The Navier-Stokes Rosetta Stone
Published 2023-12-13
Introduction
The Rosetta stone is an important rock that helped us understand Egyptian hieroglyphics. I know that I certainly find it easier to understand a notation when it’s written beside a notation I already know. And I also know that sometimes, understanding an equation is as hard as understanding Egyptian hieroglyphics.
In this post, I will give several important equations in fluid mechanics in three forms: vector calculus notation, Einstein notation, and with the Cartesian components written out. I start with the more general Cauchy equation and explain when we make certain assumptions to get the compressible and incompressible Navier-Stokes equations.
The general roadmap is:
- Define: tensors, operations, material derivative, stress tensor, viscous stress tensor.
- Continuity equation: compressible, then incompressible ($\rho$ is constant).
- Momentum equation:
- Cauchy’s momentum equation for a general continuum,
- Expand it in terms of isotropic/deviatoric stresses.
- Apply Newtonian fluid to get compressible Navier-Stokes.
- Apply constant density to 3. to get incompressible Navier-Stokes.
References I used to compile this post:
- My favourite fluid mechanics book: Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, McGraw Hill, 4th Edition
- Pope, Turbulent Flows, Cambridge University Press.
- Wikipedia Page on Navier-Stokes Equations
Prerequisites
We typically work with tensors up to rank 2 in fluids. A rank 0 tensor is a scalar. A rank 1 tensor is a vector. A rank 2 tensor can be visualized as a matrix. A tensor is a geometric object - the fact that something is a tensor allows us to define how it transforms. It’s a physical object. For example, a scalar is independent of the coordinate system - you can’t rotate a scalar. But you can rotate a vector or rank 2 tensor.
| Vector (rank 1 tensor) | Velocity vector | Matrix (rank 2 tensor) | Identity matrix |
|---|---|---|---|
| $\displaystyle \vec{a}$ | $\displaystyle \vec{V}$ | $\displaystyle A$ | $\displaystyle I$ |
| $\displaystyle a_i$ | $\displaystyle v_i$ | $\displaystyle A_{ij}$ | $\displaystyle \delta_{ij}$ |
| $\displaystyle \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}$ | $\displaystyle \begin{bmatrix} u \\ v \\ w \end{bmatrix}$ | $\displaystyle \begin{bmatrix} A_{xx} & A_{xy} & A_{xz} \\ A_{yx} & A_{yy} & A_{yz} \\ A_{zx} & A_{zy} & A_{zz} \end{bmatrix}$ | $\displaystyle \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ |
Some operations between tensors are defined, whereas other are not. While the tensors might be the words and letters of fluid mechanics, the operations are the sentence structure, grammar, spelling, and punctuation. An important set of operations involves a tensor and the $\nabla$ operator. If we take the divergence of a tensor, we reduce its rank by 1. If we take the gradient of a tensor, we increase its rank by 1. For example, the gradient of a scalar is a vector - its a vector who’s magnitude indicates the rate of change, and the components of the vector indicates the amount of change along each coordinate axis. The divergence of rank 0 tensor (scalar) is not defined. The rank of a scalar can’t be reduced anymore - we can’t have a rank -1 tensor.
| Dot product | Divergence of a vector | Vector-matrix product |
|---|---|---|
| $\displaystyle \vec{a}\cdot \vec{b}$ | $\displaystyle \vec{\nabla}\cdot \vec{a}$ | $\displaystyle \vec{a}\cdot B$ |
| $\displaystyle a_i b_i$ | $\displaystyle \frac{\partial a_i}{\partial x_i}$ | $\displaystyle a_i B_{ij}$ |
| $\displaystyle a_x b_x + a_y b_y + a_z b_z$ | $\displaystyle \frac{\partial a_x}{\partial x} + \frac{\partial a_y}{\partial y} + \frac{\partial a_z}{\partial z}$ | $\displaystyle \begin{bmatrix} a_x B_{xx} & a_yB_{xy} & a_zB_{xz} \\ a_x B_{yx} & a_yB_{yy} & a_zB_{yz} \\ a_x B_{zx} & a_yB_{zy} & a_zB_{zz} \end{bmatrix}$ |
| Divergence of a rank 2 tensor | Tensor product or outer product | Gradient of a vector |
|---|---|---|
| $\displaystyle \vec{\nabla}\cdot A$ | $\displaystyle \vec{a}\vec{b} \equiv \vec{a}\otimes\vec{b}$ | $\displaystyle \vec{\nabla} \vec{a}$ |
| $\displaystyle \frac{\partial A_{ij}}{\partial x_i}$ | $\displaystyle a_i b_j$ | $\displaystyle \frac{\partial a_i}{\partial x_j}$ |
| $\displaystyle \begin{bmatrix} \frac{\partial B_{xx}}{\partial x} + \frac{\partial B_{yx}}{\partial y} + \frac{\partial B_{zx}}{\partial z} \\ \frac{\partial B_{xy}}{\partial x} + \frac{\partial B_{yy}}{\partial y} + \frac{\partial B_{zy}}{\partial z} \\ \frac{\partial B_{xz}}{\partial x} + \frac{\partial B_{yz}}{\partial y} + \frac{\partial B_{zz}}{\partial z} \end{bmatrix}$ | $\displaystyle \begin{bmatrix} a_x b_x & a_x b_y & a_x b_z \\ a_y b_x & a_y b_y & a_y b_z \\ a_z b_x & a_z b_y & a_z b_z \end{bmatrix}$ | $\displaystyle \begin{bmatrix} \frac{\partial a_x}{\partial x} & \frac{\partial a_x}{\partial y} & \frac{\partial a_x}{\partial z} \\ \frac{\partial a_y}{\partial x} & \frac{\partial a_y}{\partial y} & \frac{\partial a_y}{\partial z} \\ \frac{\partial a_z}{\partial x} & \frac{\partial a_z}{\partial y} & \frac{\partial a_z}{\partial z} \end{bmatrix}$ |
The material/total/substantial derivative is an important quantity in continuum mechanics. It is denoted with a capital $D$ (the usual derivative is a lowercase $d$). It is the rate of change of a quantity per unit time in a frame of reference that moves with the fluid (Lagrangian perspective). It says that when we follow a chunk of fluid (i.e. Lagrangian perspective) ,the total time rate of change equals the local time rate of change of the Eulerian field, plus the spatial rate of change multiplied by how fast we’re moving through space. For more details about the material derivative, see my lecture on the material derivative.
| $\displaystyle \frac{D\vec{V}}{Dt}=\frac{\partial\vec{V}}{\partial t}+(\vec{V}\cdot \vec{\nabla})\vec{V}$ |
| $\displaystyle \frac{Dv_j}{Dt}=\frac{\partial v_j}{\partial t}+v_i \frac{\partial v_j}{\partial x_i}$ |
| $\displaystyle \begin{aligned}\frac{Du}{Dt} &= \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \\ \frac{Dv}{Dt} &= \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \\ \frac{Dw}{Dt} &= \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\end{aligned}$ |
A stress tensor is a rank 2 tensor that gives us the individual stresses on each face of a fluid element. It’s convenient to separate this tensor into pressure (isotropic) and viscous (eventually, this will be deviatoric) components.
| $\displaystyle \sigma = -P I + \tau$ |
| $\displaystyle \sigma_{ij}= -P\delta_{ij} + \tau_{ij}$ |
| $\displaystyle \sigma = -\begin{bmatrix}P & 0 & 0 \\ 0 & P & 0 \\ 0 & 0 & P \end{bmatrix} + \begin{bmatrix}\tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz}\end{bmatrix}$ |
Remark.
Any rank 2 tensor (for example, the Cauchy stress tensor) can be decomposed into an isotropic tensor and a deviatoric tensor:
\[A_{ij} = A^{iso}_{ij} + A^{dev}_{ij} = \frac{1}{3}A_{kk}\delta_{ij} + \left(A_{ij} - \frac{1}{3}A_{kk}\delta_{ij}\right)\]The isotropic tensor is the same in all directions. It is a scalar multiple of the identity tensor. The remaining part of the tensor (after we subtract away the isotropic part) is the deviatoric part.
The viscous stress tensor is a rank 2 tensor that gives us the individual viscous stresses on each face of a fluid element. Viscous stresses are created due to velocity gradients - if a certain part of the fluid is moving faster than another part, then a viscous stress which slows the faster part and speeds up the slower part will be created. Viscosity is a fluid property that tells us how much viscous stress is created by a velocity gradient. In fact, there are two types of viscosity - bulk viscosity $\zeta$ and dynamic viscosity $\mu$. Bulk viscosity is related to the rate of change of volume: notice that the diagonals give the volumetric strain rate (lecture here).
There are several reasons that we typically only consider shear viscosity in fluids. First, bulk viscosity is only important in highly compressible fluids, because for an incompressible fluid, conservation of mass means that the volumetric strain rate is zero (see again my lecture on this). Second, for many fluids, the bulk viscosity is negligible. For this reason, we typically only consider the shear stress, making the viscous stress tensor $\tau$ deviatoric.
Remark.
The part of the viscous stress tensor related to bulk viscosity is an isotropic tensor.
Here, I give the full viscous stress tensor. For most fluids analysis, the bulk viscosity is negligible, so we typically only consider the second (deviatoric) part.
| $\displaystyle \tau = \zeta (\vec{\nabla}\cdot \vec{V})I + \mu [\vec{\nabla}\vec{V} + (\vec{\nabla}\vec{V})^\text{T} - \tfrac{2}{3}(\vec{\nabla} \cdot \vec{V})I ]$ |
| $\displaystyle \tau_{ij} = \zeta \frac{\partial v_k}{\partial x_k}\delta_{ij} + \mu \left[\frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} - \frac{2}{3}\frac{\partial v_k}{\partial x_k} \delta_{ij}\right]$ |
| $\displaystyle \begin{aligned} \tau = &\zeta \begin{bmatrix} \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) & 0 & 0 \\ 0 & \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) & 0 \\ 0 & 0 & \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) \end{bmatrix} \\ &+ \mu \begin{bmatrix} 2\frac{\partial u}{\partial x} - \frac{2}{3}\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) & \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} & \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & 2\frac{\partial v}{\partial y} - \frac{2}{3}\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) & \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} \\ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} & \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} & 2\frac{\partial w}{\partial z} - \frac{2}{3}\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) \end{bmatrix} \end{aligned}$ |
Remark.
For most fluids, the viscous stress tensor is deviatoric.
Conservation of mass - the continuity equation
The compressible continuity equation is the more general case - we allow the density $\rho$ to change.
| $\displaystyle \frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot (\rho \vec{V}) = 0$ |
| $\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial (\rho v_j)}{\partial x_j} = 0$ |
| $\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0$ |
If we take density $\rho$ to be constant (physically, we can’t actually compress the fluid), then we get the incompressible continuity equation.
| $\displaystyle \vec{\nabla} \cdot \vec{V} = 0$ |
| $\displaystyle \frac{\partial v_j}{\partial x_j} = 0$ |
| $\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$ |
Conservation of momentum
| $\displaystyle \frac{\partial (\rho \vec{V})}{\partial t} + \vec{\nabla} \cdot (\rho \vec{V} \vec{V}) = \rho \vec{g} + \vec{\nabla} \cdot \sigma$ |
| $\displaystyle \frac{\partial (\rho v_j)}{\partial t} + \frac{\partial }{\partial x_i}(\rho v_j v_i) = \rho g_j + \frac{\partial \sigma_{ij}}{\partial x_i}$ |
| $\displaystyle \begin{aligned} \frac{\partial(\rho u)}{\partial t} + \frac{\partial(\rho uu)}{\partial x} + \frac{\partial (\rho uv)}{\partial y} + \frac{\partial (\rho uw)}{\partial z} &= \rho g_x + \frac{\partial \sigma_{xx}}{\partial x} + \frac{\sigma_{yx}}{\partial y} + \frac{\sigma_{zx}}{\partial z} \\ \frac{\partial (\rho v)}{\partial t } + \frac{\partial (\rho u v)}{\partial x } + \frac{\partial (\rho v v)}{\partial y} + \frac{\partial (\rho v w)}{\partial z} &= \rho g_y + \frac{\partial \sigma_{xy}}{\partial x} + \frac{\sigma_{yy}}{\partial y} + \frac{\sigma_{zy}}{\partial z} \\ \frac{\partial (\rho w)}{\partial t} + \frac{\partial (\rho u w)}{\partial x} + \frac{\partial(\rho v w)}{\partial y} + \frac{\partial (\rho w w)}{\partial z} &= \rho g_z + \frac{\partial \sigma_{xz}}{\partial x} + \frac{\sigma_{yz}}{\partial y} + \frac{\sigma_{zz}}{\partial z} \end{aligned}$ |
| $\displaystyle \rho(\frac{\partial\vec{V}}{\partial t}+(\vec{V}\cdot \vec{\nabla})\vec{V}) = - \vec{\nabla} P + \vec{\nabla} \cdot \tau + \rho \vec{g}$ |
| $\displaystyle \rho (\frac{\partial v_j}{\partial t} +v_i \frac{\partial v_j}{\partial x_i}) = -\frac{\partial P}{\partial x_j} + \frac{\partial \tau_{ij}}{\partial x_i} + \rho g_j$ |
| $\displaystyle \begin{aligned} \rho (\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z})&= - \frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x}+ \frac{\partial \tau_{yx}}{\partial y}+ \frac{\partial \tau_{zx}}{\partial z} + \rho g_x \\\\ \rho (\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z})&= - \frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x}+ \frac{\partial \tau_{yy}}{\partial y}+ \frac{\partial \tau_{zy}}{\partial z} + \rho g_y \\\\ \rho (\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z})&= - \frac{\partial P}{\partial z} + \frac{\partial \tau_{xz}}{\partial x}+ \frac{\partial \tau_{yz}}{\partial y}+ \frac{\partial \tau_{zz}}{\partial z} + \rho g_z \end{aligned}$ |
Compressible Navier-Stokes equations
If we substitute our stress tensor into Cauchy’s equation, we get the compressible Navier-Stokes equations.
| $\displaystyle \rho (\frac{\partial\vec{V}}{\partial t}+(\vec{V}\cdot \vec{\nabla})\vec{V}) = - \vec{\nabla} P + \mu \nabla^2 \vec{V} + \tfrac{1}{3}\mu \vec{\nabla}(\vec{\nabla}\cdot\vec{V}) + \rho \vec{g}$ |
| $\displaystyle \rho (\frac{\partial v_j}{\partial t} +v_i \frac{\partial v_j}{\partial x_i}) = - \frac{\partial P}{\partial x_j} + \mu \frac{\partial^2v_j}{\partial x_i \partial x_i} + \frac{1}{3}\mu \frac{\partial^2 v_i}{\partial x_j \partial x_i} + \rho g_j$ |
| $\displaystyle \begin{aligned} \rho(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) &= - \frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{3}\mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z}\right) + \rho g_x \\\\ \rho(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}) &= - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} +\frac{\partial^2 v}{\partial y^2} +\frac{\partial^2 v}{\partial z^2} \right) + \frac{1}{3}\mu \left( \frac{\partial^2 u}{\partial y\partial x} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 w}{\partial y \partial z}\right) + \rho g_y \\\\ \rho(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}) &= - \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2} +\frac{\partial^2 w}{\partial z^2} \right) + \frac{1}{3}\mu \left( \frac{\partial^2 u}{\partial z \partial x} + \frac{\partial^2 v}{\partial z \partial y} + \frac{\partial^2 w}{\partial z^2}\right) + \rho g_z \end{aligned}$ |
Incompressible Navier-Stokes equations
If we again take density $\rho$ to be constant, then we get the incompressible Navier-Stokes equations.
| $\displaystyle \rho (\frac{\partial\vec{V}}{\partial t}+(\vec{V}\cdot \vec{\nabla})\vec{V}) = - \vec{\nabla} P + \mu \nabla^2 \vec{V} + \rho \vec{g}$ |
| $\displaystyle \rho (\frac{\partial v_j}{\partial t} +v_i \frac{\partial v_j}{\partial x_i}) = - \frac{\partial P}{\partial x_j} + \mu \frac{\partial^2v_j}{\partial x_i \partial x_i} + \rho g_j$ |
| $\displaystyle \begin{aligned} \rho{\left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right)} &= - \frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) + \rho g_x \\\\ \rho{\left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right)} &= - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} +\frac{\partial^2 v}{\partial y^2} +\frac{\partial^2 v}{\partial z^2} \right) + \rho g_y \\\\ \rho{\left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right)} &= - \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2} +\frac{\partial^2 w}{\partial z^2} \right) + \rho g_z \end{aligned}$ |
The incompressible Navier-Stokes equations can be written in terms of the kinematic viscosity $\nu = \frac{\mu}{\rho}$. If we divide the Navier-Stokes equations by $\rho$, we get the following.
| $\displaystyle \frac{\partial\vec{V}}{\partial t}+(\vec{V}\cdot \vec{\nabla})\vec{V} = - \frac{1}{\rho}\vec{\nabla} P + \nu \nabla^2 \vec{V} + \vec{g}$ |
| $\displaystyle \frac{\partial v_j}{\partial t} +v_i \frac{\partial v_j}{\partial x_i} = - \frac{1}{\rho}\frac{\partial P}{\partial x_j} + \nu \frac{\partial^2v_j}{\partial x_i \partial x_i} + g_j$ |
| $\displaystyle \begin{aligned} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} &= - \frac{1}{\rho}\frac{\partial P}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) + g_x \\\\ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} &= - \frac{1}{\rho}\frac{\partial P}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} +\frac{\partial^2 v}{\partial y^2} +\frac{\partial^2 v}{\partial z^2} \right) + g_y \\\\ \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} &= - \frac{1}{\rho}\frac{\partial P}{\partial z} + \nu \left( \frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2} +\frac{\partial^2 w}{\partial z^2} \right) + g_z \end{aligned}$ |
Often, we move $1/\rho$ into the pressure gradient term, and define a new pressure $p \equiv P/\rho$. This new pressure is sometimes called the kinematic pressure (it’s just a scalar multiple of the mechanical pressure). Therefore, the kinematic flavour of the Navier-Stokes equations describe the relationship between velocity and kinematic pressure, with a single fluid property $\nu$ (the kinematic viscosity).