I go on with some basic concepts and classical results in fluid dynamics [numbering is in accordance with the previous notes]. Throughout this section, I consider compressible barotropic **ideal fluids** with the pressure law or incompressible **ideal fluids** with constant density (and hence, the pressure is an unknown function in the incompressible case).

To unify the two cases in the presentation, we write

with (incompressible) or (compressible). The momentum equation then reads

See Section 3.4 in the previous notes.

**4.1. Vorticity**

Recall that the particle trajectory is defined by . To compare the dynamics of two particles that are infinitesimally close to one another, one looks at , for sufficiently small . One has

in which by convention, . We write

the sum of symmetric and anti-symmetric matrices. For constant symmetric matrices , the map represents stretching or compressing, depending on the sign of eigenvalues of . For the anti-symmetric part, we write

with denoting *the vorticity of the fluid motion*, defined by

For constant vorticity , the solution map of the ODE , with , preserves its length and is in fact a rigid rotation around the -axis with an angle . That is, vorticity is responsible for rotation in the fluid motion.

Lemma 4Assume sufficient integrability of and . There holds

*Proof:* One computes

By using the identity and the zero boundary condition on , the lemma is proved.

**4.2. Evolution of vorticity**

With , we can compute

That is, the momentum equation now reads

Theorem 5 (Bernoulli theorem)In the steady flow of a incompressible fluid with constant density or compressible barotropic fluid, the Bernoulli function is constant on the streamlines of the flow.

*Proof:* It follows directly from (12) that , which is the statement of the theorem.

Taking the curl of the momentum equation (12) and using the fact that , , and

one gets *the evolution of vorticity in the fluid:*

For incompressible fluids, the vorticity equation reads

For compressible barotropic fluids, we use the continuity equation into the vorticity equation, yielding

Remark 1 (Vorticity in 2D)In 2D, vorticity is a scalar function, and the right-hand side term . This proves that the vorticity or is transported along the flow.

Remark 2 (Vortex stretching)In 3D, the term is sometimes referred to as the vortex stretching term. The presence of this term makes the dynamics of vorticity in 3D greatly different and much delicate than that of vorticity in 2D. In fact, there is a well-known Beale-Kato-Majda’s criterium which asserts that the sup norm of vorticity controls the breakdown of smoothness of the solutions. In the other words, smooth solutions can be continued past a time as long as remains finite. In 2D, , the global smooth solutions exist. It remains an outstanding open problem for 3D to prove or disprove the breakdown of the smoothness of solutions in finite time (which is a million dollar problem, by the way).

Theorem 6 (Helmholtz law for vorticity)Let be the Lagrangian map: .

- For incompressible fluids with constant density, vorticity moves with the fluid in the sense
- For compressible barotropic fluids, moves with the fluid.

*Proof:* The theorem follows from a direct computation. Indeed, for the incompressible flows, we compute

and

This yields the identity (16), as it holds initially at time . Similar computation holds for the compressible case, upon using the continuity equation.

Remark 3For ideal incompressible fluids with constant density, there is no vorticity in the fluids, if none initially (unless there are non-potential forces taken into account in the flows). This is unlike in the viscous fluids where vorticity can be instantaneously created due to the presence of a boundary.

Theorem 6bisVortex lines are material, that is, they move with the flow: image of a vortex line under the Lagrangian map is also a vortex line.

*Proof:* By definition, vortex lines are curves in space, for each time, that are tangent to the vorticity at every point on the curve. Precisely, let be a vortex line, with parametrization for . By definition,

for some scalar function . Let , the image of under the map . The tangent vector at each point on reads

in which the last identity was due to the Helmholtz theorem. This proves that remains a vortex line.

**4.3. Circulation**

Let be a simple, smooth, oriented closed contour which is a deformation of a circle. The fluid circulation of the velocity field on the contour is defined by the line integral

in which the last identity is due to the Stokes theorem. That is, the circulation yields a computation of vorticity flux through the surface , whose boundary is the contour .

Theorem 7 (Kelvin’s circulation theorem)For compressible barotropic fluids or incompressible fluids with constant density, the circulation of is invariant under the flow:

*Proof:* We parametrize the contour by for , with . Then, , and therefore we can compute

since is a closed curve.

**4.4. Potential flows**

We consider the ideal incompressible fluids with constant density. Potential flows are those with zero vorticity: . Equivalently, we can write

The incompressibility yields that is a harmonic function; namely,

That is, the time dependence arises only through the boundary condition on . For a given , is uniquely determined, up to a constant, from the integral formula

in which denotes the Green kernel of the Laplacian with the Neumann boundary condition. It is useful to recall that the Green kernel in the three dimensional case satisfies

In 2D, the same bound holds, except for the case where a factor of enters in the estimate. As for the pressure , the momentum equation (12) yields

That is, up to a constant, the pressure is computed through the Bernoulli equation for unsteady flows (cf. Theorem 5 for steady flows):

Theorem 8 (d’Alembert paradox)For any bounded smooth solid obstacle that moves with a constant velocity within incompressible, irrotational fluids that are at rest infinitely far away, the total force exerted on the obstacle is zero.

*Proof:* To fix the fluid domain (to be stationary), we work with the moving frame . Hence, the obstacle is at rest and the fluid is moving at the far field with constant velocity . By assumption (and the fact that is simply connected), the fluid is of the form , with solving

and

The boundary condition on is to assure that the fluid does not enter the obstacle. Up to a constant, we write

with being the harmonic function with nonzero boundary condition . By a view of (18), we have

for . In fact, we have the expansion near infinity for , and hence for :

for some constant . In addition, the Bernoulli relation yields The total force acting on the obstacle is computed by the surface integral

Recall that on the boundary . For sufficiently large , we compute

in which both incompressibility and irrationality were used. Now, when and , the above integral is zero, by the divergence theorem. Using (19) and the fact that , we have

Again, the incompressibility was used. The theorem follows, by taking .

Remark 4The d’Alembert’s paradox asserts that any solid body emerged in stationary potential flows feels neither lift nor drag acting on it (in the layman words, birds can’t fly!). The paradox is precisely due to the neglect of viscous effects. It was Prandtl in the beginning of the 20th century who showed the significance of viscosity near the boundary, despite it being sufficiently small. However, the mathematical justification of the Prandtl boundary layer theory remains an open problem (I shall discuss this subject later on in the course).

**4.5. 2D steady potential flows**

We study 2D steady flows which are both incompressible and irrotational. Denote the velocity vector field by and the 2D coordinates by . By the incompressibility and irrationality , we write

for some potential function and some stream function (which represent the trajectories of particles). Here, . Both and are harmonic functions. To make use of the power of complex analysis, we introduce *complex potential*

for . The equations (20) are the Cauchy-Riemann equations for . The *complex velocity* of the fluid is defined by

Conversely, given any complex potential function , the pair , constructed by and , defines a steady potential flow.

We are interested in computing the forces exerted by the fluid on a solid body. Let be a bounded open domain in and let . Assume that is not self-intersecting. The force on the body is equal to the pressure force acting on the boundary , and is computed by the line integral

Here, we identify a 2D vector with a complex number in the usual way . Blasius, who was in fact a student of Prandtl, then gave a force calculation in term of complex potential via a complex contour integral as follows.

Theorem 9 (Blasius theorem)Consider the incompressible, steady, and irrotational fluids in the exterior of a solid bounded body in . Then, the force exerted by the fluid on the body is computed by

*Proof:* By the Bernoulli relation for potential flows, we have and so

On the other hand, since or on the boundary , one computes

which yields the theorem.

Next, we obtain the following.

Theorem 10 (Kutta-Joukowski theorem)Consider the incompressible, steady, and irrotational fluids in the exterior of a solid bounded body in . Assume that the fluid moves with a constant velocity at infinity. Then, the force exerted by the fluid on is computed by

where is the circulation around ; see (17).

*Proof:* Let be the complex potential associated with the fluid. Since is analytic and the complex velocity , there holds the Laurent’s asymptotic expansion near infinity

in which . To compute , one uses the Cauchy’s theorem, yielding

in which the boundary condition was used. Similarly, using the Laurent series, we compute

The theorem is proved, upon using the Blasius theorem.

Remark 5The force exerted by the fluid on the solid body is normal to the direction of the flow and is proportional to the circulation around . In particular, the body experiences no drag, that is, no force opposing the flow. This is of course again a paradox, in contradiction to observation and experiment. In addition, in the absence of the circulation, there is no net force at all acting on the body (neither drag nor lift). This is the 2D version of the d’Alembert paradox, which occurs due to the neglect of viscous effects in the fluid.

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