Diffusion equation January 28 2009 Overview The Diffusion Equation Review last week Diffusion equation Physical meaning and derivation Relation to Laplace equation Solution by separation of variables Sturm Liouville orthogonal eigenfunction expansion for initial conditions Larry Caretto Mechanical Engineering 501B Seminar in Engineering Analysis January 28 2009 Only possible for homogenous boundary conditions Treatment of nonhomogenous boundary conditions 2 Review Sturm Liouville Review Orthogonal Functions Homogenous equations for a x b d dy r x dx dx q x p x y 0 dy k1 y a k 2 dx 0 Defined as inner product integral with p x from Sturm Liouville equation y y y x y x p x dx b x a i l 1 y b l 2 dy dx 0 x b Solutions ym are complete set of orthogonal eignenfunctions that can be used to expand any function i j am ym f ym ym p x y m x f x dx a b p x y m x ym x dx a 3 4 Review Even Odd Functions Review Fourier Series Based on periodic functions defined over L x L n x n x f x a0 an cos bn sin L L n 1 1 1 n x f x dx an f x cos dx 2L L L L L L 1 n x bn f x sin dx L L L L 5 ME 501B Engineering Analysis 2 Get coefficients in eigenfunction expansions b m 0 a0 yi ij a f x a m y m x L j Odd function f x f x like sine Even function g x g x like cosine cosine sine times cosine sine Product of odd times even is odd ForL odd f x f x dx 0 L For even g x L L L 0 g x dx 2 g x dx 6 1 Diffusion equation January 28 2009 Review Half Interval Series Review Half Interval Series II For even functions there are no sine terms in the Fourier series For odd functions there are no cosine terms in the Fourier series In these cases can use integrals from 0 to L for the coefficients Can also use equations from 0 to L for all functions but get periodic behavior outside this region Look at function below for L 2 Full series gives periodic results Full Series 7 8 Review Half Interval Series III Review Half Interval Series IV Expanding function for 0 x L 2 with a sine series gives odd periodic repetition Expanding function for L x 0 with a sine series gives odd periodic repetition Sine Series Sine Series 9 Review Half Interval Series V Cosine series for 0 x L 2 with gives even periodic repetition 10 Review Half Interval Series VI Cosine series for L x 0 gives even periodic repetition of different region Cosine Series Cosine Series 11 ME 501B Engineering Analysis 12 2 Diffusion equation January 28 2009 Derivation Basics Diffusion and Laplace Equations Partial differential equations related to conservation principles of fluxes governed by potentials Heat transfer from temperature gradient Mass diffusion from concentration gradient Current from electrostatic potential Magnetic fluxes Ideal fluid flow from velocity potential Laplace is steady state case of diffusion Consider a flux f that is a gradient of a potential u f u x The net outflow is f x dx A f x A Dividing by V A x gives the net outflow per unit volume f x dx f x x which is f x as x approaches zero Substituting the potential definition gives the net outflow per unit volume as u x x 13 14 Derivation Basics II Derivation Basics III A conserved quantity has zero net outflow for steady processes with no source terms For transient processes a net outflow causes a decrease in u This gives u t u x x u y y u z z For constant properties and we have the diffusion equation This gives u x x 0 or u x x u y y u z z 0 constant properties gives Laplace s equation 2u 2u 2u u 2 2 2 2u t x y z 2u 2u 2u 2u 0 x 2 y 2 z 2 Can be two dimensional Changes with coordinate systems Can have one or two space dimensions Changes with coordinate systems 15 Derivation Basics IV Multidimensional Equations Diffusion equation gives transient process for systems where steady state is Laplace s equation 2u 2u 2u u 2 2 2 2u t x y z transient General diffusion equation for three dimensions 2u 0 steady state Can have different space dimensions called the diffusion coefficient or diffusivity has dimensions of length 2 time 17 ME 501B Engineering Analysis 16 Cartesian Cylindrica l Sphere 2u 2u u 2u t 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 18 3 Diffusion equation January 28 2009 Variable Properties Diffusion Equation Solutions Included in outer space derivative Governs heat conduction and species diffusion for t 0 and 0 x xmax u u div grad u t u u u u z y z x y x t u u u 1 u 1 r 2 Cylindrical z z r r t r r u 1 1 u u u 1 sin 2 2 2 2 r 2 Sphere r sin r r sin t r r Cartesian u x t is temperature species concentration Initial condition u x 0 u0 x Boundaries u 0 t uL t u xmax t uR t Diffusivity is material property length 2 time u 2u 2 t x 19 Separation of Variables Separation of Variables Works Assume u x t X x T t u X x T t T t X x t t t 2 X x 2 X x T t 2u 2 T t 2 x 2 x x Divide by X x T t 1 1 T t 1 2 X x X x x 2 T t t Result is function of t equal to function of 21x Solve ODEs to Get u x t d 2 X x 2 X x 0 dx 2 22 Look at case where u 0 t u xmax t 0 Since u 0 t T t X 0 and u xmax t T t X xmax we can only satisfy these boundary conditions if X 0 X xmax 0 T t Ae t 2 X x B sin x C cos x u x t T t X x Ae t B sin x C cos x 2 e t C1 sin x C2 cos x 2 23 ME 501B Engineering Analysis Assumed solution u x t X x T t gives a function of x equal a function of t Since x and t are independent both sides must equal a constant for this to be true 1 1 T t 1 2 X x 2 T t t …