A note on reverse engineering the Navier-Stokes equations

eulernavierstokesThe Navier-Stokes equations are the fundamental equations of fluid mechanics, analogous to, say, Maxwell’s equations in the case of electromagnetism. They are important for applications in science and engineering but they are difficult to solve analytically so real-life applications rely almost exclusively on computer-aided methods. In fact, the equations are still not yet fully understood mathematically. Some basic questions about the existence and nature of possible solutions remain unanswered. Because of their importance, the Clay Mathematics Institute has offered a prize of one million dollars to anyone who can clarify some specific fundamental questions about them (see the CMI web page for details).

The equations are named after Claude-Louis Navier (1785-1836) and George Gabriel Stokes (1819-1903) who worked on their development independently. Some other leading contemporary mathematicians were also involved and I was intrigued to learn that all of these mathematicians used Euler’s equations of motion as a starting point in their derivations of the Navier-Stokes equations. Euler’s equations had been derived much earlier in 1757 by the great mathematician Leonhard Euler (1707-1783). In tensor notation, using the Einstein summation convention, the Euler equations take the form

\rho g^i - \frac{\partial p}{\partial x^i} = \rho \big(\frac{\partial v^i}{\partial t} + v^j \frac{\partial v^i}{\partial x^j}\big)

I explored a derivation of these equations from Newton’s second law in a previous post. To better understand how the two sets of equations are related, it occurred to me that it might be an interesting exercise, and probably not too difficult, to try to `reverse engineer’ the Navier-Stokes equations using Euler’s equations as a starting point. I want to briefly record my exploration of this idea in the present note.

I am interested in the incompressible fluid version of the Navier-Stokes equations which can be written in tensor form, again using the Einstein summation convention, as

\rho g^i - \frac{\partial p}{\partial x^i} + \mu \frac{\partial}{\partial x^j}\big(\frac{\partial v^i}{\partial x^j}\big) = \rho \big(\frac{\partial v^i}{\partial t} + v^j \frac{\partial v^i}{\partial x^j}\big)

where \mu > 0 is a viscosity coefficient specific to the fluid in question. Putting the Euler equations and the Navier-Stokes equations side by side like this makes it clear that the key difference between them is the addition of the viscosity-related terms on the left-hand side of the Navier-Stokes equations. To clarify things a bit more it is helpful to see the equations laid out fully. The Euler equations constitute a three-equation system which looks like this:

\rho g^1 - \frac{\partial p}{\partial x^1} = \rho \big(\frac{\partial v^1}{\partial t} + v^1 \frac{\partial v^1}{\partial x^1} + v^2 \frac{\partial v^1}{\partial x^2} + v^3 \frac{\partial v^1}{\partial x^3}\big)

\rho g^2 - \frac{\partial p}{\partial x^2} = \rho \big(\frac{\partial v^2}{\partial t} + v^1 \frac{\partial v^2}{\partial x^1} + v^2 \frac{\partial v^2}{\partial x^2} + v^3 \frac{\partial v^2}{\partial x^3}\big)

\rho g^3 - \frac{\partial p}{\partial x^3} = \rho \big(\frac{\partial v^3}{\partial t} + v^1 \frac{\partial v^3}{\partial x^1} + v^2 \frac{\partial v^3}{\partial x^2} + v^3 \frac{\partial v^3}{\partial x^3}\big)

The Navier-Stokes equations also constitute a three-equation system which looks like this:

\rho g^1 - \frac{\partial p}{\partial x^1} + \mu \bigg(\frac{\partial}{\partial x^1}\big(\frac{\partial v^1}{\partial x^1}\big) + \frac{\partial}{\partial x^2}\big(\frac{\partial v^1}{\partial x^2}\big) + \frac{\partial}{\partial x^3}\big(\frac{\partial v^1}{\partial x^3}\big)\bigg) = \rho \big(\frac{\partial v^1}{\partial t} + v^1 \frac{\partial v^1}{\partial x^1} + v^2 \frac{\partial v^1}{\partial x^2} + v^3 \frac{\partial v^1}{\partial x^3}\big)

\rho g^2 - \frac{\partial p}{\partial x^2} + \mu \bigg(\frac{\partial}{\partial x^1}\big(\frac{\partial v^2}{\partial x^1}\big) + \frac{\partial}{\partial x^2}\big(\frac{\partial v^2}{\partial x^2}\big) + \frac{\partial}{\partial x^3}\big(\frac{\partial v^2}{\partial x^3}\big)\bigg) = \rho \big(\frac{\partial v^2}{\partial t} + v^1 \frac{\partial v^2}{\partial x^1} + v^2 \frac{\partial v^2}{\partial x^2} + v^3 \frac{\partial v^2}{\partial x^3}\big)

\rho g^3 - \frac{\partial p}{\partial x^3} + \mu \bigg(\frac{\partial}{\partial x^1}\big(\frac{\partial v^3}{\partial x^1}\big) + \frac{\partial}{\partial x^2}\big(\frac{\partial v^3}{\partial x^2}\big) + \frac{\partial}{\partial x^3}\big(\frac{\partial v^3}{\partial x^3}\big)\bigg) = \rho \big(\frac{\partial v^3}{\partial t} + v^1 \frac{\partial v^3}{\partial x^1} + v^2 \frac{\partial v^3}{\partial x^2} + v^3 \frac{\partial v^3}{\partial x^3}\big)

Looking at the pattern of the indices appearing in the viscosity-related terms on the left-hand side of the Navier-Stokes equations, one is immediately reminded of the indices identifying the positions of the elements of a matrix and we can in fact arrange the viscosity-related terms in matrix form as

\begin{bmatrix} \frac{\partial}{\partial x^1}\big(\frac{\partial v^1}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^1}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^1}{\partial x^3}\big)\\ \ \\ \frac{\partial}{\partial x^1}\big(\frac{\partial v^2}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^2}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^2}{\partial x^3}\big)\\ \ \\ \frac{\partial}{\partial x^1}\big(\frac{\partial v^3}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^3}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^3}{\partial x^3}\big)\end{bmatrix}

From the point of view of reverse engineering the Navier-Stokes equations this is highly suggestive because in deriving the Euler equations we encountered one key matrix, the rank-2 stress tensor \tau^{ij}. Could the above matrix be related to the 3 \times 3 stress tensor? The answer is yes, and in fact we have that

\begin{bmatrix} \frac{\partial}{\partial x^1}\big(\frac{\partial v^1}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^1}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^1}{\partial x^3}\big)\\ \ \\ \frac{\partial}{\partial x^1}\big(\frac{\partial v^2}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^2}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^2}{\partial x^3}\big)\\ \ \\ \frac{\partial}{\partial x^1}\big(\frac{\partial v^3}{\partial x^1}\big) & \frac{\partial}{\partial x^2}\big(\frac{\partial v^3}{\partial x^2}\big) & \frac{\partial}{\partial x^3}\big(\frac{\partial v^3}{\partial x^3}\big)\end{bmatrix} = \begin{bmatrix} \frac{\partial \tau^{11}}{\partial x^1} & \frac{\partial \tau^{12}}{\partial x^2} & \frac{\partial \tau^{13}}{\partial x^3}\\ \ \\ \frac{\partial \tau^{21}}{\partial x^1} & \frac{\partial \tau^{22}}{\partial x^2} & \frac{\partial \tau^{23}}{\partial x^3}\\ \ \\ \frac{\partial \tau^{31}}{\partial x^1} & \frac{\partial \tau^{32}}{\partial x^2} & \frac{\partial \tau^{33}}{\partial x^3} \end{bmatrix}

In deriving the Euler equations what we did was to assume that the fluid was friction-free so that there were no shear forces acting on any of the faces of a given fluid volume element. This means that the fluid was being modelled as being inviscid, i.e., having no viscosity. As a result, we kept only the diagonal elements of the stress tensor and re-interpreted these as pressures. Approximating the net pressure in each direction on a fluid volume element using a Taylor series expansion we were then led to include only the first-order partials of the diagonal terms of the stress tensor (interpreted as the first-order partials of pressure) in the Euler equations.

In the case of the Navier-Stokes equations we are no longer assuming that the fluid is inviscid so we are including all the first-order partials of the stress tensor in the equations. Thus, the Navier-Stokes equations could be written as

\rho g^1 - \frac{\partial p}{\partial x^1} + \mu \big(\frac{\partial \tau^{11}}{\partial x^1} + \frac{\partial \tau^{12}}{\partial x^2} + \frac{\partial \tau^{13}}{\partial x^3}\big) = \rho \big(\frac{\partial v^1}{\partial t} + v^1 \frac{\partial v^1}{\partial x^1} + v^2 \frac{\partial v^1}{\partial x^2} + v^3 \frac{\partial v^1}{\partial x^3}\big)

\rho g^2 - \frac{\partial p}{\partial x^2} + \mu \big(\frac{\partial \tau^{21}}{\partial x^1} + \frac{\partial \tau^{22}}{\partial x^2} + \frac{\partial \tau^{23}}{\partial x^3}\big) = \rho \big(\frac{\partial v^2}{\partial t} + v^1 \frac{\partial v^2}{\partial x^1} + v^2 \frac{\partial v^2}{\partial x^2} + v^3 \frac{\partial v^2}{\partial x^3}\big)

\rho g^3 - \frac{\partial p}{\partial x^3} + \mu \big(\frac{\partial \tau^{31}}{\partial x^1} + \frac{\partial \tau^{32}}{\partial x^2} + \frac{\partial \tau^{33}}{\partial x^3}\big) = \rho \big(\frac{\partial v^3}{\partial t} + v^1 \frac{\partial v^3}{\partial x^1} + v^2 \frac{\partial v^3}{\partial x^2} + v^3 \frac{\partial v^3}{\partial x^3}\big)

Detailed consideration of the forces acting on an infinitesimal volume element in the case of a viscous fluid lead to expressions for the stress tensor components as functions of the first-order partials of the components of the velocity vector of the form

\tau^{ij} = \tau^{ji} = \mu \big(\frac{\partial v^i}{\partial x^j} + \frac{\partial v^j}{\partial x^i}\big)

The first-order partials of these are then of the form

\frac{\partial \tau^{ij}}{\partial x^j} = \mu \frac{\partial }{\partial x^j}\big(\frac{\partial v^i}{\partial x^j}\big) + \mu \frac{\partial }{\partial x^j}\big(\frac{\partial v^j}{\partial x^i}\big)

But note that with incompressible flow the mass density of the fluid is a constant so the divergence of the velocity vanishes (this is implied directly by the continuity equation with constant mass density). Therefore the second term on the right-hand side is zero since

\frac{\partial }{\partial x^j}\big(\frac{\partial v^j}{\partial x^i}\big) = \frac{\partial }{\partial x^i}\big(\frac{\partial v^j}{\partial x^j}\big)

= \frac{\partial }{\partial x^i}\big(\frac{\partial v^1}{\partial x^1}\big) + \frac{\partial }{\partial x^i}\big(\frac{\partial v^2}{\partial x^2}\big) + \frac{\partial }{\partial x^i}\big(\frac{\partial v^3}{\partial x^3}\big) = \frac{\partial }{\partial x^i} \text{div}(\vec{v}) = 0

Thus we are left with

\frac{\partial \tau^{ij}}{\partial x^j} = \mu \frac{\partial }{\partial x^j}\big(\frac{\partial v^i}{\partial x^j}\big)

which are the extra terms appearing on the left-hand side of the Navier-Stokes equations.