This Spring ’16 semester, I am teaching a graduate Math 505 course, whose goal is to introduce the basic concepts and the fundamental mathematical problems in Fluid Mechanics for students both in math and engineering. The difficulty is to assume no background in both fluids and analysis of PDEs from the students. That’s it!

Anyhow, materials for my course are based on various books and lecture notes, one of which is the great lecture notes by V. Sverak (selected topics on fluid mechanics, 2011). More information, some pdf notes, and so on can be found from my course webpage!

**1. Continuum assumption**

Fluids are made up of many many discrete molecules that interact with one another. Throughout the course, we shall assume that fluid molecules are small enough to be infinitesimally close to one another (and so, of course, the number of molecules is infinite). That is, we shall work with the continuum models of fluids. The mathematical justification of the continuum dynamics of fluids (macroscopic description) from the deterministic Hamiltonian dynamics of discrete molecules (microscopic description) remains an outstanding unsolved problem (see, however, Quastel-Yau ’98 for stochastic particles). This is the Hilbert’s 6th problem. One could also formally derive the continuum models through the mesoscopic description as suggested by Boltzmann. Again, a rigorous derivation remains incomplete (cf., L. Saint-Raymond).

**2. Eulerian and Lagrangian description of fluid motion**

Let be the fluid domain, . By the continuum assumption, each point is viewed as a fluid “particle”. The motion of fluids is described by the velocity vector field

at each particle and at a time . This is the so-called *Eulerian description* of fluids, introduced by Euler in 1755. There is another common way to describe fluid motion, *the Lagrangian description*, which keeps track of the trajectory of particles. For each “initial” particle , denote by the new position of the particle at the time , which is defined by the ODEs

That is, the map , as runs in , keeps track of the trajectory of the initial particle , whereas the Lagrangian map gives the new position of the particle when time evolves. By a view of the standard ODE theory (e.g., the standard Picard’s iteration), for each , a trajectory exists locally in time near zero, if is continuous in and . In addition, if is Lipschitz in , the local trajectory is unique.

In the case when has a boundary, it is natural to assume that the fluids do not cross the boundary. That is,

in which denotes the outer normal unit vector at . With the above boundary condition, it is easy to see that the trajectories , when exist, either remain on or never cross it. On the other words, fluids in the interior of remain in the interior, and those on the boundary remain on the boundary.

**2.1. Continuity equation**

Let be the density distribution of fluids. That is, the mass of fluids in the infinitesimal volume is equal to , and the total of mass in an arbitrary domain is defined by

Let be the image of under the map . The conservation of mass reads

Using the change of variables for and denoting the Jacobian determinant , we have

Since was arbitrary, the conservation of mass implies

This is the conservation of mass in the Lagrangian coordinate. To write the conservation in the Eulerian coordinate, we take the time derivative of (2). A direct computation yields

Lemma 1There holds

*Proof:* Exercise.

Here, denotes the usual gradient vector. In addition, for any quantity , the rate of change of quantity along each particle trajectory is computed by

The derivative is often referred to as the material derivative.

Hence, (2) yields

for all and . Here, assuming sufficient regularity of , the map is a diffeomorphism from to itself. This shows that for all points , there is a unique so that . That is, the above equation yields

For an arbitrary fluid subdomain , using the continuity equation and the divergence theorem, we compute

which asserts that the rate of change of the total mass in is equal to the total density flux, , of the fluid through the boundary . The minus sign is due to the fact that is the outer normal unit vector. Of course, this is the alternate way to derive the continuity equation (3).

**2.2. Transport theorem**

It is straightforward to check that

Lemma 3 (Transport theorem)Let be a velocity vector field, with on , and let be the corresponding material derivative. For solving the continuity equation (3), there holds

for any smooth function .

*Proof:* Exercise. The lemma shows that the integral is conserved in time, provided solving the transport equation , or equivalently

That is, is constant along the particle trajectory , associated with the velocity field . In particular, solves the transport equation, and thus the transport theorem yields the conservation of the total mass in . In fact, if we let be the characteristic function on , then solves the transport equation (in the weak sense), and the transport theorem reassures the conservation of mass; see (1).

**2.3. Incompressibility**

We say the fluid flow is incompressible if the Lagrangian map is a volume-preserving map, for all time . Precisely, there holds

for all subdomains . Here, denotes the image of under the map . Exactly as done in the previous section, this is equivalent to

For incompressible flows, the density is constant along the particle trajectories , or equivalently

for all . In particular, *homogeneity (i.e., constant density)* of incompressible fluids propagates in time. Gas or air are compressible flows, whereas water is modeled by the incompressible flow.

For the incompressible flows, it is easy to check that the quantity

is conserved in time, for arbitrary smooth function so that the integral is well-defined.

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**3. Momentum equations**

Certainly, the continuity equation does not constitute a complete set of equations to describe fluids, since the velocity field itself is an unknown. In this section, we derive the momentum equations. The derivation uses the continuum version of the Newton’s second law:

in which is the density, *the acceleration* or the rate of change of the fluid velocity, and the total force acting on the fluid. Here, in (5), the forces are understood as the net force acting on fluid parcels.

**3.1. Acceleration**

By definition, *the acceleration* is defined by

The above holds for all and , and so for all points . That is, the acceleration of fluid motion at each is

For *free particles*, that is, for fluids that experience neither internal nor external forces , the velocity field satisfies

which is the inviscid Burgers equation. In the Lagrangian coordinates, this shows that the velocity field is constant along the particle trajectories and so the trajectories are simply straight lines

It is easy to construct smooth initial data so that two trajectories with different initial velocity meet in a finite time, which results in the discontinuity of the velocity field.

In addition, using the transport theorem, Lemma 3, with , one has for *free particles* the conservation of mass, momentum, and energy

**3.2. Examples of forces**

An example of forces includes gravity, Coriolis, or electromagnetic forces that acts on the fluid. For instance, the gravity force is often taken to be

where denotes the upward vertical direction. In what follows, we shall ignore these forces.

**3.3. Cauchy stress tensor**

Cauchy stress tensor , a -tensor , accounts for the force acting on the boundary of fluid parcels. That is, for any fluid subdomain , the net force produced by the stress tensor is defined by

which yields the net force (due to the Cauchy stress)

in the (arbitrary) fluid domain , by a view of the divergence theorem. Here, by convention, the -component of the vector is .

**3.4. Ideal fluids**

By definition, ideal fluid is defined by ideally setting the Cauchy stress tensor to be of the form

in which is the so-called the pressure of the fluid and denotes the identity matrix. By (5) and (6), the momentum equations *for ideal fluids* then read

The equations, together with the continuity equations, are referred to as *the Euler equations.*

**3.5. Viscous fluids**

To account for friction, one needs to take into account of the additional viscous stress tensor . For Newtonian fluids, the viscous stress is assumed to be proportional to the gradient of velocity field:

in which denote the LamÃ© viscosity coefficients. A direct computation yields the net viscous force

**3.6. Compressible flows**

Combining, the conservation of mass and momentum yields the compressible Euler (when no viscosity) and Navier-Stokes equations

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:

In the literature, the full set of compressible flows takes into account of the conservation of energy as well.

Let us compute the rate of change of the total energy. The kinetic energy satisfies

whereas the potential energy satisfies

or equivalently, . For instance, in the case of the – law pressure , we take . The total energy satisfies

with defined as in (9). In particular, the total energy is decreasing in time. For *ideal fluids*, the total energy is constant in time (for smooth solutions).

**3.7. Incompressible flows**

When the flow is assumed to be incompressible, the Euler and Navier-Stokes equations are

Unlike in the compressible case, this set of equations is complete and the pressure itself is an unknown function. The most popular model is when fluid is incompressible and homogenous (), and is often referred to simply as the Euler () and Navier-Stokes equations.

We have the (kinetic) energy equality

Here, we note that since satisfies the transport equation, with the incompressible velocity vector field, the potential energy is conserved in time; see Section 2.

**3.8. Reynolds number**

Consider the incompressible homogenous Navier-Stokes equations. Let be the time unit, the length unit, and the velocity unit, with . Let us introduce the change of variables

We then arrive at the non-dimensional Navier-Stokes equations:

with being called the physical Reynolds number. The problem of small viscosity limit or high Reynolds number has a very long story. Indeed, it is one of the most classical subjects in fluid dynamics. It interests most prominent physicists such as Lord Rayleigh, W. Orr, A. Sommerfeld, Heisenberg, W. Tollmien, H. Schlichting, among many others. It was already noted by Reynolds himself in his seminal experiment (1883) that the Reynolds number governs the transition from laminar to turbulent flows. The studies became active around 1930, motivated by the study of the boundary layer around wings. In airplanes design, it is crucial to study the boundary layer around the wing, and more precisely the transition between the laminar and turbulent regimes, and even more crucial to predict the point where boundary layer splits from the boundary. A large number of papers has been devoted to the estimation of the critical Reynolds number of classical shear flows (Blasius profile, exponential suction/blowing profile, etc…). It was Heisenberg in 1924 who first estimated the critical Reynolds number of parallel shear flows. C. C Lin and then Tollmien around 1940s completed the picture with lower and upper stability branches, respectively for parallel flows and boundary layers. Most of the physical literature, together with many mathematical insights, on the subject is well documented by Drazin and Reid in their famous book on hydrodynamics instability. I will be sure to come back to this topic near the end of the course.

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