BENG 221: Mathematical Methods in Bioengineering
Lecture 12
Diffusion in Polar and Cylindrical Coordinates
References
Haberman APDE, Sec. 7.7, 7.8 and 7.9.
http://en.wikipedia.org/wiki/Cylindrical_coordinate_system
http://en.wikipedia.org/wiki/Bessel_function
http://en.wikipedia.org/wiki/Fourier–Bessel_series
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Cylindrical Value Boundary Conditions
h01 = 5.783/R2
h02 = 30.47/R2
h03 = 74.89/R2
h04 = 139/R2
h05 = 222.9/R2
h11 = 14.68/R2
h12 = 49.22/R2
h13 = 103.5/R2
h14 = 177.5/R2
h15 = 271.3/R2
h21 = 26.37/R2
h22 = 70.85/R2
h23 = 135/R2
h24 = 218.9/R2
h25 = 322.6/R2
h31 = 40.71/R2
h32 = 95.28/R2
h33 = 169.4/R2
h34 = 263.2/R2
h35 = 376.7/R2
h41 = 57.58/R2
h42 = 122.4/R2
h43 = 206.6/R2
h44 = 310.3/R2
h45 = 433.8/R2
Cartesian Box Value Boundary Conditions
h11 = 19.74/L2
h12 = 49.35/L2
h13 = 98.7/L2
h14 = 167.8/L2
h15 = 256.6/L2
h21 = 49.35/L2
h22 = 78.96/L2
h23 = 128.3/L2
h24 = 197.4/L2
h25 = 286.2/L2
h31 = 98.7/L2
h32 = 128.3/L2
h33 = 177.7/L2
h34 = 246.7/L2
h35 = 335.6/L2
h41 = 167.8/L2
h42 = 197.4/L2
h43 = 246.7/L2
h44 = 315.8/L2
h45 = 404.7/L2
h51 = 256.6/L2
h52 = 286.2/L2
h53 = 335.6/L2
h54 = 404.7/L2
h55 = 493.5/L2
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Cylindrical Flux Boundary Conditions
h01 = 0/R2
h02 = 14.68/R2
h03 = 49.22/R2
h04 = 103.5/R2
h05 = 177.5/R2
h11 = 3.39/R2
h12 = 28.42/R2
h13 = 72.87/R2
h14 = 137/R2
h15 = 220.9/R2
h21 = 9.328/R2
h22 = 44.97/R2
h23 = 99.39/R2
h24 = 173.5/R2
h25 = 267.2/R2
h31 = 17.65/R2
h32 = 64.24/R2
h33 = 128.7/R2
h34 = 212.7/R2
h35 = 316.4/R2
h41 = 28.28/R2
h42 = 86.16/R2
h43 = 160.8/R2
h44 = 254.9/R2
h45 = 368.5/R2
Cartesian Box Flux Boundary Conditions
h00 = 0/L2
h01 = 9.87/L2
h02 = 39.48/L2
h03 = 88.83/L2
h04 = 157.9/L2
h10 = 9.87/L2
h11 = 19.74/L2
h12 = 49.35/L2
h13 = 98.7/L2
h14 = 167.8/L2
h20 = 39.48/L2
h21 = 49.35/L2
h22 = 78.96/L2
h23 = 128.3/L2
h24 = 197.4/L2
h30 = 88.83/L2
h31 = 98.7/L2
h32 = 128.3/L2
h33 = 177.7/L2
h34 = 246.7/L2
h40 = 157.9/L2
h41 = 167.8/L2
h42 = 197.4/L2
h43 = 246.7/L2
h44 = 315.8/L2