FormBox[Cell[], TraditionalForm]Flow Around a Cylinder

    The flow around a cylinder is described by the vector field
        Overscript[v, ⇀](x, y) = U(1 - R^2 (x^2 - y^2)/(x^2 + y^2)^2, R (-2x y)/(x^2 + y^2)^2) defined for (x^2 + y^2)^(1/2) ≥R.
The corresponding pressure for this flow is
        p(x, y) = -U^2/2 (R^4 - 2R^2(x^2 - y^2) + (x^2 + y^2)^2)/(x^2 + y^2)^2
(up to a constant). We can plot the pressure field as follows.

Off[Plot3D :: "plnc"] ;            ... ;                

RowBox[{RowBox[{Cell[p(x,y) = ]}], , -(U^2 (R^4 - 2 R^2 (x^2 - y^2) + (x^2 + y^2)^2))/(2 (x^2 + y^2)^2)}]

[Graphics:../HTMLFiles/index_420.gif]

Note the areas of high pressure just before and after the obstacle; the lowest pressure occurs on the sides where the fluid moves the fastest.
    We can now compute the force that the fluid exerts on the obstacle; Overscript[F, ⇀] = -∲_C p Overscript[n, ⇀] ds. The symmetry of the situation immediately tells us that Overscript[F, ⇀] = 0, and this is easily verified.

x[θ_] := R Cos[θ] ;           &nbs ... sp;     (* Evaluate the integral *)       ⇀ Print[   F    = , F] ;

⇀    F    =  {0, 0}

This somewhat surprising result is caused by the fact that the absence of viscosity is equivalent to the absence of friction; in particular there is no friction between the fluid and the obstacle, and hence no force is exerted.