Pressure‐driven flow in thin straight tubes of non‐uniform cross‐section

We analyze stationary Stokes flow of a Navier–Stokes fluid in a thin tube with a variable cross‐section. The objective is to derive a simplified model by examining the asymptotic behavior as the tube's thickness approaches zero. The flow is driven by a pressure gradient between the inlet and outlet, which is modeled by prescribing the normal component of the stress tensor at the tube's ends. Using multiple‐scale asymptotic expansions, we first obtain an approximate solution. This inner approximation is accurate for thin tubes, except in thin boundary layers near the inlet and outlet. To address this, we refine the approximation by introducing boundary layer correctors. Finally, we rigorously prove an error estimate for the difference between the exact solution and the improved approximate solution.