Variables Diffusion 1 T t 2 T t t 1 2 X x X x x 2 T t Ae 2 t x 0 L Boundary Conditions Laplace 2 1 Y y Y y y 2 2 1 2 X x X x x 2 Diffusion Laplace X x B sin x C cos x X x B sin x C cos x C 0 for u 0 y 0 C 0 for u 0 t 0 n n for u L y 0 for u L t 0 L L n x X n x Bn sin n x X n x Bn sin L L Y y A sinh y D cosh y 5 ME 501B Engineering Analysis 4 6 1 Laplace equation solutions February 11 2009 y 0 Boundary Condition Diffusion No equivalent condition because of open boundary in time General Solution Fitted Condition Diffusion Laplace Laplace u x t Y 0 A sinh 0 D cosh 0 0 Must have D 0 C e 2n t n n 1 n y Yn y An sinh L u x y sin n x Cn sinh n y sin n x n 1 n n L u x 0 u0 x C n n 1 sin n x n n L u x H u N x C n 1 n sin n r sinh n H 7 Eigenvalue Expansion for Cn Diffusion n x 0 u0 x sin L dx L 2 n x 0 sin L dx L 2 n x u0 x sin dx L 0 L 8 Review Constant Boundary Laplace If uN x U the solution for u x y is L n x x sin dx L L n H 2 n x sinh sin dx L 0 L L 2 n x uN x sin dx L 0 L n H sinh L L u Cm 4U u x y N 0 2n 1 x 2n 1 y sin sinh L L n H 2 1 n 1 2n 1 sinh L u U depends on x L y L and H L Plot on next page for H L 1 9 10 Review Gradients as Fluxes Laplace and diffusion equation are based on conservation of fluxes which are negative gradients of potential Laplace equation gives equilibrium where net flux from region should be zero Provided notes showing that net outflow is zero for first Laplace solution H Net outflow 0 11 ME 501B Engineering Analysis u H u L u L u x x 0 dy x x L dy y y 0 dx y y H dx 0 0 0 0 12 2 Laplace equation solutions February 11 2009 Review Gradient Boundary Review Zero Gradient Solution Zero gradient at x 0 u 2 x 2 u 2 y 2 General solution u x H u N x ux 0 y 0 L 2 sinh n H L 0 Cn y H u x H uN x u N x cos n x dx Solution for uN x U a constant u L y 0 u x 0 0 n n 0 u u L y u x 0 0 x 0 y x 0 2n 1 2L 2n 1 n 2L u x y Cn cos n x sinh n y 0 0 x L 0 y H x L u x y y 0 2U L 1 n cos n x sinh n y n sinh n H n 0 n 2n 1 2L 13 14 Exercise Rectangular Laplace Summary Solve problem from last class with boundary condition at y H changed to a gradient Use previous solution of similar problem as starting point in this case have most of the work done Problem homogenous boundary conditions except at y H Separation of variables solution gives u x y X x Y y Asin x Bcos x Csinh y Dcosh y Start here Boundary conditions at x 0 and x L give A or B and eigenvalue Eigenfunction expansion at y H gives coefficients in infinite series solution of all eigenfunctions u 0 u 0 x 0 u x y Cn sin n x n cosh n y y n 1 ME 501B Engineering Analysis 16 Get eigenfunction expansion L n n Cn L 2 g N x sin n x dx cosh n H n L 0 n n L For constant gradient gN x GN L n n L Apply y H boundary condition u x H g N x Cn sin n x n cosh n H y n 1 n n L Exercise Solution II Take the gradient y 0 n 1 Start with potential solution n 1 x L u x y Cn sin n x sinh n y Exercise Solution u x y Cn sin n x sinh n y 2u 2u 0 x 2 y 2 u 0 15 y H u y gN x n n 17 Cn 2 GN sin n x dx 0 cosh n H n L u x y 4GN L n 1 3 5 2GN n cos n x cosh n H n L sin n x sinh n y cosh n H 2n L L 2GN 0 n 1 1 n cosh n H n L n n L 18 3 Laplace equation solutions February 11 2009 More than One Nonzero Boundary Laplace s equation for 0 x L and 0 y H with boundary conditions shown Do not have homogenous boundary conditions in any coordinate direction Solution is sum of two simpler solutions y H u uN x u 0 u uE y u 0 y 0 x 0 x L 19 Superposition Solution II u x 0 u1 x 0 u2 x 0 0 0 0 u x H u1 x H u2 x H uN 0 uN u 0 y u1 0 y u2 0 y 0 0 0 u L y u1 …