Math 597F, Notes 1: Euler and Navier-Stokes equations

ttn12 fluid dynamics, Math 597F: topics on boundary layers

This is the first lecture of my Math 597F topics course. In this lecture, I will derive the partial differential equations, known as Euler and Navier-Stokes equation, that are widely used to model the dynamics of a fluid.  To begin, let {\rho(x,t) \in \mathbb{R}, v(x,t)\in \mathbb{R}^n} be the density function and velocity vector field of the fluid at a position {x\in \mathbb{R}^n} and a time {t}. Consider a fluid particle with initial position {x}. Its trajectory {X(t;x)} as time evolves is governed by the following ODE equation:

\displaystyle \frac{d}{dt}X(t; x ) = v (X(t;x), t), \qquad X(0; x) = x,

in which {x} serves as a parameter. An important quantity is the Jacobian of {X(t;x)} with respect to {x}: define

\displaystyle J(t;x): = \mbox{det} \nabla_x X(t;x).

Here the subscript denotes partial derivatives with respect to {x}.

Lemma 1 {\frac{d}{dt} J(t;x) = (\mathrm{div}_x v ) J(t;x) .}

Proof: Let {X(t;x) = (X_1, \cdots, X_n)} be a particle trajectory. Using the ODE equation for a particle trajectory, we compute

\displaystyle \begin{aligned} \frac{d}{dt}J(t;x) &= \sum_k \mbox{det} (\nabla_x X_1, \cdots, \frac{d}{dt}\nabla_x X_k, \cdots ,\nabla_x X_n ) \\ &= \sum_k \mbox{det} (\nabla_x X_1, \cdots, \sum_\ell \nabla_x X_\ell \frac{\partial}{\partial {x_\ell}} v_k, \cdots ,\nabla_x X_n ) \\ &= \sum_k \frac{\partial}{\partial {x_k}} v_k \mbox{det} (\nabla_x X_1, \cdots, \nabla_x X_k, \cdots ,\nabla_x X_n ) \\& = (\mathrm{div}_x v )J (t;x). \end{aligned}

\Box

Euler and Navier-Stokes equations are constituted from basic principles of mechanics: conservation of mass, momentum, and energy. Let {\Omega_0} be an arbitrary initial fluid domain in {\mathbb{R}^n} and let {\Omega_t} be the corresponding fluid domain at time {t}, that is the image of {\Omega_0} via the trajectory map {x\mapsto X(t; x)}. Conservation of mass asserts that the total of mass is constant in time: precisely,

\displaystyle \frac{d}{dt} \int_{\Omega_t} \rho(X,t)\; dX = 0.

Using a change of variables {X = X(t;x)} with {dX = J(t;x) dx}, one gets

\displaystyle 0 = \frac{d}{dt} \int_{\Omega_0} \rho(X(t;x),t) J(t;x) \; dx = \int_{\Omega_0} \Big( \rho_t + v \cdot \nabla \rho + \rho \mathrm{div} v \Big) J(t;x)\; dx .

By grouping the last two terms into {\mathrm{div}(\rho v)} and using the fact that {\Omega_0} was arbitrary, this yields the conservation of mass equation, or known as continuity equation:

\displaystyle \rho_t + \mathrm{div} (\rho v) = 0.\ \ \ \ \ (1)

Next, conservation of momentum asserts that the rate of change of total momentum is equal to the total external force {f} (gravity, Coriolis, electromagnetic forces,…) acting on the fluid, plus internal force {F} acting on the surface of the fluid domain. Precisely,

\displaystyle \frac{d}{dt}\int_{\Omega_t} \rho v \; dX = \int_{\partial \Omega_t} F \cdot n \; dS_t + \int_{\Omega_t} \rho f\; dX.

Again, the change of variables {X = X(t;x)}, together with a use of the divergence theorem, yields at once the conservation of momentum equation:

\displaystyle (\rho v)_t + \mathrm{div} (\rho v\otimes v) = \nabla \cdot F + \rho f,\ \ \ \ \ (2)

in which {v \otimes v = (v_kv_j)}. Classically, the internal force consists of the scalar pressure {p} and the viscous force: precisely, {F = - p I + \tau}, where {\tau} denotes the viscous stress tensor. For Newtonian fluid, the viscous stress is assumed to be proportional to the gradient of velocity field: {\tau = \lambda \mathrm{div} v I + \nu (\nabla u + (\nabla u)^t)}, with Lamé viscosity coefficients {\lambda, \nu\ge 0}. Ideal fluid is when there is no viscosity.

The third basic principle is conservation of energy, which for sake of presentation, I will skip to derive it here.

Compressible Euler and Navier-Stokes equations. The set of equations (1) and (2), with {F = - p I + \tau}, is known in the literature as compressible Navier-Stokes equations modeling the dynamics of a fluid (of constant temperature). These equations are called Euler equations, when {\lambda = \nu =0}. For this set of equations to be complete, a pressure law is needed. For instance, a barotropic gas is the fluid flow where the pressure is an (invertible) function of density: {p = p(\rho)}.

Incompressibility. For our purpose, we focus on the incompressible fluid. A fluid is said to be incompressible if the volume of fluid domain is invariant in time: precisely,

\displaystyle \mathrm{Vol} (\Omega_t) = \int_{\Omega_t} \; dX = \int_{\Omega_0} \; dx = \mathrm{Vol} (\Omega_0).

Taking time derivative of this relation and using the arbitrariness of {\Omega_0} yield {\mathrm{div} v =0}.

Stratified incompressible fluid. For instance, when the external force {f = - g e_z}, where {g} denotes the gravity and {e_z} denotes the vertical direction. The following set of equations describes a stratified incompressible fluid:

\displaystyle \left \{ \begin{aligned} \rho_t + \mathrm{div} (\rho v) & =0 \\ (\rho v)_t + \mathrm{div} (\rho v \otimes v) + \nabla p &= \nu \Delta v - \rho g e_z \\ \nabla \cdot v &=0. \end{aligned}\right. \ \ \ \ \ (3)

Unlike in the compressible case, this set of equations is complete and the pressure itself is an unknown function in the incompressible fluid.

The Euler and Navier-Stokes equations. The most popular case is when fluid is incompressible and homogenous ({\rho =1}), and is often simply referred as the Euler and Navier-Stokes equations:

\displaystyle \left \{ \begin{aligned} v_t + \mathrm{div} (v \otimes v) + \nabla p &= \nu \Delta v \\ \nabla \cdot v &=0. \end{aligned}\right. \ \ \ \ \ (4)

Again, the set of equations is complete, with unknown functions {v \in \mathbb{R}^n, p\in \mathbb{R}}. Euler equations are when viscosity vanishes: {\nu =0}.

This topics course is all about the singular perturbation problem or vanishing viscosity problem: {\nu \rightarrow 0}. This problem is closely tied with the turbulent theory. If we now factor out the physical units: the time unit {T}, the length unit {L}, and the velocity unit {U}, which of course satisfy the relation {L = UT}, and introduce the change of variables

\displaystyle t \mapsto \frac tT , \qquad x\mapsto \frac xL, \qquad v\mapsto \frac vU,

we arrive at the non-dimensional Navier-Stokes equations:

\displaystyle \left \{ \begin{aligned} v_t + \mathrm{div} (v \otimes v) + \nabla p &= \frac{1}{Re} \Delta v, \\ \nabla \cdot v &=0, \end{aligned}\right. \ \ \ \ \ (5)

with {Re = \frac{UL}{\nu}} being called the physical Reynolds number. As already mentioned in the course overview, the Reynolds number governs the transition of laminar to turbulent flows. Mathematically, the problem of small viscosity and high Reynolds number is equivalent.

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6 thoughts on “Math 597F, Notes 1: Euler and Navier-Stokes equations”

  1. Very nice introduction! Thank you.
    In your notes, $\latex x$ is Lagrangian coordinate, and $\latex X$ is Euler coordinate. So in the change of variables, should it be $\latex X\mapsto \frac{X}{L}$?

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