---
title: "Complete governing equations of fluid movement"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:complete-governing-equations-fluid"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/complete-governing-equations-fluid
site: BCT Engineering
---

# Complete governing equations of fluid movement

Computational Techniques in Civil Engineering · Chapter 5: Finite difference method

- Asked in: 2070 Bhadra, 2 marks
- Video: [Description and Derivation of the Navier-Stokes Equations](https://www.youtube.com/watch?v=NjoMoH51UZc) — LearnMechE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/complete-governing-equations-fluid
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Write down the complete governing equations describing the movement of fluid.

## Short answer

The 1D unsteady Saint-Venant equations: continuity ∂A/∂t+∂Q/∂x=q, momentum ∂Q/∂t+∂(Q²/A)/∂x+gA∂y/∂x−gAS₀+gAS_f=0, with S_f=n²Q|Q|/(A²R^{4/3}). In wide rectangular form: ∂y/∂t+V∂y/∂x+y∂V/∂x=0, ∂V/∂t+V∂V/∂x+g∂y/∂x=g(S₀−S_f).

## Answer

The movement of water in an open channel (1D unsteady flow) is described by the Saint-Venant equations, a pair of equations from conservation of mass and momentum.

**Continuity (conservation of mass):**
$$
\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q
$$
**Momentum (conservation of momentum):**
$$
\underbrace{\frac{\partial Q}{\partial t}}_{\text{local acc.}}+\underbrace{\frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right)}_{\text{convective acc.}}+\underbrace{gA\frac{\partial y}{\partial x}}_{\text{pressure force}}-\underbrace{gAS_0}_{\text{gravity}}+\underbrace{gAS_f}_{\text{friction}}=0
$$
where $A$ = flow area, $Q$ = discharge, $y$ = flow depth, $q$ = lateral inflow per unit length, $S_0$ = bed slope, $S_f=\dfrac{n^2Q|Q|}{A^2R^{4/3}}$ = friction slope (Manning), $g$ = gravity. These are the Saint-Venant equations (1D unsteady open-channel flow).

In terms of velocity $V$ and depth $y$ for a wide rectangular channel:
$$
\frac{\partial y}{\partial t}+V\frac{\partial y}{\partial x}+y\frac{\partial V}{\partial x}=0,\qquad
\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g(S_0-S_f)
$$

Assumptions: one-dimensional flow, hydrostatic pressure, small bed slope, uniform velocity over the section, and friction given by the steady-flow (Manning) formula. The two equations have two unknowns, $Q$ (or $V$) and $y$ (or $A$), as functions of $x$ and $t$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. 1D unsteady open-channel flow is governed by:
   A. Navier-Stokes only
   B. Saint-Venant equations (continuity + momentum)
   C. Bernoulli's equation alone
   D. Darcy's law
   Answer: B. Mass and momentum conservation form the Saint-Venant pair. (Key point: Name)

2. In the momentum equation, gA dy/dx represents:
   A. friction force
   B. pressure force
   C. gravity
   D. convective acceleration
   Answer: B. It is the hydrostatic pressure force term. (Key point: Momentum terms)

3. Which is an assumption behind the equations?
   A. Three-dimensional flow
   B. Hydrostatic pressure and small bed slope
   C. Large bed slope
   D. Friction neglected
   Answer: B. Also one-dimensional flow and steady-flow Manning friction. (Key point: Assumptions)

## Related in Computational Techniques in Civil Engineering

- [Governing equations; forward, backward, central differencing](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Agoverning-eq-forward-backward-central) — Computational Techniques in Civil Engineering, chapter 5
- [Governing equations, kinematic wave, Courant number](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Agoverning-eq-kinematic-courant) — Computational Techniques in Civil Engineering, chapter 5

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