# Neumann Values

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

Neumann values prescribe the flux over the boundary edge.

 In[1]:= X\[CapitalOmega] = ImplicitRegion[ y >= 0 && (x - 3/2)^2 + y^2 >= 1/25 && x^2 + y^2 >= 1 && x^2 + y^2 <= 4 && y <= x Tan[\[Pi]/8], {x, y}];

Visualize the simulation region.

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Specify Dirichlet boundary conditions to set temperatures inside the pipe and outside.

 In[3]:= XSubscript[\[CapitalGamma], D] = {DirichletCondition[u[x, y] == 200., x^2 + y^2 == 1], DirichletCondition[u[x, y] == 15., x^2 + y^2 == 4]};

Specify a flux into the cooling duct in the pipe.

 In[4]:= XSubscript[\[CapitalGamma], N] = NeumannValue[-1000., (x - 3/2)^2 + y^2 == 1/25];

Solve the Laplace equation.

 In[5]:= Xuif = NDSolveValue[{-10 \!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) == Subscript[\[CapitalGamma], N], Subscript[\[CapitalGamma], D]}, u, {x, y} \[Element] \[CapitalOmega]];

Plot the solution.

 In[6]:= Xcp = ContourPlot[uif[x, y], {x, y} \[Element] \[CapitalOmega], Mesh -> None, ColorFunction -> "TemperatureMap", Contours -> 21, AspectRatio -> Automatic, Frame -> False, ImageSize -> 300]
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## Mathematica

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