RegionCapillary =
      RegionUnion[Rectangle[{0, 0.1}, {0.9, 0.2}],
         Rectangle[{0, 0.8}, {0.9, 0.9}],
         Rectangle[{0.8, 0.2}, {0.9, 0.9}] ];
RegionShell = Rectangle[{0, 0}, {1, 1}];

<< NDSolve`FEM`
StokesOp := {
   -Inactive[Laplacian][ u[x, y], {x, y}] + D[ p[x, y], x],
   -Inactive[Laplacian][ v[x, y], {x, y}] + D[ p[x, y], y],
   (* incompressibility div u = 0 *)
   Div[{u[x, y], v[x, y]}, {x, y}]
   } ;

bcs := {
         DirichletCondition[p[x, y] == 10^3, x == 0 && y > 0.5],
         DirichletCondition[p[x, y] == 0, x == 0 && y < 0.5],
         DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, True]
         };

Clear[xVel, yVel, pressure];
AbsoluteTiming[{xVel, yVel, pressure} =
         NDSolveValue[{StokesOp == {0, 0, 0}, bcs}, {u, v,
     p}, {x, y} [Element] RegionCapillary,

    Method -> {"FiniteElement",
      "InterpolationOrder" -> {u -> 2, v -> 2, p -> 1},
                  "MeshOptions" -> {"MaxCellMeasure" -> 10^-4}}
            ]];

DensityPlot[Sqrt[
        xVel[x, y]^2 + yVel[x, y]^2], {x, y} [Element]
   RegionCapillary,
      PlotRange -> All, PlotLabel -> "velocity magnitude",
      PlotPoints -> 30] // Quiet

em = First@pressure["Coordinates"];
(*em["Wireframe"]*)
mr = MeshRegion[em];

Pe = 10^2;
advectionOp :=
 If[{x, y} [Element] mr,
   Pe {xVel[x, y], yVel[x, y]}.Inactive[Grad][c[x, y], {x, y}], 0] -
  Inactive[Laplacian][c[x, y], {x, y}]

bcs := {
   DirichletCondition[c[x, y] == 0, x == 0 && 0.8 < y < 0.9],
   DirichletCondition[c[x, y] == 1, x > 0]
   };

AbsoluteTiming[
  csol = NDSolveValue[{advectionOp ==
      NeumannValue[0, x == 0 && ((0 < y < 0.8) || (0.9 < y < 1))],
     bcs}, c, {x, y} [Element] RegionShell,
    Method -> {"FiniteElement", "InterpolationOrder" -> {c -> 2},
      "MeshOptions" -> {"MaxCellMeasure" -> 10^-5}}
    ]];
DensityPlot[csol[x, y], {x, y} [Element] RegionShell,
 PlotRange -> All, PlotLabel -> "x velocity", PlotPoints -> 30]

advectionOp :=
     If[{x, y} [Element] mr,
       Pe {xVel[x, y], yVel[x, y]}.Inactive[Grad][c[x, y], {x, y}], 0] -
      Inactive[Laplacian][c[x, y], {x, y}]