Analytical and Simplified Solutions for Common Flow Problems

TL;DR

The full Navier-Stokes equations are usually too hard to solve directly, but for many civil and water engineering problems — pipe flow, flow between plates, thin sheet flow — smart simplifications make them solvable by hand. You'll learn to reduce the equations using symmetry and steady-state assumptions, derive Hagen-Poiseuille and Couette-Poiseuille flow, and apply them to real design checks like pipe sizing and drainage tube calibration.

1. The Mental Model

Navier-Stokes is one equation trying to describe every possible fluid motion, which is why it's usually unsolvable in closed form. But if you kill enough of the terms — no acceleration, no swirl, only one direction of flow — what's left is often just a simple second-order ODE you can integrate twice. The skill here isn't memorizing formulas; it's knowing which terms honestly vanish for your geometry. Simplification isn't cheating the physics — it's finding the physics that's actually happening.

2. The Core Material

The incompressible Navier-Stokes equation in vector form is:

$$\rho\left(\frac{\partial \vec{u}}{\partial t} + \vec{u}\cdot\nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$$

Every "simplified solution" in this note comes from crossing out terms on the left and right based on physical reasoning about the flow geometry: is it steady? Is it one-directional? Is it fully developed (no change along the flow direction)? Once you strip those terms away, you're left with a balance between pressure, viscosity, and gravity that you can actually integrate.

2.1 Flow Between Parallel Plates (Couette-Poiseuille)

Picture flow between two flat, infinite plates separated by gap $h$ — a decent model for flow in a narrow joint, a lubrication gap, or a thin fracture in rock (relevant to seepage through jointed bedrock under a dam). Assume:

• Steady flow: $\partial u/\partial t = 0$
• Fully developed: $u = u(y)$ only, no variation along $x$
• No flow in $y$ or $z$: $v = w = 0$

Plugging these into the $x$-momentum equation k