在通道中的斯托克斯流 

指定通道区域.

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X \[CapitalOmega] = ImplicitRegion[ 0 <= x <= 2 && 0 <= y <= 0.5 && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), {x, y}]; RegionPlot[\[CapitalOmega], AspectRatio -> Automatic, ImageSize -> 300]

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定义斯托克斯流运算符.

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X stokesFlowOperator = {Inactive[ Div][({{-1, 0}, {0, -1}}.Inactive[Grad][u[x, y], {x, y}]), {x, y}] + \!\(\*SuperscriptBox[\(w\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, y], Inactive[ Div][({{-1, 0}, {0, -1}}.Inactive[Grad][v[x, y], {x, y}]), {x, y}] + \!\(\*SuperscriptBox[\(w\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[x, y], \!\(\*SuperscriptBox[\(v\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[x, y] + \!\(\*SuperscriptBox[\(u\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, y]};

在左边界向正 方向指定流入剖面. 在右手边界指定流出压力. 在其余边界的流体速度为0.

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X Subscript[\[CapitalGamma], D] = { DirichletCondition[u[x, y] == 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.], DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 0 < x < 2], DirichletCondition[w[x, y] == 0., x == 2]};

求解方程. 为了使数值解稳定，给定插值阶数选项. 当流体压力 为一阶插值时，流体速度 和 用为二阶插值.

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X {xVel, yVel, pressure} = NDSolveValue[{stokesFlowOperator == {0, 0, 0}, Subscript[\[CapitalGamma], D]}, {u, v, w}, {x, y} \[Element] \[CapitalOmega], Method -> {"FiniteElement", "InterpolationOrder" -> {u -> 2, v -> 2, w -> 1}}];

用流线绘制流速图.

显示完整的 Wolfram 语言输入隐藏输入

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X rmf = RegionMember[\[CapitalOmega]]; Show[BoundaryDiscretizeRegion[\[CapitalOmega]], StreamPlot[{xVel[x, y], yVel[x, y]}, {x, 0, 2}, {y, 0, 1/2}, RegionFunction -> Function[{x, y}, rmf[{x, y}]], AspectRatio -> Automatic], ImageSize -> 300 ]

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