Couette and Poiseuille solutions

Let the plates lie at y = 0 and y = h. The lower plate is stationary and the upper plate moves in the positive x direction at speed U. Assume steady incompressible Newtonian flow with constant viscosity, no streamwise body force, and a fully developed velocity u(y) in the x direction. There is no cross-stream velocity. Under these assumptions the advective acceleration vanishes exactly, because the velocity has no x dependence; this is a geometric cancellation rather than an appeal to small Reynolds number.

The remaining streamwise equation balances the pressure gradient against the second derivative of velocity. Define G = −dp/dx, positive when pressure falls downstream. Integrate twice and impose no-slip at both plates. The result contains a linear wall-driven contribution and a parabolic pressure-driven contribution. Their superposition is valid here because the reduced equation and boundary conditions are linear; it does not imply that arbitrary solutions of the full nonlinear Navier–Stokes equation can be added.

The volumetric flux per unit width is the integral of u across the gap, with units of square meters per second. Multiplying by a physical channel width gives volumetric flow rate only when side-wall effects are negligible. The average speed is this per-width flux divided by h. For pure plane Poiseuille flow, the maximum occurs at the channel midpoint and is one and a half times the average. For pure Couette flow, the average is one-half of the moving-wall speed.

Differentiate the profile to find the shear-stress component tau_xy. It varies linearly with y when G is nonzero, supplying the viscous-force density needed to balance the constant pressure gradient. Remember that traction also depends on surface orientation. On the fluid domain, the outward normals at the upper and lower boundaries point in opposite directions. Consequently the same positive stress component would correspond to opposite streamwise tractions on those two surfaces. This sign check prevents confusing stress components with forces on walls.

A circular pipe has a different transverse differential operator and a different profile. For stationary walls and radius R, cylindrical Poiseuille flow has u(r) = G(R² − r²)/(4mu), total flow rate pi G R⁴/(8mu), and maximum speed twice the cross-sectional mean. Do not insert a pipe diameter into the plane-channel formula. The numerical factors encode geometry and boundary conditions. In a finite-width rectangular duct, both transverse directions matter unless the width is large compared with the plate separation.

Fully developed means the streamwise velocity profile has stopped changing along the channel, not that the fluid stopped moving or that pressure stopped decreasing. Near an inlet, acceleration and development invalidate the simple reduction. At sufficiently strong disturbances the assumed laminar state may also be inappropriate, even though its algebra remains an exact solution of the ideal equations. Temperature-dependent viscosity, non-Newtonian rheology, slip, and significant edge effects require revisions. Use the exact profile as a benchmark for a stated regime, then measure where the real system departs.