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.

show complete Wolfram Language inputhide input

In[2]:=

X Show[Graphics[{Circle[{0, 0}, 2], Circle[{0, 0}, 1], Circle[#, 1/5] & /@ {{-1, -1}, {-1, 1}, {1, -1}, {1, 1}, {3/2, 0}, {-3/2, 0}, {0, 3/2}, {0, -3/2}}}], RegionPlot[\[CapitalOmega]], ImageSize -> 300]

Out[2]=

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

In[3]:=

X Subscript[\[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]:=

X Subscript[\[CapitalGamma], N] = NeumannValue[-1000., (x - 3/2)^2 + y^2 == 1/25];

Solve the Laplace equation.

In[5]:=

X uif = 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]:=

X cp = ContourPlot[uif[x, y], {x, y} \[Element] \[CapitalOmega], Mesh -> None, ColorFunction -> "TemperatureMap", Contours -> 21, AspectRatio -> Automatic, Frame -> False, ImageSize -> 300]

Out[6]=

show complete Wolfram Language inputhide input

In[7]:=

X Show[{Graphics[{Circle[{0, 0}, 2], Circle[{0, 0}, 1], Circle[#, 1/5] & /@ {{-1, -1}, {-1, 1}, {1, -1}, {1, 1}, {3/2, 0}, {-3/2, 0}, {0, 3/2}, {0, -3/2}}}, PlotRange -> {{0.7, 2.2}, {-0.5, 1}}], cp}]

Out[7]=