---
title: "Navier-Stokes equations for Newtonian fluid"
subject: "Fluid Mechanics"
chapter: 4
key: "semester_4/fluid_mechanics_mechanical:navier-stokes-newtonian-fluid"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/navier-stokes-newtonian-fluid
site: BCT Engineering
---

# Navier-Stokes equations for Newtonian fluid

Fluid Mechanics · Chapter 4: Basic Equations for Fluid Flow

- Asked in: year not given, 8 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_4/fluid_mechanics_mechanical/navier-stokes-newtonian-fluid
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-4

## Question

Explain how the Navier-Stokes equations are obtained for a Newtonian incompressible fluid. Write the x-component, identify each term, state the assumptions, and show how the equation reduces to (i) Euler's equation and (ii) the hydrostatic equation.

## Short answer

The Navier-Stokes equations follow from Newton's second law for a fluid element, with viscous stresses proportional to strain rates. The x-component contains inertia, pressure, gravity, and viscous terms. Setting viscosity to zero gives Euler's equation; setting velocity to zero yields the hydrostatic equation.

## Answer

#### Basis
The Navier-Stokes (N-S) equations are Newton's second law written for a fluid element, with the viscous stresses related to the velocity gradients by Stokes' relations for a Newtonian fluid:

$$
\tau_{xy} = \mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right), \qquad \sigma_{xx} = -p + 2\mu\frac{\partial u}{\partial x}
$$

#### Steps
1. Take a fluid element $dx\,dy\,dz$ with normal and shear stresses on its faces.
2. Net surface force in the $x$-direction $=$ (net normal stress gradient + shear stress gradients) $\times\ dx\,dy\,dz$.
3. Add the body force $\rho g_x\,dx\,dy\,dz$.
4. Equate the sum to mass $\times$ acceleration, $\rho\,\dfrac{Du}{Dt}\,dx\,dy\,dz$.
5. Substitute the Newtonian stress relations, and use incompressible continuity ($\nabla\cdot\vec V = 0$) with constant $\mu$ to simplify.

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

In vector form: $\rho\dfrac{D\vec V}{Dt} = \rho\vec g - \nabla p + \mu\nabla^2\vec V$.

| Term | Meaning |
|---|---|
| $\rho\,\partial u/\partial t$ | Local inertia force (unsteadiness) |
| $\rho(\vec V\cdot\nabla)u$ | Convective inertia force |
| $\rho g_x$ | Body force (gravity) |
| $-\partial p/\partial x$ | Pressure force |
| $\mu\nabla^2u$ | Viscous force |

The y and z components are written in the same way.

#### Assumptions
Newtonian fluid, incompressible flow, constant viscosity and a continuum. The equations are exact under these assumptions: they apply to laminar flow and to the instantaneous velocities in turbulent flow.

#### Reduction
- **Euler's equation:** for an inviscid fluid, $\mu = 0$:

$$
\rho\frac{D\vec V}{Dt} = \rho\vec g - \nabla p
$$

- **Hydrostatic equation:** for a fluid at rest, $\vec V = 0$, and with $z$ upward, $\partial p/\partial z = -\rho g$ and $\partial p/\partial x = \partial p/\partial y = 0$.

The N-S equations have exact solutions only for simple cases, such as laminar flow between parallel plates and in circular pipes (Hagen-Poiseuille flow).

## Check yourself

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

1. The Navier-Stokes equations are obtained by:
   A. applying Newton's second law to a fluid element and relating viscous stresses to velocity gradients through Stokes' law for a Newtonian fluid
   B. applying Bernoulli's equation to a streamline
   C. applying conservation of energy
   D. assuming the fluid is inviscid
   Answer: A. Surface forces plus body force equal mass times acceleration, with Newtonian stress relations substituted. (Key point: Basis of Navier-Stokes)

2. In the x-component, the term mu (d^2u/dx^2 + d^2u/dy^2 + d^2u/dz^2) represents:
   A. local inertia force
   B. convective inertia force
   C. pressure force
   D. viscous force
   Answer: D. mu del^2 u is the viscous force; the left side holds the local and convective inertia terms. (Key point: Terms of the equation)

3. Euler's equation follows from Navier-Stokes by setting:
   A. V = 0
   B. mu = 0 (inviscid fluid)
   C. rho = 0
   D. the pressure gradient to zero
   Answer: B. Removing the viscous term gives rho DV/Dt = rho g - grad p. (Key point: Reduction to Euler's equation)

4. Setting V = 0 in the Navier-Stokes equation gives:
   A. dp/dz = -rho g, dp/dx = dp/dy = 0
   B. dp/dz = rho g
   C. p = constant everywhere
   D. rho DV/Dt = 0 only with viscosity
   Answer: A. With no motion, the inertia and viscous terms vanish, leaving a balance of pressure and gravity. (Key point: Reduction to hydrostatic equation)

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