---
title: "Navier-Stokes and Bernoulli equations with terms"
subject: "Fluid Mechanics"
chapter: 5
key: "semester_3/fluid_mechanics:navier-stokes-bernoulli-terms"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/navier-stokes-bernoulli-terms
site: BCT Engineering
---

# Navier-Stokes and Bernoulli equations with terms

Fluid Mechanics · Chapter 5: Hydrodynamics

- Asked in: 2078 Kartik, 4 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_3/fluid_mechanics/navier-stokes-bernoulli-terms
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-5

## Question

Write the Navier-Stokes and Bernoulli's equation (derivation not required). Explain each terms in the equations with physical meaning.

## Short answer

Navier-Stokes terms: local and convective acceleration, body force, pressure gradient, viscous force per unit volume. Bernoulli’s terms: pressure head p/γ, velocity head V²/2g, elevation z sum to constant total head along a streamline for steady inviscid incompressible flow.

## Answer

#### Navier-Stokes equation

For an incompressible Newtonian fluid with constant viscosity, in vector form:

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

x-component in Cartesian form:

$$
\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) = \rho g_x - \frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}\right)
$$

(The y and z equations are similar.) Together with continuity, $\nabla\cdot\vec V = 0$, it describes the flow.

| Term | Physical meaning |
|---|---|
| $\rho\,\partial\vec V/\partial t$ | Local (unsteady) inertia force per unit volume |
| $\rho(\vec V\cdot\nabla)\vec V$ | Convective inertia force per unit volume (change of velocity with position) |
| $\rho\vec g$ | Body (gravity) force per unit volume |
| $-\nabla p$ | Pressure force per unit volume |
| $\mu\nabla^2\vec V$ | Viscous force per unit volume |

#### Bernoulli's equation

For steady, incompressible, frictionless flow along a streamline:

$$
\frac{p}{\gamma} + \frac{V^2}{2g} + z = \text{constant}
$$

| Term | Meaning |
|---|---|
| $p/\gamma$ | Pressure head (flow work per unit weight) |
| $V^2/2g$ | Velocity (kinetic) head |
| $z$ | Elevation (potential) head |
| Sum | Total head, i.e. total energy per unit weight, constant along the streamline |

Each term has the unit of length (metres of fluid). Multiplying by $\gamma$ gives the pressure form, static + dynamic + hydrostatic pressure: $p + \tfrac12\rho V^2 + \gamma z = $ constant.

## Check yourself

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

1. Which term in Navier-Stokes represents viscous force per unit volume?
   A. -del p
   B. rho g
   C. mu del^2 V
   D. rho dV/dt
   Answer: C. mu del^2 V is the viscous force. (Key point: Viscous term)

2. What does rho (V.del)V represent?
   A. Unsteady inertia
   B. Pressure force
   C. Body force
   D. Convective inertia force due to change of velocity with position
   Answer: D. Local part is rho dV/dt; convective part arises from position change. (Key point: Convective term)

3. Bernoulli p/gamma + V^2/2g + z = const applies to:
   A. Steady, incompressible, frictionless flow along a streamline
   B. Unsteady viscous flow
   C. Across streamlines in any flow
   D. Compressible flow with friction
   Answer: A. These are the Bernoulli assumptions. (Key point: Bernoulli assumptions)

4. What is V^2/2g?
   A. Pressure head
   B. Velocity (kinetic) head
   C. Elevation head
   D. Friction head
   Answer: B. Each term has length units; V^2/2g is kinetic energy per unit weight. (Key point: Head terms)

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