How to solve stokes flow in 3D?

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I am trying to extend stokes flow example to 3D but I get an error. Not sure what's wrong.



For example we define the region and this works:



[CapitalOmega] = 
 ImplicitRegion[-0.5 <= z <= 0.5 && 0 <= x <= 2 && 
 0 <= y <= 
 0.5 && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, 
 z];
RegionPlot3D[[CapitalOmega]]


Then we define the stokes flow operator,I am not sure this is right: 



stokesFlowOperator = Inactive[

 Div][(-1, 0, 0, -1.Inactive[Grad][
 u[x, y, z], x, y, z]), x, 
 y, z] + 

!(*SuperscriptBox[(w), 
TagBox[
RowBox["(", 
RowBox["1", ",", "0"], ")"],
Derivative],
MultilineFunction->None])[x, y], 
 Inactive[

 Div][(-1, 0, 0, -1.Inactive[Grad][
 v[x, y, z], x, y, z]), x, 
 y, z] + 

!(*SuperscriptBox[(w), 
TagBox[
RowBox["(", 
RowBox["0", ",", "1"], ")"],
Derivative],
MultilineFunction->None])[x, y, z], nactive[

 Div][(-1, 0, 0, -1.Inactive[Grad][
 v[x, y, z], x, y, z]), x, 
 y, z] + 

!(*SuperscriptBox[(w), 
TagBox[
RowBox["(", 
RowBox["0", ",", "1", ",", "1"], ")"],
Derivative],
MultilineFunction->None])[x, y, z],

!(*SuperscriptBox[(v), 
TagBox[
RowBox["(", 
RowBox["0", ",", "1", ",", "0"], ")"],
Derivative],
MultilineFunction->None])[x, y, z] + 

!(*SuperscriptBox[(u), 
TagBox[
RowBox["(", 
RowBox["1", ",", "0", ",", "1"], ")"],
Derivative],
MultilineFunction->None])[x, y, z];
Subscript[[CapitalGamma], 
 D] = DirichletCondition[
 u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.], 
 DirichletCondition[u[x, y, z] == 0., v[x, y, z] == 0., 
 0 < x < 2], DirichletCondition[w[x, y, z] == 0., x == 2];


This lines fails:



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


And I get this error:



NDSolveValue::dsvar: 0.35` cannot be used as a variable.
Set::shape: Lists xVel,yVel,zVel,pressure and NDSolveValue[False,DirichletCondition[u[x,y,0.35]==0.35 +7.13861 Plus[<<2>>] y,x==0.],DirichletCondition[u[x,y,0.35]==0.,v[x,y,0.35]==0.,0<x<2],DirichletCondition[w[x,y,0.35]==0.,x==2],u,v,w,x,y,0.35[Element]ImplicitRegion[-0.5<=z<=0.5&&0<=x<=2&&0<=y<=0.5&&!(x>=1&&y<=0.1)&&!(x>=1&&y>=0.4),x,y,z],Method->FiniteElement,InterpolationOrder->u->2,v->2,w->1] are not the same shape.


I am not sure what I am doing wrong when trying to extend to 3D.






differential-equations







share|improve this question

























    up vote
    3
    down vote

    favorite












    I am trying to extend stokes flow example to 3D but I get an error. Not sure what's wrong.



    For example we define the region and this works:



    [CapitalOmega] = 
     ImplicitRegion[-0.5 <= z <= 0.5 && 0 <= x <= 2 && 
     0 <= y <= 
     0.5 && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, 
     z];
    RegionPlot3D[[CapitalOmega]]


    Then we define the stokes flow operator,I am not sure this is right: 



    stokesFlowOperator = Inactive[

     Div][(-1, 0, 0, -1.Inactive[Grad][
     u[x, y, z], x, y, z]), x, 
     y, z] + 

    !(*SuperscriptBox[(w), 
    TagBox[
    RowBox["(", 
    RowBox["1", ",", "0"], ")"],
    Derivative],
    MultilineFunction->None])[x, y], 
     Inactive[

     Div][(-1, 0, 0, -1.Inactive[Grad][
     v[x, y, z], x, y, z]), x, 
     y, z] + 

    !(*SuperscriptBox[(w), 
    TagBox[
    RowBox["(", 
    RowBox["0", ",", "1"], ")"],
    Derivative],
    MultilineFunction->None])[x, y, z], nactive[

     Div][(-1, 0, 0, -1.Inactive[Grad][
     v[x, y, z], x, y, z]), x, 
     y, z] + 

    !(*SuperscriptBox[(w), 
    TagBox[
    RowBox["(", 
    RowBox["0", ",", "1", ",", "1"], ")"],
    Derivative],
    MultilineFunction->None])[x, y, z],

    !(*SuperscriptBox[(v), 
    TagBox[
    RowBox["(", 
    RowBox["0", ",", "1", ",", "0"], ")"],
    Derivative],
    MultilineFunction->None])[x, y, z] + 

    !(*SuperscriptBox[(u), 
    TagBox[
    RowBox["(", 
    RowBox["1", ",", "0", ",", "1"], ")"],
    Derivative],
    MultilineFunction->None])[x, y, z];
    Subscript[[CapitalGamma], 
     D] = DirichletCondition[
     u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.], 
     DirichletCondition[u[x, y, z] == 0., v[x, y, z] == 0., 
     0 < x < 2], DirichletCondition[w[x, y, z] == 0., x == 2];


    This lines fails:



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


    And I get this error:



    NDSolveValue::dsvar: 0.35` cannot be used as a variable.
    Set::shape: Lists xVel,yVel,zVel,pressure and NDSolveValue[False,DirichletCondition[u[x,y,0.35]==0.35 +7.13861 Plus[<<2>>] y,x==0.],DirichletCondition[u[x,y,0.35]==0.,v[x,y,0.35]==0.,0<x<2],DirichletCondition[w[x,y,0.35]==0.,x==2],u,v,w,x,y,0.35[Element]ImplicitRegion[-0.5<=z<=0.5&&0<=x<=2&&0<=y<=0.5&&!(x>=1&&y<=0.1)&&!(x>=1&&y>=0.4),x,y,z],Method->FiniteElement,InterpolationOrder->u->2,v->2,w->1] are not the same shape.


    I am not sure what I am doing wrong when trying to extend to 3D.






    differential-equations







    share|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      I am trying to extend stokes flow example to 3D but I get an error. Not sure what's wrong.



      For example we define the region and this works:



      [CapitalOmega] = 
       ImplicitRegion[-0.5 <= z <= 0.5 && 0 <= x <= 2 && 
       0 <= y <= 
       0.5 && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, 
       z];
      RegionPlot3D[[CapitalOmega]]


      Then we define the stokes flow operator,I am not sure this is right: 



      stokesFlowOperator = Inactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       u[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["1", ",", "0"], ")"],
      Derivative],
      MultilineFunction->None])[x, y], 
       Inactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       v[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z], nactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       v[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z],

      !(*SuperscriptBox[(v), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1", ",", "0"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z] + 

      !(*SuperscriptBox[(u), 
      TagBox[
      RowBox["(", 
      RowBox["1", ",", "0", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z];
      Subscript[[CapitalGamma], 
       D] = DirichletCondition[
       u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.], 
       DirichletCondition[u[x, y, z] == 0., v[x, y, z] == 0., 
       0 < x < 2], DirichletCondition[w[x, y, z] == 0., x == 2];


      This lines fails:



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


      And I get this error:



      NDSolveValue::dsvar: 0.35` cannot be used as a variable.
      Set::shape: Lists xVel,yVel,zVel,pressure and NDSolveValue[False,DirichletCondition[u[x,y,0.35]==0.35 +7.13861 Plus[<<2>>] y,x==0.],DirichletCondition[u[x,y,0.35]==0.,v[x,y,0.35]==0.,0<x<2],DirichletCondition[w[x,y,0.35]==0.,x==2],u,v,w,x,y,0.35[Element]ImplicitRegion[-0.5<=z<=0.5&&0<=x<=2&&0<=y<=0.5&&!(x>=1&&y<=0.1)&&!(x>=1&&y>=0.4),x,y,z],Method->FiniteElement,InterpolationOrder->u->2,v->2,w->1] are not the same shape.


      I am not sure what I am doing wrong when trying to extend to 3D.






      differential-equations







      share|improve this question













      I am trying to extend stokes flow example to 3D but I get an error. Not sure what's wrong.



      For example we define the region and this works:



      [CapitalOmega] = 
       ImplicitRegion[-0.5 <= z <= 0.5 && 0 <= x <= 2 && 
       0 <= y <= 
       0.5 && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, 
       z];
      RegionPlot3D[[CapitalOmega]]


      Then we define the stokes flow operator,I am not sure this is right: 



      stokesFlowOperator = Inactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       u[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["1", ",", "0"], ")"],
      Derivative],
      MultilineFunction->None])[x, y], 
       Inactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       v[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z], nactive[

       Div][(-1, 0, 0, -1.Inactive[Grad][
       v[x, y, z], x, y, z]), x, 
       y, z] + 

      !(*SuperscriptBox[(w), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z],

      !(*SuperscriptBox[(v), 
      TagBox[
      RowBox["(", 
      RowBox["0", ",", "1", ",", "0"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z] + 

      !(*SuperscriptBox[(u), 
      TagBox[
      RowBox["(", 
      RowBox["1", ",", "0", ",", "1"], ")"],
      Derivative],
      MultilineFunction->None])[x, y, z];
      Subscript[[CapitalGamma], 
       D] = DirichletCondition[
       u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.], 
       DirichletCondition[u[x, y, z] == 0., v[x, y, z] == 0., 
       0 < x < 2], DirichletCondition[w[x, y, z] == 0., x == 2];


      This lines fails:



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


      And I get this error:



      NDSolveValue::dsvar: 0.35` cannot be used as a variable.
      Set::shape: Lists xVel,yVel,zVel,pressure and NDSolveValue[False,DirichletCondition[u[x,y,0.35]==0.35 +7.13861 Plus[<<2>>] y,x==0.],DirichletCondition[u[x,y,0.35]==0.,v[x,y,0.35]==0.,0<x<2],DirichletCondition[w[x,y,0.35]==0.,x==2],u,v,w,x,y,0.35[Element]ImplicitRegion[-0.5<=z<=0.5&&0<=x<=2&&0<=y<=0.5&&!(x>=1&&y<=0.1)&&!(x>=1&&y>=0.4),x,y,z],Method->FiniteElement,InterpolationOrder->u->2,v->2,w->1] are not the same shape.


      I am not sure what I am doing wrong when trying to extend to 3D.






      differential-equations




      differential-equations






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          2 Answers
          2






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          up vote
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          These are not the Stokes equations. You must have built in several errors when translating the 2D equations into 3D. The following works although it complains because there are no boundary conditions for the pressure that would make it unique. Also, I am don't know at all whether these are the boundary conditions that you would like to apply.



          a = IdentityMatrix[3];
          stokesFlowOperator = 
           Inactive[Div][a.Inactive[Grad][u[x, y, z], x, y, z], x, y, z] -
           D[p[x, y, z], x],
           Inactive[Div][a.Inactive[Grad][v[x, y, z], x, y, z], x, y, z] -
           D[p[x, y, z], y],
           Inactive[Div][a.Inactive[Grad][w[x, y, z], x, y, z], x, y, z] -
           D[p[x, y, z], z],
           Div[u[x, y, z], v[x, y, z], w[x, y, z], x, y, z]
           ;
          ΓD = 
           DirichletCondition[u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.],
           DirichletCondition[u[x, y, z] == 0., 0 < x < 2],
           DirichletCondition[v[x, y, z] == 0., 0 < x < 2],
           DirichletCondition[w[x, y, z] == 0., x == 2]
           ;


          xVel, yVel, zVel, pressure = NDSolveValue[
           
           stokesFlowOperator == 0, 0, 0, 0,
           ΓD
           ,
           u, v, w, p,
           x, y, z ∈ Ω,
           Method -> "FiniteElement", 
           "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1
           ]





          share|improve this answer





























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            down vote













            Totally wrong. I will show an example with a Poiseuille profile at the inlet.



            H = 1/2; L = 2;
            [CapitalOmega] = 
             ImplicitRegion[
             0 <= z <= H && 0 <= x <= L && 
             0 <= y <= 
             H && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, z];
            RegionPlot3D[[CapitalOmega]]
            Um = 45/100; [Nu] = 1;
            U0[y_, z_] := 16*Um*y*z*(H - y)*(H - z)/H^4

            U, V, W, P = 
             NDSolveValue[-[Nu]*Laplacian[u[x, y, z], x, y, z] + 
            !(*SuperscriptBox[(p), 
            TagBox[
            RowBox["(", 
            RowBox["1", ",", "0", ",", "0"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z], -[Nu]*
             Laplacian[v[x, y, z], x, y, z] + 
            !(*SuperscriptBox[(p), 
            TagBox[
            RowBox["(", 
            RowBox["0", ",", "1", ",", "0"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z], -[Nu]*
             Laplacian[w[x, y, z], x, y, z] + 
            !(*SuperscriptBox[(p), 
            TagBox[
            RowBox["(", 
            RowBox["0", ",", "0", ",", "1"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z], 
            !(*SuperscriptBox[(u), 
            TagBox[
            RowBox["(", 
            RowBox["1", ",", "0", ",", "0"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z] + 
            !(*SuperscriptBox[(v), 
            TagBox[
            RowBox["(", 
            RowBox["0", ",", "1", ",", "0"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z] + 
            !(*SuperscriptBox[(w), 
            TagBox[
            RowBox["(", 
            RowBox["0", ",", "0", ",", "1"], ")"],
            Derivative],
            MultilineFunction->None])[x, y, z] == 0, 0, 0, 0, 
             DirichletCondition[u[x, y, z] == U0[y, z], v[x, y, z] == 0, 
             w[x, y, z] == 0, x == 0], 
             DirichletCondition[u[x, y, z] == 0, v[x, y, z] == 0, 
             w[x, y, z] == 0, 0 < x < L],
             DirichletCondition[p[x, y, z] == 0, x == L], u, v, w, 
             p, x, y, z [Element] [CapitalOmega], 
             Method -> "FiniteElement", 
             "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1, 
             "MeshOptions" -> "MaxCellMeasure" -> 
             0.0001]; 
            ContourPlot[U[x, y, H/2], x, 0, L, y, 0, H, 
             AspectRatio -> Automatic, Contours -> 20, PlotPoints -> 50]


            fig1






            share|improve this answer




















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              2 Answers
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              2 Answers
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              up vote
              3
              down vote













              These are not the Stokes equations. You must have built in several errors when translating the 2D equations into 3D. The following works although it complains because there are no boundary conditions for the pressure that would make it unique. Also, I am don't know at all whether these are the boundary conditions that you would like to apply.



              a = IdentityMatrix[3];
              stokesFlowOperator = 
               Inactive[Div][a.Inactive[Grad][u[x, y, z], x, y, z], x, y, z] -
               D[p[x, y, z], x],
               Inactive[Div][a.Inactive[Grad][v[x, y, z], x, y, z], x, y, z] -
               D[p[x, y, z], y],
               Inactive[Div][a.Inactive[Grad][w[x, y, z], x, y, z], x, y, z] -
               D[p[x, y, z], z],
               Div[u[x, y, z], v[x, y, z], w[x, y, z], x, y, z]
               ;
              ΓD = 
               DirichletCondition[u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.],
               DirichletCondition[u[x, y, z] == 0., 0 < x < 2],
               DirichletCondition[v[x, y, z] == 0., 0 < x < 2],
               DirichletCondition[w[x, y, z] == 0., x == 2]
               ;


              xVel, yVel, zVel, pressure = NDSolveValue[
               
               stokesFlowOperator == 0, 0, 0, 0,
               ΓD
               ,
               u, v, w, p,
               x, y, z ∈ Ω,
               Method -> "FiniteElement", 
               "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1
               ]





              share|improve this answer


























                up vote
                3
                down vote













                These are not the Stokes equations. You must have built in several errors when translating the 2D equations into 3D. The following works although it complains because there are no boundary conditions for the pressure that would make it unique. Also, I am don't know at all whether these are the boundary conditions that you would like to apply.



                a = IdentityMatrix[3];
                stokesFlowOperator = 
                 Inactive[Div][a.Inactive[Grad][u[x, y, z], x, y, z], x, y, z] -
                 D[p[x, y, z], x],
                 Inactive[Div][a.Inactive[Grad][v[x, y, z], x, y, z], x, y, z] -
                 D[p[x, y, z], y],
                 Inactive[Div][a.Inactive[Grad][w[x, y, z], x, y, z], x, y, z] -
                 D[p[x, y, z], z],
                 Div[u[x, y, z], v[x, y, z], w[x, y, z], x, y, z]
                 ;
                ΓD = 
                 DirichletCondition[u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.],
                 DirichletCondition[u[x, y, z] == 0., 0 < x < 2],
                 DirichletCondition[v[x, y, z] == 0., 0 < x < 2],
                 DirichletCondition[w[x, y, z] == 0., x == 2]
                 ;


                xVel, yVel, zVel, pressure = NDSolveValue[
                 
                 stokesFlowOperator == 0, 0, 0, 0,
                 ΓD
                 ,
                 u, v, w, p,
                 x, y, z ∈ Ω,
                 Method -> "FiniteElement", 
                 "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1
                 ]





                share|improve this answer
























                  up vote
                  3
                  down vote










                  up vote
                  3
                  down vote









                  These are not the Stokes equations. You must have built in several errors when translating the 2D equations into 3D. The following works although it complains because there are no boundary conditions for the pressure that would make it unique. Also, I am don't know at all whether these are the boundary conditions that you would like to apply.



                  a = IdentityMatrix[3];
                  stokesFlowOperator = 
                   Inactive[Div][a.Inactive[Grad][u[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], x],
                   Inactive[Div][a.Inactive[Grad][v[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], y],
                   Inactive[Div][a.Inactive[Grad][w[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], z],
                   Div[u[x, y, z], v[x, y, z], w[x, y, z], x, y, z]
                   ;
                  ΓD = 
                   DirichletCondition[u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.],
                   DirichletCondition[u[x, y, z] == 0., 0 < x < 2],
                   DirichletCondition[v[x, y, z] == 0., 0 < x < 2],
                   DirichletCondition[w[x, y, z] == 0., x == 2]
                   ;


                  xVel, yVel, zVel, pressure = NDSolveValue[
                   
                   stokesFlowOperator == 0, 0, 0, 0,
                   ΓD
                   ,
                   u, v, w, p,
                   x, y, z ∈ Ω,
                   Method -> "FiniteElement", 
                   "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1
                   ]





                  share|improve this answer














                  These are not the Stokes equations. You must have built in several errors when translating the 2D equations into 3D. The following works although it complains because there are no boundary conditions for the pressure that would make it unique. Also, I am don't know at all whether these are the boundary conditions that you would like to apply.



                  a = IdentityMatrix[3];
                  stokesFlowOperator = 
                   Inactive[Div][a.Inactive[Grad][u[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], x],
                   Inactive[Div][a.Inactive[Grad][v[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], y],
                   Inactive[Div][a.Inactive[Grad][w[x, y, z], x, y, z], x, y, z] -
                   D[p[x, y, z], z],
                   Div[u[x, y, z], v[x, y, z], w[x, y, z], x, y, z]
                   ;
                  ΓD = 
                   DirichletCondition[u[x, y, z] == z + 4*0.3*y*(0.5 - y)/(0.41)^2, x == 0.],
                   DirichletCondition[u[x, y, z] == 0., 0 < x < 2],
                   DirichletCondition[v[x, y, z] == 0., 0 < x < 2],
                   DirichletCondition[w[x, y, z] == 0., x == 2]
                   ;


                  xVel, yVel, zVel, pressure = NDSolveValue[
                   
                   stokesFlowOperator == 0, 0, 0, 0,
                   ΓD
                   ,
                   u, v, w, p,
                   x, y, z ∈ Ω,
                   Method -> "FiniteElement", 
                   "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1
                   ]






                  share|improve this answer














                  share|improve this answer



                  share|improve this answer








                  edited 13 mins ago

























                  answered 1 hour ago









                  Henrik Schumacher

                  40.5k256121




                  40.5k256121




















                      up vote
                      2
                      down vote













                      Totally wrong. I will show an example with a Poiseuille profile at the inlet.



                      H = 1/2; L = 2;
                      [CapitalOmega] = 
                       ImplicitRegion[
                       0 <= z <= H && 0 <= x <= L && 
                       0 <= y <= 
                       H && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, z];
                      RegionPlot3D[[CapitalOmega]]
                      Um = 45/100; [Nu] = 1;
                      U0[y_, z_] := 16*Um*y*z*(H - y)*(H - z)/H^4

                      U, V, W, P = 
                       NDSolveValue[-[Nu]*Laplacian[u[x, y, z], x, y, z] + 
                      !(*SuperscriptBox[(p), 
                      TagBox[
                      RowBox["(", 
                      RowBox["1", ",", "0", ",", "0"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z], -[Nu]*
                       Laplacian[v[x, y, z], x, y, z] + 
                      !(*SuperscriptBox[(p), 
                      TagBox[
                      RowBox["(", 
                      RowBox["0", ",", "1", ",", "0"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z], -[Nu]*
                       Laplacian[w[x, y, z], x, y, z] + 
                      !(*SuperscriptBox[(p), 
                      TagBox[
                      RowBox["(", 
                      RowBox["0", ",", "0", ",", "1"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z], 
                      !(*SuperscriptBox[(u), 
                      TagBox[
                      RowBox["(", 
                      RowBox["1", ",", "0", ",", "0"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z] + 
                      !(*SuperscriptBox[(v), 
                      TagBox[
                      RowBox["(", 
                      RowBox["0", ",", "1", ",", "0"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z] + 
                      !(*SuperscriptBox[(w), 
                      TagBox[
                      RowBox["(", 
                      RowBox["0", ",", "0", ",", "1"], ")"],
                      Derivative],
                      MultilineFunction->None])[x, y, z] == 0, 0, 0, 0, 
                       DirichletCondition[u[x, y, z] == U0[y, z], v[x, y, z] == 0, 
                       w[x, y, z] == 0, x == 0], 
                       DirichletCondition[u[x, y, z] == 0, v[x, y, z] == 0, 
                       w[x, y, z] == 0, 0 < x < L],
                       DirichletCondition[p[x, y, z] == 0, x == L], u, v, w, 
                       p, x, y, z [Element] [CapitalOmega], 
                       Method -> "FiniteElement", 
                       "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1, 
                       "MeshOptions" -> "MaxCellMeasure" -> 
                       0.0001]; 
                      ContourPlot[U[x, y, H/2], x, 0, L, y, 0, H, 
                       AspectRatio -> Automatic, Contours -> 20, PlotPoints -> 50]


                      fig1






                      share|improve this answer
























                        up vote
                        2
                        down vote













                        Totally wrong. I will show an example with a Poiseuille profile at the inlet.



                        H = 1/2; L = 2;
                        [CapitalOmega] = 
                         ImplicitRegion[
                         0 <= z <= H && 0 <= x <= L && 
                         0 <= y <= 
                         H && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, z];
                        RegionPlot3D[[CapitalOmega]]
                        Um = 45/100; [Nu] = 1;
                        U0[y_, z_] := 16*Um*y*z*(H - y)*(H - z)/H^4

                        U, V, W, P = 
                         NDSolveValue[-[Nu]*Laplacian[u[x, y, z], x, y, z] + 
                        !(*SuperscriptBox[(p), 
                        TagBox[
                        RowBox["(", 
                        RowBox["1", ",", "0", ",", "0"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z], -[Nu]*
                         Laplacian[v[x, y, z], x, y, z] + 
                        !(*SuperscriptBox[(p), 
                        TagBox[
                        RowBox["(", 
                        RowBox["0", ",", "1", ",", "0"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z], -[Nu]*
                         Laplacian[w[x, y, z], x, y, z] + 
                        !(*SuperscriptBox[(p), 
                        TagBox[
                        RowBox["(", 
                        RowBox["0", ",", "0", ",", "1"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z], 
                        !(*SuperscriptBox[(u), 
                        TagBox[
                        RowBox["(", 
                        RowBox["1", ",", "0", ",", "0"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z] + 
                        !(*SuperscriptBox[(v), 
                        TagBox[
                        RowBox["(", 
                        RowBox["0", ",", "1", ",", "0"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z] + 
                        !(*SuperscriptBox[(w), 
                        TagBox[
                        RowBox["(", 
                        RowBox["0", ",", "0", ",", "1"], ")"],
                        Derivative],
                        MultilineFunction->None])[x, y, z] == 0, 0, 0, 0, 
                         DirichletCondition[u[x, y, z] == U0[y, z], v[x, y, z] == 0, 
                         w[x, y, z] == 0, x == 0], 
                         DirichletCondition[u[x, y, z] == 0, v[x, y, z] == 0, 
                         w[x, y, z] == 0, 0 < x < L],
                         DirichletCondition[p[x, y, z] == 0, x == L], u, v, w, 
                         p, x, y, z [Element] [CapitalOmega], 
                         Method -> "FiniteElement", 
                         "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1, 
                         "MeshOptions" -> "MaxCellMeasure" -> 
                         0.0001]; 
                        ContourPlot[U[x, y, H/2], x, 0, L, y, 0, H, 
                         AspectRatio -> Automatic, Contours -> 20, PlotPoints -> 50]


                        fig1






                        share|improve this answer






















                          up vote
                          2
                          down vote










                          up vote
                          2
                          down vote









                          Totally wrong. I will show an example with a Poiseuille profile at the inlet.



                          H = 1/2; L = 2;
                          [CapitalOmega] = 
                           ImplicitRegion[
                           0 <= z <= H && 0 <= x <= L && 
                           0 <= y <= 
                           H && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, z];
                          RegionPlot3D[[CapitalOmega]]
                          Um = 45/100; [Nu] = 1;
                          U0[y_, z_] := 16*Um*y*z*(H - y)*(H - z)/H^4

                          U, V, W, P = 
                           NDSolveValue[-[Nu]*Laplacian[u[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["1", ",", "0", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], -[Nu]*
                           Laplacian[v[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "1", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], -[Nu]*
                           Laplacian[w[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "0", ",", "1"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], 
                          !(*SuperscriptBox[(u), 
                          TagBox[
                          RowBox["(", 
                          RowBox["1", ",", "0", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] + 
                          !(*SuperscriptBox[(v), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "1", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] + 
                          !(*SuperscriptBox[(w), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "0", ",", "1"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] == 0, 0, 0, 0, 
                           DirichletCondition[u[x, y, z] == U0[y, z], v[x, y, z] == 0, 
                           w[x, y, z] == 0, x == 0], 
                           DirichletCondition[u[x, y, z] == 0, v[x, y, z] == 0, 
                           w[x, y, z] == 0, 0 < x < L],
                           DirichletCondition[p[x, y, z] == 0, x == L], u, v, w, 
                           p, x, y, z [Element] [CapitalOmega], 
                           Method -> "FiniteElement", 
                           "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1, 
                           "MeshOptions" -> "MaxCellMeasure" -> 
                           0.0001]; 
                          ContourPlot[U[x, y, H/2], x, 0, L, y, 0, H, 
                           AspectRatio -> Automatic, Contours -> 20, PlotPoints -> 50]


                          fig1






                          share|improve this answer












                          Totally wrong. I will show an example with a Poiseuille profile at the inlet.



                          H = 1/2; L = 2;
                          [CapitalOmega] = 
                           ImplicitRegion[
                           0 <= z <= H && 0 <= x <= L && 
                           0 <= y <= 
                           H && ! (x >= 1 && y <= 0.1) && ! (x >= 1 && y >= 0.4), x, y, z];
                          RegionPlot3D[[CapitalOmega]]
                          Um = 45/100; [Nu] = 1;
                          U0[y_, z_] := 16*Um*y*z*(H - y)*(H - z)/H^4

                          U, V, W, P = 
                           NDSolveValue[-[Nu]*Laplacian[u[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["1", ",", "0", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], -[Nu]*
                           Laplacian[v[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "1", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], -[Nu]*
                           Laplacian[w[x, y, z], x, y, z] + 
                          !(*SuperscriptBox[(p), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "0", ",", "1"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z], 
                          !(*SuperscriptBox[(u), 
                          TagBox[
                          RowBox["(", 
                          RowBox["1", ",", "0", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] + 
                          !(*SuperscriptBox[(v), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "1", ",", "0"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] + 
                          !(*SuperscriptBox[(w), 
                          TagBox[
                          RowBox["(", 
                          RowBox["0", ",", "0", ",", "1"], ")"],
                          Derivative],
                          MultilineFunction->None])[x, y, z] == 0, 0, 0, 0, 
                           DirichletCondition[u[x, y, z] == U0[y, z], v[x, y, z] == 0, 
                           w[x, y, z] == 0, x == 0], 
                           DirichletCondition[u[x, y, z] == 0, v[x, y, z] == 0, 
                           w[x, y, z] == 0, 0 < x < L],
                           DirichletCondition[p[x, y, z] == 0, x == L], u, v, w, 
                           p, x, y, z [Element] [CapitalOmega], 
                           Method -> "FiniteElement", 
                           "InterpolationOrder" -> u -> 2, v -> 2, w -> 2, p -> 1, 
                           "MeshOptions" -> "MaxCellMeasure" -> 
                           0.0001]; 
                          ContourPlot[U[x, y, H/2], x, 0, L, y, 0, H, 
                           AspectRatio -> Automatic, Contours -> 20, PlotPoints -> 50]


                          fig1







                          share|improve this answer












                          share|improve this answer



                          share|improve this answer










                          answered 1 hour ago









                          Alex Trounev

                          2,655312




                          2,655312



























                               

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