= Solution
Let $t(n)$ denote <traction>: the force per unit area exerted across an oriented internal surface with unit normal $n$. Define $\sigma_{ij}$ as component $i$ of the <traction> on a coordinate plane whose normal is $e_j$. <Balance of linear momentum> on a shrinking tetrahedron makes the surface-force terms, of order area, dominate its inertia and <body forces>, of order volume. Resolving its oblique face into the three coordinate projected areas gives Cauchy's <traction> formula
$$
\boxed{t(n)=\sigma n,\qquad t_i(n)=\sigma_{ij}n_j.}
$$
Opposite faces have opposite <tractions>, as follows independently from a shrinking pillbox. This fixes both the index and normal conventions.
For a <material volume> $V(t)$, the <Reynolds transport theorem> says $d\int_{V(t)}f\,dV/dt=\int_{V(t)}[f_t+\nabla\cdot(fu)]dV$. Apply it componentwise to $f=\rho u_i$ and use <mass conservation>. With no <body force> in this calculation, <balance of linear momentum> and the <divergence theorem> give
$$
\int_{V(t)}\rho\frac{Du_i}{Dt}\,dV
=\int_{\partial V(t)}\sigma_{ij}n_j\,dS
=\int_{V(t)}\partial_j\sigma_{ij}\,dV.
$$
Arbitrariness of the <material volume> yields $\boxed{\rho Du_i/Dt=\partial_j\sigma_{ij}}$, with $D/Dt=\partial_t+u_j\partial_j$.
For constant <dynamic viscosity> and incompressible <velocity>, $\partial_ju_j=0$, so
$$
\partial_j\sigma_{ij}=-\partial_ip+\mu\partial_j\partial_ju_i+
\mu\partial_i(\partial_ju_j)=-\partial_ip+\mu\nabla^2u_i.
$$
Thus $\boxed{\rho(u_t+u\cdot\nabla u)=-\nabla p+\mu\nabla^2u}$.
In fully developed steady <pipe flow> $u=w(x,y)e_z$, the convective acceleration vanishes. The transverse equations give $p_x=p_y=0$ and the axial equation gives $p_z=\mu(w_{xx}+w_{yy})$. One side depends only on $z$ and the other only on $(x,y)$, so both equal a constant. Hence
$$
\boxed{w_{xx}+w_{yy}=p_z/\mu,\qquad p_z=\text{constant}.}
$$
If $n$ is outward from the fluid cross-section, the wall-on-fluid axial shear per length is $\mu\oint\partial_nw\,ds=\mu\int_A\Delta w\,dA=Ap_z$. The fluid-on-pipe force is its negative:
$$
\boxed{\text{drag per unit length}=-Ap_z.}
$$
For forward <pressure>-driven flow $p_z<0$, this force points along the flow. Its value depends on <cross-sectional area> and <pressure> drop, even though the detailed <velocity> profile depends on the shape.
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