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,L]3[0,L]^3[0,L]3 with Dirichlet boundary conditions. The velocity field u(x,y,z,t)=(u,v,w)\mathbf{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)\lambda_{nml} = \pi^2/L^2 (n^2 + m^2 + l^2)λnml=π2/L2(n2+m2+l2). The analytical solution explicitly describes the initial acceleration (“velocity explosion”) during the force application (0≤t≤t00 \le t \le t_00≤t≤t0) and the subsequent exponential decay for t>t0t > t_0t>t0. A Python simulation accompanies the analytical derivation, computing the velocity modulus ∣u(x,y,z,t)∣|\mathbf{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.