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“Notes of Fluid Mechanics”
Piero Olla
ISAC-CNR and INFN, Sez. Cagliari, I–09042 Monserrato, Italy
June 9, 2022
Contents
1 Continuous limit 2
1.1 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Fluid kinematics 5
2.1 Lagrangian and Eulerian description of a flow . . . . . . . . . . . . . . . . . . . . 5
2.2 Lagrangian transport of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Vorticity, rate of strain and compression rate . . . . . . . . . . . . . . . . . . . . 10
2.4 Application to Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Conservation of mass and momentum 14
3.1 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Constitutive laws 17
4.1 The Navier-Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Condition of local thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . . 18
4.2.1 Digression into plasma physics . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Microscopic interpretation of pressure and viscosity . . . . . . . . . . . . . . . . . 20
4.4 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.5 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Conservation of energy 24
5.1 Kinetic energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Isoentropic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.4 Propagation of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.5 Bernoulli’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.6 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Hydrostatics 31
6.1 Stability under convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1
7 Compressible flows 35
7.1 The Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.2 The Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.2.1 Viscous Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.2.2 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8 Ideal and viscous flows 44
8.1 Potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 Fluid inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.3 Gravity waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8.3.1 Viscous corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.4 Viscous flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.5 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9 Vorticity dynamics 55
9.1 Kelvin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.2 Helicity conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
9.3 Two-dimensional flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9.4 Invariance under relabeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.5 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
10 Turbulence 65
10.1 Homogeneous isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.1.1 Time structure of the inertial range . . . . . . . . . . . . . . . . . . . . . 70
10.1.2 Transport of a passive scalar . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.1.3 Two-dimensional turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 72
10.2 Suggested reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1 Continuous limit
Weare interested in the description of fluids at macroscopic scales such that effects
from the discrete nature of the medium can be disregarded. In other words, we are
focusing on phenomena at scales l, much larger than the typical molecular separation
a0. The typical number Nl of molecules in a volume Vl of linear size l, therefore,
will be very large,
N ∼(l=a )3 ≫ 1; (1.1)
l 0
and the relative fluctuation δNl=Nl very small; we can thus approximate instanta-
neous quantities with their average with respect to fluctuations from discrete effects,
Nl ≃ hNli.
The lenght l defines the spatial scale of variation of the macroscopic variables
of interest in the fluid. This allows us to define macroscopic quantities through a
process of of coarse graining at an intermediate scale a, a0 ≪ a ≪ l, such that the
variation of macroscopic quantities at scale a is small, and the relative fluctuation
magnitude of these quantities is small as well.
2
Let us start with the density n. We first define a corse-grained density
na(x;t) = Na(x;t) ≃ hNa(x;t)i = hna(x;t)i (1.2)
Va Va
and then exploit the condition a0 ≪ a ≪ l to formally carry out the continuous
limit
n(x;t) = limna(x;t): (1.3)
a→0
We follow the same procedure with the current density J and fluid velocity
u(x;t). We indicate with vi(t) the instantaneous velocities of the molecules in
Va = Va(x) and define
n(x;t)u(x;t) = J(x;t) = limV−1 Xvi(t); (1.4)
a→0 a
i∈Va
Weshall focus in this course on systems composed of a single species of molecules.
The density n and the current density J will then be proportional, through the
molecular mass m, to the mass density ρ and the mass current density Jm:
ρ = mn; J =mJ: (1.5)
m
Macroscopic quantities such as the density n and the fluid velocity u are sums
of microscopic contributions by the individual molecules. If the interaction of the
molecules is not too strong, it is possible to consider the microscopic contributions
as statistical independent. This hypthesis allows us to estimate the fluctuation
amplitude of macroscopic quantities. Suppose we have N molecules contributing to
the sum; indicate with xi the contribution by the ith molecule and with X = PN xi
i
the macroscopic quantity. We suppose the xi to be identically distributed variables
with average hx i = µ and RMS h(x −µ)2i1=2 = σ . We have for the average of X:
i x i x
µX =Nµx (1.6)
and for its RMS:
σ2 =h(X −µX)2i=Xh(xi−µ)(xj −µ)i: (1.7)
X
ij
Statistical independence, however, implies that the xi’s are uncorrelated:
h(x −µ)(x −µ)i = σ2δ : (1.8)
i j ij
Hence only terms with i = j in Eq. (1.7) contribute to σ2 and we are left with
X
σ2 =Nσ2: (1.9)
X x
3
Thus, for large N,
δX ∼ σX ∼N−1=2: (1.10)
X µX
As an application, let us evaluate the fluctuation in the occupation number Na in
a volume Va. Indicate with N the number of molecules in the fluid that could
potentially lie in Va and introduce a random variable xi that is = 1 if i ∈ Va,
=0otherwise. We consider identically distributed molecules, indicating with p the
probability that a given molecule lies in Va at a given time. We immediately find
µ =p and; σ2 = hx2i−µ2 = p−p2; (1.11)
x x i x
and therefore, from Eqs. (1.6) and (1.9),
µ =pN; σ2 =(p−p2)N; (1.12)
Na Na
which implies
p 2
δN ∼ (p −p )N = (1−p)1=2N−1=2 ∼ N−1=2: (1.13)
N pN
We can carry out the same reasoning with the other macroscopic quantities we
have introduced in this section, and we obtain
δna ∼ δJa ∼ δua ∼ δNa ∼ (na3)−1=2: (1.14)
n J u Na
The condition for a continuous limit can then be reformulated as
nl3 ≫ 1: (1.15)
Weconcludethesectionbyintroducingaconceptthatwillaccompanyusthrough-
out the course: that of fluid element (or fluid parcel). A “fluid element” is simply
a portion of the fluid that, on time scales of interest, is not significantly deformed
by the gradients of u(x;t). We define more precisely fluid element by the condition
that points on its surface move with the local fluid velocity u(x;t). This requirement
guarantees that the mass in the volume remains constant, even though molecules
continuously cross the volume boundary. We use the notation VL to make clear that
a given volume (non necessarily a fluid element) is transported by the flow. Note
that the motion of a fluid element is identical to that of a solid particle small enough
to be transported by the fluid without exerting any feedback force. We call such an
object a passive tracer.
1.1 Suggested reading
• L.D. Landau and E.M. Lifshitz, “Statistical Physics” Vol. 5, Secs. 1, 2 and
114 (Pergamon Press, 1980)
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