 Solution

Integrate axial force balance $\nabla \cdot \sigma =0$ over the fluid between the two remote tube sections, excluding the particle. The wall's axial traction contributesAt each remote section the Hagen-Poiseuille flow has $u_x$ independent of $x$, so ${\sigma }_{xx}=-p$. The downstream and upstream caps therefore contribute $-\pi a^{2}p(x_2)+\pi a^{2}p(x_1)$. The particle contributes zero net axial force because it is force-free. Adding the three contributions proves the wall-shear and pressure-drop balance in a tube:Here $p$ is the physical pressure. The sign follows from the outward normals on the two caps; no convention for a positive pressure-drop magnitude has yet been introduced.