Potential Flow and d’Alembert’s Paradox

Euler’s equations formulated by the great mathematician Leonard Euler around 1750 is a mathematical model of slightly viscous incompressible fluid flow in the form of two differential equations expressing (i) Newton’s 2nd Law and (ii) incompressibility together with a slip boundary condition stating that the fluid meets a solid impenetrable boundary with vanishingly small friction.

Euler claimed that all of slightly viscous incompressible fluid flow (like water or air at subsonic speeds) was captured in his model, and so all secrets of fluid mechanics  could be discovered simply by solving his equations,  which however Euler had to leave to “future geometers” because he could not do it himself.  

But there was one class of special solutions to Euler’s equations (i)-(ii) as (iii) stationary (time-independent) and (iv) rotation-free (without vortices) solutions.  These were named potential solutions  because the 2d flow velocity (vx,vy) appears as the gradient (du/dx,du/dy,du/dz) of a harmonic function u(x,y,z) satisfying

  • Laplace u = d^2u/dx^2 + d^2u/dy^2 + d^2u/dz^2 = 0 in the flow domain

with a homogeneous Neumann condition at a solid boundary.

Although promising as special solutions of Euler’s equations, potential solution were discovered (by Euler’s adversary d’Alembert) to not represent true physics, because the drag (resistance to motion) of a body moving through a fluid was predicted to be zero, as shown in Example 1 and Example 2 displaying the same high pressure in flow attachment in front as in separation in the back (in red). 

This was referred to d’Alembert’s paradox of bluff body flow presented in 1755. The result was that Euler’s grand idea of describing all of fluid mechanics in his two equations (i) and (ii) was left in the air, and it did not surface until 2008 when it was discovered (by Hoffman and Johnson) that bluff body flow is captured by computational solution of Eulers equation as 

  • potential flow modified by 3d rotational slip separation. 

It was thus shown that potential solutions are unstable at rear separation and there changes into a physical solution of Euler’s equations without high pressure thus with drag (and lift) can as shown in Example 3 and Example 4.