The app evaluates the steady, incompressible, axisymmetric linearized Euler response to an arbitrary annular axial loading qx(r), with finite Gaussian force width ε.
q̂(k) = ∫₀ᴿ qx(s) J₀(ks) s ds
p(x,r) = ½ ∫₀∞ q̂(k) J₀(kr) Sε(k,x) k dk
u(x,r) = Hε(x) qx(r)/(ρU₀) − p(x,r)/(ρU₀)
v(x,r) = −[2ρU₀]⁻¹ ∫₀∞ q̂(k) J₁(kr) Eε(k,x) k dk
w(x,r) = Hε(x) qθ(r)/(ρU₀)
The Gaussian kernels are evaluated in a numerically stable scaled-complementary-error-function form. At ε → 0, they recover the infinitesimally thin disk kernels.
Scope. The closed field is a linear semi-analytical base solution. Heavy-loading cases may require nonlinear wake-pressure, contraction, and radial-equilibrium corrections.