ClearAll[u, \[Theta], z];
R = 0.06;
\[Rho] = 1.2*10^3;
g = 9.8;
\[Gamma] = 0.05;
\[Mu] = 0.1;
\[CapitalOmega] = 2 Pi*10;
L = 0.2;
a = Sqrt[(\[Mu]*\[CapitalOmega])/(\[Rho]*g*R)];
B = (\[Gamma]/(\[Rho]*g*R^2))^-1;
ped = {f[\[Theta], z] ==
D[u[\[Theta], z], z, z] + D[u[\[Theta], z], \[Theta], \[Theta]] +
u[\[Theta], z] + (u[\[Theta], z]*Sin[\[Theta]])/B,
D[u[\[Theta], z] - u[\[Theta], z]^3/3 +
a*((5*u[\[Theta], z]^4)/12*Cos[\[Theta]] - u[\[Theta], z]^2/
2 + (B*u[\[Theta], z]^3)/3*
D[f[\[Theta], z], \[Theta]]), \[Theta]] ==
D[(B*u[\[Theta], z]^3)/3*D[f[\[Theta], z], z], z]};
bc = {u[0, z] == u[2 Pi, z],
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[0, z] ==
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[2 Pi, z], f[0, z] == f[2 Pi, z],
u[\[Theta], 0] == 0,
u[\[Theta], L] == 0, (u^(0, 1))[\[Theta], 0] ==
0, (u^(0, 1))[\[Theta], L] == 0};
NDSolve[{pde, bc}, u, {\[Theta], 0, 2 Pi}, {z, 0, L}]
截图如下

额我傻了 g重复定义了 pde符号没有匹配 然而还是没有结果
ClearAll[u, \[Theta], z];
R = 0.06;
\[Rho] = 1.2*10^3;
g = 9.8;
\[Gamma] = 0.05;
\[Mu] = 0.1;
\[CapitalOmega] = 2 Pi*10;
L = 0.2;
a = Sqrt[(\[Mu]*\[CapitalOmega])/(\[Rho]*g*R)];
B = (\[Gamma]/(\[Rho]*g*R^2))^-1;
u[\[Theta], z];
h[\[Theta], z] := D[u[\[Theta], z], \[Theta]];
k[\[Theta], z] := D[u[\[Theta], z], z];
pde = {f[\[Theta], z] ==
D[u[\[Theta], z], z, z] + D[u[\[Theta], z], \[Theta], \[Theta]] +
u[\[Theta], z] + (u[\[Theta], z]*Sin[\[Theta]])/B,
D[u[\[Theta], z] - u[\[Theta], z]^3/3 +
a*((5*u[\[Theta], z]^4)/12*Cos[\[Theta]] - u[\[Theta], z]^2/
2 + (B*u[\[Theta], z]^3)/3*
D[f[\[Theta], z], \[Theta]]), \[Theta]] ==
D[(B*u[\[Theta], z]^3)/3*D[f[\[Theta], z], z], z]};
bc = {u[0, z] == u[2 Pi, z], h[0, z] == h[2 Pi, z],
f[0, z] == f[2 Pi, z], u[\[Theta], 0] == 0, u[\[Theta], L] == 0,
k[\[Theta], 0] == k[\[Theta], L] == 0};
NDSolve[{pde, bc}, u, {\[Theta], 0, 2 Pi}, {z, 0, L}]
NDSolve::ndode: 输入不是一个常微分方程. >>