3D Navier–Stokes Exact Solution with Dirichlet Boundary Conditions: Analytical Formulas and Python Implementation

This work presents an exact analytical solution of the three-dimensional incompressible Navier–Stokes equations in a cubic domain 0, L30, L³0, L3 with Dirichlet boundary conditions. The velocity field u (x, y, z, t) = (u, v, w) u (x, y, z, t) = (u, v, w) u (x, y, z, t) = (u, v, w) responds to a short impulsive external force f (x, y, z, t) f (x, y, z, t) f (x, y, z, t) and decays smoothly afterward. Using a Dirichlet sine series expansion, each mode satisfies a linear ordinary differential equation with Laplacian eigenvalues λnml=π2/L2 (n2+m2+l2) ₍₌₋ = ²/L² (n² + m² + l²) λnml=π2/L2 (n2+m2+l2). The analytical solution explicitly describes the initial acceleration (“velocity explosion”) during the force application (0≤t≤t00 t t₀0≤t≤t0) and the subsequent exponential decay for t>t0t > t₀t>t0. A Python simulation accompanies the analytical derivation, computing the velocity modulus ∣u (x, y, z, t) ∣|u (x, y, z, t) |∣u (x, y, z, t) ∣ on a discrete grid, illustrating the dynamic response and decay of the flow. This model provides a fully reproducible framework suitable for educational demonstrations, computational validation, and further analytical studies of 3D fluid dynamics. Keywords: 3D Navier–Stokes, Dirichlet boundary conditions, analytical solution, impulsive forcing, sine series, incompressible flow, Python simulation.