Transient Neumann Values 

Compute the cooling effect of a duct with a cooling liquid in an axisymmetric cross section of a pipe.

Specify a solution domain.

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Specify a Neumann value that depends on time and stationary Dirichlet conditions.

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Solve a steady-state problem corresponding to the boundary conditions at to find an initial condition. The "Continuation" mesh generator is used to better resolve the corners and edges of the domain .

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Solve a heat equation with a temperature set on the outer boundaries and a time-dependent flux over the inner boundary, using the steady-state solution as an initial condition. To ensure the best initialization, the same spatial mesh is used as for the steady-state solution.

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Plot the solution.

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