Diffusion Equations in Cylindrical Coordinates

Outline Diffusion Equations in Cylindrical Coordinates • Review last class – Gradient and convection boundary condition • • • • • Larry Caretto Mechanical Engineering 501B Seminar in Engineering Analysis February 4, 2009 Diffusion equation in radial coordinates Solution by separation of variables Result is form of Bessel’s equation Review Bessel functions Eigenfunction expansion in Bessel functions 2 Review Homework Problem Review Homework Problem II • Page 561 problem 5: find u(x,t) for 0 ≤ x ≤ L • General solution is sum of all eigenfunctions ∂u ∂u =α 2 ∂t ∂x ∂u ∂x 2 ∂u = ∂x x =0 = 0 u ( x,0) = f ( x) x= L ∞ u ( x, t ) = ∑ Ane −λ nαt cos(λ n x) n =0 2 0 for cosines • General orthogonal relationship for An An = • Start with separation of variables solution [C1 sin(λx) + C2 cos(λx)] u ( x, t ) = e −λ2αt [C1 cos(λx) − C2 sin(λx)] ∂u ∂x = λe −λ2αt L λn = nπ L ⎛ nπx ⎞ ⎟dx L ⎠ L 2 ⎛ nπx ⎞ ∫ cos ⎜⎝ L ⎟⎠dx 0 ∫ f ( x) cos⎜⎝ 0 • Final cosine result has A0 L L 1 2 ⎛ nπx ⎞ A0 = ∫ f ( x)dx An = ∫ f ( x) cos⎜ ⎟dx L0 L0 ⎝ L ⎠ • For ∂u/∂x|x=0 = 0, C1 = 0 • For ∂u/∂x|x=L = 0, λ = nπ/L for integer n 3 Review Convection Problem T∞ Heat Flux qx=L Heat Flux qx=-L x = -L Symmetry Condition x=0 ∂T ∂x x=L =0 x =0 ξ= T∞ Conduction-convection heat balance at x = L ∂T q x = L = −k = h(Tx = L − T∞ ) ∂x x = L αt x τ= 2 L L Θ= Review Dimensionless Problem • Diffusion equation Solution Region Physical Region 4 T − T∞ T0 5− T∞ ∂T ∂ 2T =α 2 ∂t ∂x ∂Θ ∂ 2 Θ = ∂τ ∂ξ 2 • Initial T(x,0) = f0(x) Θ(ξ ,0) = condition ∂T • Symmetry boundary ∂x = 0 x=0 condition • Convection boundary condition k ∂T ∂x + h(Tx = L − T∞ ) = 0 x=L f 0 (ξL) − T∞ T0 − T∞ ∂Θ =0 ∂ξ ξ =0 ∂Θ hL + Θξ =1 = 0 ∂ξ ξ =1 k Homogenous6 boundary condition 1 Review Finding λn Review Initial Conditions 25 • Usual formula for Cm but λm ≠ mπ kλ/ hL 20 15 1 k/hL = 1 10 Functions 5 Cm = Line Cotangent Intersections 0 ∫ Θ0 (ξ) cos(λ mξ)dξ 0 1 ∫ cos 2 1 = ( λ m ξ ) dξ 2λ m ∫ Θ0 (ξ) cos(λ mξ)dξ 0 cos(λ m ) sin(λ m ) + λ m 0 -5 • Constant initial temperature, T0, gives Θ0 = 1 -10 1 -15 cot λ -20 Cm = -25 0 2 4 6 8 10 12 14 16 18 20 x 2λ m ∫ (1) cos(λ mξ) dξ 0 cos(λ m ) sin(λ m ) + λ m = 2 sin(λ m ) cos(λ m ) sin(λ m ) + λ m 7 8 Review Solution Cm • General solution 1 m=1 cos(λ m ) sin(λ m ) + λ m 0 1 e −λ m τ cos(λ mξ) 2 0.8 • Solution for T(x,0) = T0 (Θ0 = 1) −λ2m τ ∞ 2 sin(λ m )e cos(λ m ξ) m =1 cos(λ m ) sin( λ m ) + λ m • Need root-finding method to obtain eigenvalues, λm = (hL/k) cot λm Θ(ξ, τ) = ∑ tau = 1 0.1 0 0.1 0.2 0.3 Dimensionless Temperatur 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 – Also gets constants C1 and C2 which determines functions that are in solution tau = 1.5 0 tau = 3 tau = 4 • Apply boundary conditions to determine eigenvalues tau = 0.75 0.2 tau = 2 – Time solution will be exponential tau = 0.5 0.3 tau = 1.5 0.3 • Solve by separation of variables tau = 0.35 0.5 0.4 0.4 – Define u(x,t) = v(x,t) + w(x) – Use u – uref in original equation 0.7 0.6 0.5 • Create Sturm-Lioville problem for nonzero spatial boundary conditions 3 . 00 =0 tau 5 00 0. 01 0. 0. 02 tau = 1 Review Summary = 0. 0 5 tau = 0.75 0.6 x/L = = u ta ta u tau = 0.2 = tau = 0.5 0 u ta ta u tau = 0.02 0 1 tau = 0.1 tau = 0.05 0.7 0.1 Review Solution for hL/k = 10 0.9 tau = 0.1 0.2 9 0.8 tau = 0.2 0.9 Dimensionless Temperatur ∞ 2λ m ∫ Θ0 (ξ) cos(λ mξ) dξ Θ(ξ, τ) = ∑ Review Solution for hL/k = 1 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 11 12 2 Review Summary II Cylindrical Diffusion Equation • Write solution as sum of all possible eigenfunctions with individual constants • Use eigenfunction expansion to match initial conditions – If a solution for u(x,t) = v(x,t) + w(x) is used the eigenfunction expansion must be for u0(x) – w(x) • Solution is sum of all eigenfunctions with constants determined from matching initial conditions • General diffusion equation for three dimensions Cartesian Cylindrica l Sphere ∇2u = ∇ 2u = ∂u = α∇2u ∂t ∂ 2u ∂ 2u ∂ 2u + + ∂x 2 ∂y 2 ∂z 2 ∇ 2u = One-dimensional radial equation 1 ∂ ⎛ ∂u ⎞ 1 ∂ 2u ∂ 2u + ⎜r ⎟ + r ∂r ⎝ ∂r ⎠ r 2 ∂θ 2 ∂z 2 1 ∂ ⎛ 2 ∂u ⎞ 1 ∂ 2u 1 ∂ 2u cot φ ∂u + 2 + ⎜r ⎟+ 2 2 2 r ∂φ r ∂r ⎝ ∂r ⎠ r sin φ ∂θ 2 r 2 ∂φ 2 13 Radial Diffusion Equation New Variable for u(R,t) = uR • u(r,t) = v(r,t) + uR • Governs diffusion (heat conduction) in cylinder for t ≥ 0 and 0 ≤ r ≤ R – u(r,t) is temperature, species concentration – Initial condition u(r,0) = u0(r) – No boundary condition at r = 0 except that u(0,t) is finite • ∂u/∂r|r=0 = 0 by symmetry • Diffusivity, α, is material property (length)2/(time) 14 ∂u 1 ∂ ∂u =α r ∂t r ∂r ∂r – v(r,t) satisfies diffusion equation – v(R,t) = 0 and v(0,t) is finite • Gives a Sturm-Liouville problem for radial function • Since uR is a constant, u(r,t) is a solution to the original problem – It satisfies the differential equation and the boundary conditions ∂u ∂ (v + u R ) 1 ∂ ∂u 1 ∂ ∂ (v + u R ) = =α =α r r r ∂r ∂r r ∂r ∂r ∂t ∂t 15 Separation of Variables Solve ODEs to Get v(r,t) • Assume v(r,t) = P(r)T(t) ∂u ∂[P(r )T (t )] ∂T (t ) = = P(r ) = ∂t ∂t ∂t 1 ∂ ∂u 1 ∂ ∂[P(r )T (t )] 1 ∂ ∂P(r ) α r =α r = αT (t ) r r ∂r ∂r r ∂r r ∂r ∂r ∂r • Divide by αP(r)T(t) 1 1 ∂T (t ) α T (t ) ∂t = 16 • Have exponential ODEs in time and Bessel’s equation for radial function dT (t ) − λ2αt = −λ2αT (t ) T ( t ) = Ae dt • This is form of d dP (r ) r + λ2 rP (r ) = 0 Bessel’s equation dr dr • Solution of Bessel’s equation is P(r) = BJ0(λr) + CY0(λr) 1 1 ∂ ∂P(r ) r = −λ2 P(r ) r ∂r ∂r – J0 and Y0 are (order zero) Bessel functions of first and second kind, respectively 17 18 3 Review Bessel’s Equation Gamma Functions d 2 y ( x) 1 dy ( x) x 2 −ν 2 + + y ( x) = 0 dx 2 x dx x2 • Function Γ(x) generalizes factorials to non-integer arguments • Arises in mechanical and thermal problems in circular geometries • The value of ν is a known parameter • Use power series solution technique known as Frobenius method • Have two linearly independent solutions ∞ • Definition Γ( x) = ∫ e −t t x −1dt 0 • Analog of (n+1)! = (n+1)n! Γ( x + 1) = xΓ( x) • For integer x, Γ(n+1) = n! = nΓ(n) • Used for coefficients of Bessel functions with noninteger order 19 20 Bessel Functions, First Kind Bessel Function Plot Bessel Functions of the First Kind for Integer Orders • Separate expressions for integer and noninteger values of ν • Use n for integer values of ν ⇒ ( −1) m x 2 m + n J n ( x ) = ∑ 2 m+n m! ( m + n )! m =0 2 0.9 0.8 0.7 0.6 ∞ 0.5 0.4 Jn(x) ( −1) m x 2 m +ν Jν ( x ) = ∑ 2 m +ν m! Γ( m + ν + 1) m =0 2 ∞ 1 n n=1 n=4 0.2 • First few terms ⎛x⎞ J n ( x) = ⎜ ⎟ ⎝2⎠ n=0 0.3 0.1 0 2 4 ⎡1 ⎤ 1 1 ⎛x⎞ ⎛ x⎞ ⎜ ⎟ + ⎜ ⎟ + L⎥ ⎢ − ⎢⎣ n! 1! ( n + 1)! ⎝ 2 ⎠ 2! ( n + 2)! ⎝ 2 ⎠ ⎥⎦ -0.1 -0.2 -0.3 • Plots for n = 0,1, and 4 on next chart -0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x 21 22 Bessel Functions, Second Kind Bessel Function Plot Bessel Functions of the Second Kind of Integer Order • Yn(x) is defined as follows for integer ν = n ∞ ⎧ ( −1) m −1 x 2 m 2⎡ ⎛ m 1 m + n 1 ⎞ ⎫⎤ ⎛ x ⎞ n ⎜ ∑ + ∑ ⎟ ⎬⎥ ⎢ J n ( x )⎜ ln + γ ⎟ + x ∑ ⎨ 2 m + n + 2 m! (m + n )! ⎝ k =1 k k =1 k ⎠ ⎭⎦ π⎣ ⎠ ⎝ 2 m =0 ⎩ 2 ⎧ ( n − m − 1)! x ⎫⎤ + x −n ∑ ⎨ 2 m−n +2 ⎬⎥Not present in equation for Y0 m! (m + n )! ⎭⎦ m =0 ⎩ 2 n −1 0.5 0.4 0.3 0.2 2m • General solution to Bessel’s equation given by Bessel functions of order n or ν – y(x) = AJn(x) + BYn(x) for integer ν = n – y(x) = AJν(x) + BJ-ν(x) for noninteger ν 0.1 0 -0.1 Yn(x) Yn ( x ) = 0.6 n=0 n=1 n=4 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 Yn becomes infinite as x approaches zero -0.8 -0.9 -1 0 23 1 2 3 4 5 6 7 8 x 9 10 11 12 13 14 15 24 4 Bessel Equation Solutions Bessel Function Summary • Bessel’s equation, x2d2y/dx2 + xdy/dx + (x2 - ν2)y = 0, main applications are to problems in radial geometries. • Physical problem gives value for ν • The general solution to Bessel’s equation is y = C1Jν(x) + C2Yν(x) where C1 and C2 are constants that are determined by the boundary conditions on the differential equation. • If we want a solution for x = 0 we cannot use Yn(x) so a general solution that includes x = 0 is y(x) = AJn(x) • Formally define Yν(x) for non-integer ν Yν ( x ) = (cos νπ )Jν ( x ) − J −ν ( x ) sin νπ • In limit as n approaches an integer, this definition approaches Yn(x) • Gives y(x) = AJν(x) + BYν(x) for any ν 25 26 Bessel’s Equation Summary II Radial Diffusion Solution • Jν(x) and Yν(x): Bessel functions, order ν, first and second kind, respectively. • Transform Bessel’s equation whose solution is y = AJn(x) + BYn(x) • Define z = x/k so y = AJn(kz) + BYn(kz) • Transformed equation is – have oscillatory behavior – found in various tables and computer library solutions – At x = 0, J0(x) = 1 and Jn(x) = 0 for n ≠ 0 – As x approaches zero, Yn(x) approaches minus infinity ⎞ d dy ⎛ ν 2 z + ⎜⎜ − + k 2 z ⎟⎟ y = 0 dz dz ⎝ z ⎠ • Can transform some equations into the form of Bessel’s equation. • Radial diffusion equation has ν = 0 d dP r + λ2 rP = 0 • Solution is P = dr dr AJ0(λr) + BY0(λr) 27 28 Transformation Details Bessel as Sturm-Liouville dy d y • Bessel’s x 2 2 + x + (x 2 − ν 2 )y = 0 dx dx equation • Define z = x/k so that x = kz 2 • Compare transformed Bessel’s equation to Sturm-Liouville problem ⎛ 2 ⎞ Bessel’s equation d ⎡⎢ z dy ⎤⎥ + ⎜⎜ − ν + k 2 z ⎟⎟ y = 0 – dy/dx = dy/d(kz) = (1/k)dy/dx – d2y/dx2 = = d2y/d(kz)2 = (1/k2)d2y/dx2 – Substitute into Bessel’s equation and divide entire equation by z 2 1⎡ 1 dy 2 1 d y kz + kz + k 2 z 2 − ν2 ( ) ⎢ z⎣ k dz k 2 dz 2 ( z dz ⎣ dz ⎦ ⎝ Sturm-Liouville )y ⎤⎥ ⎦ ⎞ ⎞ d 2 y dy ⎛ ν 2 d dy ⎛ ν 2 + + ⎜⎜ − + k 2 z ⎟⎟ y = z + ⎜⎜ − + k 2 z ⎟⎟ y = 0 2 dz ⎝ z dz dz ⎝ z dz ⎠ ⎠ 29 z ⎠ d ⎛ dy ⎞ ⎜ r ( z ) ⎟ + [q ( z ) + λp ( z )]y = 0 dz ⎝ dz ⎠ • Bessel is Sturm-Liouville equation with r(z) = p(z) = z, q(z) = -ν2/z and λ = k2 Weight function p(z) 30 5 Radial Diffusion Solution II Radial Diffusion Solution III • Radial diffusion d r dP(r) + ⎛ 0 + λ2 r ⎞P(r) = 0 ⎜ ⎟ equation for P(r) dr dr ⎝ r ⎠ • Start with general solution P(r) = AJ0(λr) + BY0(λr) • Solution applies in region to 0 ≤ r ≤ R • Must have P(r) finite at r = 0 • Since Y0(r) → -∞ as r → 0 we must have B = 0 for P(r) to be finite at r = 0 • Condition P(R) = 0 requires J0(λ R) = 0 • Need solutions, α, of equation J0(α) = 0 • Transformed Bessel equation ⎞ d ⎡ dy ⎤ ⎛ ν 2 + ⎜ − + k 2 z ⎟⎟ y = 0 z dz ⎢⎣ dz ⎥⎦ ⎜⎝ z ⎠ • Solution to second (general ν) equation is y = AJν(kz) + BYν(kz) r is weight • Solution to radial diffusion (ν = 0) function equation is AJ0(λr) + BY0(λr) – Call solutions to J0(α) = 0 the zeros of J0 31 32 Bessel Function Zeros, αmn Zeros of Bessel Functions Bessel Functions of the First Kind for Integer Orders 1 0.9 0.7 0.6 0.5 0.4 Jn(x) J2(αm2) = 0 J0(αm0) = 0 J1(αm1) = 0 m=1 2.40483 3.83171 5.13562 m=2 5.52008 7.01559 8.41724 m=3 8.65373 10.17347 11.61984 m = 4 11.79153 13.32369 14.79595 m = 5 14.93092 16.47063 17.95982 Ref: Abramowitz and Stegun, NBS AMS 55 Note increase by about π between successive zeros If Jn(α) = 0, α is a zero of Jn. Initial point where Jn(0) = 0 is not a useful zero of Jn 0.8 n=0 n=1 0.3 n=4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x 33 34 Zeros of J0 Zeros of J1 Bessel Functions of the First Kind for Integer Orders Bessel Functions of the First Kind for Integer Orders 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 n=0 n=1 0.3 n=4 0.2 0.1 0.4 Jn(x) Jn(x) 0.4 n=0 n=1 0.3 n=4 0.2 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 0 1 α10 2 3 4 α20 5 6 7 8 x α30 9 10 11 α40 12 13 14 α50 -0.4 15 0 35 1 2 3 α11 4 5 6 α21 7 x 8 9 α31 10 11 12 α41 13 14 15 36 6 Zeros of J4 Bessel Eigenfunctions, Jn(αmn) Bessel Functions of the First Kind for Integer Orders • Different eigenfuctions are all Bessel functions of the same order, n 1 0.9 0.8 – Unlike sines and cosines we can have many different Bessel functions – For both kinds of functions, the eigenfunctions are given by the zeros – sin(nπ) = 0 or cos[(2n+1)π/2] = 0 – Jn(αmn) = 0 does not have fixed intervals like the sine and cosine – For radial solution, J0(λR) = 0 gives λmR = αm0; define km0 = αm0/R, so λm = km0 0.7 0.6 0.5 Jn(x) 0.4 n=0 n=1 0.3 n=4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 1 2 3 4 5 α14 6 7 8 9 α24 10 11 x 12 13 14 α34 15 37 38 Sine Eigenfunctions Bessel (J0) Eigenfunctions Sine nπx for Different n Values Bessel Function J0(αm0x) 1 Here R = 1 so am0 = km0 1.0 0.9 0.8 0.8 0.7 0.6 0.6 0.4 0.5 n=1 0 m= 1 m= 2 m= 4 m= 10 0.3 n=2 0.2 n=3 0.1 n = 10 -0.2 J sin npx 0.4 0.2 0.0 -0.1 -0.4 -0.2 -0.3 -0.6 -0.4 -0.5 -0.8 0 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 39 40 x Bessel Function Expansion • General eigenfunction expansions • Bessel eigenfunctions: Jn(kmnx) • Integral of xJn(kmnx)dx depends on n and definition of kmn • For n = 0 general result is ∞ f ( x ) = ∑ am y m ( x ) m =1 ∞ f ( x) = ∑ am J n ( k mn x) R ∫ x[J m =1 b am = Normalization Integral R 0 ∫ p( x) ym f ( x)dx ∫ xJ n (kmn x) f ( x)dx = a b ∫ a p ( x) ym2 dx [ R2 2 J 0 (k mn R) + J12 (k mn R) 2 ] • For J0(kmnR) = 0 this simplifies to 0 R ∫ x[J R ∫ x[J n (kmn x)] dx 2 0 0 p(x) = x is weight function (k mn x)] dx = 2 0 (k mn x)] dx = 2 0 [ R2 2 J 0 (k mn R) + J12 (k mn R) 2 ] • General result uses Struve H function 41 42 7 Integrals to Compute am Example: f(x) = 1 • Construct Bessel function expansion for f(x) = 1 in region 0 ≤ x ≤ 1 using J0 in case where J0(km0R) = 0 for R = 1 • Here we have a “simple” integral with ν – 1 = 0 so ν = 1 • Integrals with Bessel functions may give complicated results – Use symbolic int function of MATLAB • Some simple cases are possible using equations on page 194 of Kreyszig ν ν ∫ x Jν −1(kmν −1x)dx = R am = x Jν (kmν −1x) kmν −1 2 ∫ xJ 0 (km 0 x) f ( x)dx 0 [RJ1 (km0 R)]2 x ⎛ α m0 J0 ⎜ R ⎝ R 0 1 = 2∫ ⎞ ⎛x⎞ x ⎟(1) d ⎜ ⎟ ⎠ ⎝R⎠ 2 [ J1 ( α m 0 ) ] x ξ=1 • Useful only when desired power of x matches the Bessel function order 43 Example Expansion ξ= ⎡ ξJ (α ξ) ⎤ ⎡ J (α ) ⎤ R 2⎢ 1 m0 ⎥ 2 ⎢ 1 m 0 − 0⎥ α ⎣ α m 0 ⎦ ξ =0 2 m0 ⎦= = = ⎣ α m 0 J1 (α44m 0 ) [J1 (α m0 )]2 [J1 (α m0 )]2 Back to Radial Diffusion Problem • Have exponential ODEs in time and Bessel’s equation for radial function Bessel Function Expansion for f(x) = 1 1. 6 1. 4 Partial sum of 1. 2 1 terms 1 2 terms 3 terms 0.8 10 terms 0.6 50 terms 0.4 dT (t ) = −λ2αT (t ) dt v(r , t ) = T (t ) P (r ) = Ae − λ αt [BJ 0 (λr ) + CY0 (λr )] 2 • Boundary conditions: C = 0 and λmR = αm0 ∞ 0.2 u (r , t ) = ∑ Cm e −λmαt J 0 (λm r ) + u R 0 0 0.1 0.2 0.3 0.4 0.5 0. 6 0.7 0.8 0.9 1 x ∞ u0 ( r ) = u (r ,0) = ∑ Cm J 0 (λm r ) + u R m =1 Cm = (λm r )(u0 − u R )dr 0 R ∫ r[J 0 R (λm r )] dr 2 0 = ∫ rJ 0 46 Example: u0(r) = U, a Constant – Region is 0 ≤ r ≤ R and p(r) = r is weight function 0 λm R = α m 0 45 • Radial equation for P(r) is a SturmLiouville problem so we use eigenfunction expansion for initial condition ∫ rJ 2 m =1 Initial Condition R d dP (r ) r + λ2 rP (r ) = 0 dr dr λm R = α m 0 (λm r )(u0 − u R )dr 0 2 R [J1 (λm R)]2 2 47 R Cm = ∫ rJ 0 (λm r )(U − uR )dr 0 r=R ⎡ rJ1 (λm r ) ⎤ ⎥ ⎢ 2(U − u R ) ⎣ λm ⎦ r =0 = 2 R [J1 (λm R)]2 R2 [J1 (λm R)]2 2 RJ (λ R ) 2(U − u R ) 1 m λm 2(U − u R ) 2(U − u R ) = = = 2 λm RJ1 (λm R) α m 0 J1 (α m 0 ) u − u R = R 2 [J1 (λm R)] U − uR r⎞ 2(U − u R ) −α m2 0 R 2 ⎛ e J 0 ⎜α m0 ⎟ + u R R⎠ ⎝ m =1 α m 0 J 1 (α m 0 ) ∞ u (r , t ) = ∑ αt ⎛ αt r ⎞ f⎜ 2 , ⎟ ⎝R R⎠ 48 8 Radial Diffusion [u(r,t) - uR] / (U0 - uR) =0 .1 = = 0. 01 6 .00 0.5 ta u 0. 02 0. 04 tau tau 0.6 =0 .06 = =0 tau 0.7 .08 =0 ta u =0 0.8 =0 tau tau tau tau ta u 0.9 01 .00 =0 04 tau .00 =0 2 .00 1.0 .15 tau = 0.2 0.4 0.3 tau = 0.3 0.2 tau = 0.4 tau = 0.5 tau = 0.6 tau = 0.8 0.1 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/R 9