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### Hollow cylinder, transient 1-D.

R11 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and r = b.
 GR11(r, t  r, ) = exp - (t - )/a2 x where R(r) = J0()Y0() - Y0()J0()

with eigenvalues given by J0()Y0(b/a) - Y0()J0(b/a) = 0.

R12 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and G/r = 0 (Neumann)at r = b.

 GR12(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = J0()Y1() - Y0()J1()

with eigenvalues given by J0()Y1(b/a) - Y0()J1(b/a) = 0.

R13 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and kG/r + hG = 0 (convection) at r = b.

 GR13(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = S0J0() - V0Y0() with V0 = J1() + BJ0() S0 = - Y1() + BY0() B = ha/k

Eigenvalues are given by S0J0() - V0Y0() = 0 .

R21 Hollow cylinder a < r < b, with G/r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.

 GR21(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = J0()Y0() - Y0()J0()

with eigenvalues given by J1()Y0(b/a) - Y1()J0(b/a) = 0.

R22 Hollow cylinder a < r < b, with G/r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.

 GR22(r, t  r, ) = + exp - (t - )/a2 x  R(r) R(r) where R(r) = J0()Y1() - Y0()J1()

with eigenvalues given by J1()Y1(b/a) - Y1()J1(b/a) = 0.

R23 Hollow cylinder a < r < b, with G/r = 0 (Neumann) at r = a and kG/r + hG = 0 (convection) at r = b.

 GR23(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = S0J0() - V0Y0() with V0 = J1() + BJ0() S0 = - Y1() + BY0() B = ha/k

Eigenvalues are given by S0J1() - V0Y1() = 0 .

R31 Hollow cylinder a < r < b, with - kG/r + hG = 0 (convection) at r = a and G = 0 (Dirichlet) at r = b.

 GR31(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = J0()Y0() - Y0()J0() with   U0 = - J1() + BJ0() W0 = - Y1() + BY0() B = ha/k

Eigenvalues are given by U0Y0(b/a) - W0J0(b/a) = 0.

R32 Hollow cylinder a < r < b, with - kG/r + hG = 0 (convection) at r = a and G/r = 0 (Neumann) at r = b.

 GR32(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = J0()Y1() - Y0()J1() with  U0 = - J1() + BJ0() W0 = - Y1() + BY0() B = ha/k

Eigenvalues are given by U0Y1(b/a) - W0J1(b/a) = 0.

R33 Hollow cylinder a < r < b, with - kG/r + h1G = 0 (convection) at r = a and kG/r + h2G = 0 (convection) at r = b.

 GR33(r, t  r, ) = exp - (t - )/a2 x  R(r) R(r) where R(r) = S0J0() - V0Y0() with  S0 = - Y1() + B2Y0() with  U0 = - J1() - B1J0() V0 = - J1() + B2J0() W0 = - Y1() + B1Y0() B1 = ; B2 =

Eigenvalues are given by S0U0 - V0W0 = 0.

Next: Radial-spherical coordinates. Transient 1-D. Up: Cylindrical Coordinates. Transient 1-D. Previous: Solid cylinder transient 1-D.
Kevin D. Cole
2002-12-31