# The Navier-Stokes equation and a fully developed turbulence

M. Apostol
Department of Theoretical Physics, Institute of Atomic Physics,
Magurele-Bucharest MG-6, POBox MG-35, Romania
email<EMAIL_ADDRESS>
###### Abstract

In fairly general conditions we give explicit (smooth) solutions for the
potential flow. We show that, rigorously speaking, the equations of the fluid
mechanics have not rotational solutions. However, within the usual
approximations of an incompressible fluid and an isentropic flow, the
remaining Navier-Stokes equation has approximate vorticial (rotational)
solutions, generated by viscosity. In general, the vortices are unstable, and
a discrete distribution of vorticial solutions is not in mechanical
equilibrium; it forms an unstable vorticial liquid. On the other hand, these
solutions may exhibit turbulent, fluctuating instabilities for large
variations of the velocity over short distances. We represent a fully
developed turbulence as a homogeneous, isotropic and highly-fluctuating
distribution of singular centres of turbulence. A regular mean flow can be
included. In these circumstances the Navier-Stokes equation exhibits three
time scales. The equations of the mean flow can be disentangled from the
equations of the fluctuating part, which is reduced to a vanishing inertial
term. This latter equation is not satisfied after averaging out the temporal
fluctuations. However, for a homogeneous and isotropic distribution of non-
singular turbulence centres the equation for the inertial term is satisfied
trivially, _i.e._ both the average fluctuating velocity and the average
fluctuating inertial term are zero. If the velocity is singular at the
turbulence centres, we are left with a quasi-ideal classical gas of
singularities, or a solution of singularities in quasi thermal equilibrium in
the background fluid. This is an example of an emergent dynamics. We give
three examples of vorticial liquids.

Key words: potential flow; vorticity; instabilities; turbulence; gas of
singularities; singular vortices

##  1 Introduction

In fairly general conditions we give explicit (smooth) solutions for the
potential flow. As it is well known, the fluids may develop turbulence. In its
extreme manifestation the turbulent flow displays very irregular, disordered
velocities, fluctuating in time at each point in space. This is known as a
fully developed turbulence. By using such fluctuating velocities, besides a
steady mean velocity, the Navier-Stokes equation becomes an infinite hierarchy
of equations for velocity mean correlation functions, known as Reynolds’s
equations,[1] which need closure assumptions. According to the experimental
observations, it was realized that such irregular movements of the fluid
exhibit distributions of swirls (eddies, vortices), of various magnitude and
vorticities; it is likely that the large eddies transfer energy to the small
eddies, which dissipate it.[2]-[4] Statistical concepts like correlations,
homogeneity and isotropy have been introduced in the theory of turbulence,[5,
6] and dimensional analysis and similarity arguments allowed the derivation of
the energy spectrum of the turbulent eddies.[7]-[9]

Meanwhile, the relation of this statistical turbulence with the Navier-Stokes
equation remained unclear.[10]-[13] Could the Navier-Stokes equation describe
a turbulent motion? To what extent and in what sense? Has the Navier-Stokes
equation smooth and stable solutions? What is the appropriate representation
of a turbulent field of velocities?[14]

The dynamics of the vorticity has enjoyed much interest (see Refs. [15, 16]
and References therein). The results depend on model assumptions. Dynamical-
system concepts and statistical models have been invoked in studies of
turbulence, with chaotic behaviour, intermittency and coherent structures
(see, for instance, Refs. [17]-[21]). In particular, by analogy with the
quantum turbulence, "Turbulent flows may be regarded as an intricate
collection of mutually-interacting vortices", and "Vortex filaments may thus
be seen as the fundamental structure of turbulence,… ".[17]

The difficulties exhibited by the Navier-Stokes equation are related to the
viscosity, which governs the vorticity, and the inertial term, which is
quadratic in velocity. We show in this paper that the viscosity term in the
Navier-Stokes equation may produce vorticity, provided the fluid is
incompressible and the flow is isentropic. Although such an approximate
treatment may look reasonable, we can see that, rigorously speaking, the
fluids cannot exhibit vorticity. Moreover, we give arguments that the vortices
are unstable.

Further, we show in this paper that large variations of the velocity over
short distances lead to highly fluctuating, swirling instabilities, controlled
by viscosity. This is characteristic for the phenomenon of a fully developed
turbulence. In this case, the inertial term acquires a major role in
describing the flow. We represent a fully developed turbulence as a
superposition of fluctuating velocities, associated to a discrete set of
turbulence centres. A mean flow may be included. In general, the Navier-Stokes
equation, averaged over fluctuations, is not satisfied. On the other hand, a
homogeneous and isotropic distribution of (non-singular) turbulence centres
leads to vanishing averages of velocity and inertial term, such that the
Navier-Stokes equation is satisfied trivially. If the turbulence centres are
singular, we are left with a gas of singularities (or a solution of
singularities in the background fluid), which is in quasi thermal equilibrium.
The corresponding Navier-Stokes equation for the fluid of singularities is
reduced to Newton’s equation of motion, with a small friction.

We illustrate the above descripton with three examples of vorticial liquids
(filamentary liquid, coulombian and dipolar liquid).

##  2 Potential flow. Incompressible fluid

Let us consider a potential flow of an incompressible fluid. The velocity
$\boldsymbol{v}=grad\Phi$ is given by the gradient of a potential $\Phi$,
which satisfies the Laplace equation

$\Delta\Phi=0$ (1)

(incompressiblity condition $div\boldsymbol{v}=0$). The viscosity term
$\sim\Delta\boldsymbol{v}$ is zero, such that we are left with Euler’s
equation

$\frac{\partial\boldsymbol{v}}{\partial
t}+(\boldsymbol{v}grad)\boldsymbol{v}=-\frac{1}{\rho}gradp\,\,\,,$ (2)

where $\rho$ is the density and $p$ denotes the pressure. By using the well-
known identity

$(\boldsymbol{v}grad)\boldsymbol{v}=-\boldsymbol{v}\times
curl\boldsymbol{v}+grad(v^{2}/2)\,\,\,,$ (3)

equation (2) becomes

$\frac{\partial\boldsymbol{v}}{\partial t}+grad(v^{2}/2+p/\rho)=0\,\,\,,$ (4)

where $curl\boldsymbol{v}=0$. As it is well known, by using equation (1), this
equation leads to

$\frac{\partial\Phi}{\partial
t}+\frac{1}{2}(grad\Phi)^{2}+\frac{p}{\rho}=0\,\,.$ (5)

In this equation $p$ should be viewed as the variation of the pressure with
respect to equilibrium. We assume that $p$ does not depend on $\Phi$ and the
time.

In equations (1) and (5) the variables may be separated. Let
$g(\boldsymbol{r})$ be a solution of equation (1) (satisfying the boundary
conditions); the potential can be written as $\Phi=f(t)g(\boldsymbol{r})$,
where the function $f(t)$ satisfies equation (5),

$\frac{df}{dt}+\frac{1}{2g}(gradg)^{2}+\frac{p}{\rho g}=0\,\,.$ (6)

The acceptable solution of this equation (for $f(0)=0$) is

$f(t)=\frac{\sqrt{2\mid p\mid/\rho}}{\mid gradg\mid}\tanh\frac{\sqrt{\mid
p\mid/2\rho}\mid gradg\mid}{g}t$ (7)

for $p<0$. It may happen that the boundary conditions for the equation
$\Delta\Phi=0$ depend on time, thus providing the time derivative
$\dot{\Phi}$; in that case equation (5) gives the pressure.

##  3 Potential flow. Compressible fluid

Let us write down the equations of the fluid mechanics

$\begin{array}[]{c}\frac{\partial\rho}{\partial t}+\rho
div\boldsymbol{v}+\boldsymbol{v}grad\rho=0\,\,,\\\ \\\
\rho\frac{\partial\boldsymbol{v}}{\partial
t}+\rho(\boldsymbol{v}grad)\boldsymbol{v}=-gradp-\rho grad\varphi+\\\ \\\
+\eta\Delta\boldsymbol{v}+\left(\frac{1}{3}\eta+\zeta\right)grad\,div\boldsymbol{v}\,\,,\\\
\\\ \rho T\left(\frac{\partial s}{\partial
t}+\boldsymbol{v}grads\right)=\kappa\Delta
T+\sigma_{ij}^{{}^{\prime}}\partial_{j}v_{i}\,\,\,,\end{array}$ (8)

where $p$ is the internal pressure, $\varphi$ is an external potential, $\eta$
and $\zeta$ are the viscosity coefficients, $T$ is the temperature, $s$ is the
entropy per unit masss, $\kappa$ is the thermoconductivity and

$\sigma_{ij}^{{}^{\prime}}=\eta\left(\partial_{i}v_{j}+\partial_{j}v_{i}-\frac{2}{3}\delta_{ij}div\boldsymbol{v}\right)+\zeta\delta_{ij}div\boldsymbol{v}\,\,.$
(9)

is the viscosity tensor. In the Navier-Stokes equation (the second equation
(8)) the forces which determine the velocity are $gradp$ and $\rho
grad\varphi$, where $p=p(\rho,T)$ is a function of density and temperature.
For all the usual flows the relative variations $\delta\rho/\rho_{0}$ of the
density, $\delta T/T_{0}$ of the temperature, $\delta p/p_{0}$ of the
pressure, $\delta s/s_{0}$ of the entropy as well as the variation
$\delta\varphi/\varphi_{0}$ of the external potential are small, in comparison
with their equilibrium values, labelled by the suffix $0$, when the fluid is
at rest. Consequently, we may view the velocity $\boldsymbol{v}$ as a first-
order quantity, and linearize the above equations as

$\begin{array}[]{c}\frac{\partial\rho}{\partial
t}+\rho_{0}div\boldsymbol{v}=0\,\,,\\\ \\\
\rho_{0}\frac{\partial\boldsymbol{v}}{\partial
t}=-gradp-\rho_{0}grad\varphi+\\\ \\\
+\eta\Delta\boldsymbol{v}+\left(\frac{1}{3}\eta+\zeta\right)grad\,div\boldsymbol{v}\,\,,\\\
\\\ \rho_{0}T_{0}\frac{\partial s}{\partial t}=\kappa\Delta T\,\,.\end{array}$
(10)

We note that within this approximation there is no heat source, and the first-
order equation of energy conservation is reduced to an identity.

The density and entropy variations can be written as

$\begin{array}[]{c}\delta\rho=\frac{\rho_{0}}{K}\delta p-\beta\rho_{0}\delta
T\,\,,\\\ \\\ \delta s=-\frac{\beta}{\rho_{0}}\delta
p+\frac{c_{p}}{T_{0}}\delta T\,\,\,,\end{array}$ (11)

where $K$ is the isothermal modulus of compressibility
($1/K=-\frac{1}{V}(\partial V/\partial p)_{T}$, $V=1/\rho$),
$\beta=\frac{1}{V}(\partial V/\partial T)_{p}$ is the dilatation coefficient
and $c_{p}$ is the specific heat per unit mass at constant pressure, all at
equilibrium. In deriving equations (11) the Gibbs free energy $d\Phi=Vdp-sdT$
is used.

Part of the temperature variation in equation (11) is compensated by pressure
variation, as in an adiabatic process; we denote this contribution by $\delta
T_{1}$. The remaining part, denoted by $\delta T$, corresponds to the
conducted heat. Therefore, we write

$\begin{array}[]{c}\delta\rho=\frac{\rho_{0}}{K}\delta p-\beta\rho_{0}\delta
T_{1}-\beta\rho_{0}\delta T\,\,,\\\ \\\ \delta s=-\frac{\beta}{\rho_{0}}\delta
p+\frac{c_{p}}{T_{0}}\delta T_{1}+\frac{c_{p}}{T_{0}}\delta
T=\frac{c_{p}}{T_{0}}\delta T\,\,\,,\end{array}$ (12)

whence

$\frac{\beta}{\rho_{0}}\delta p=\frac{c_{p}}{T_{0}}\delta T_{1}$ (13)

and

$\delta\rho=\frac{\rho_{0}}{K}\left(1-\frac{\beta^{2}T_{0}K}{\rho_{0}c_{p}}\right)\delta
p-\beta\rho_{0}\delta T\,\,.$ (14)

In this equation we use the thermodynamic relation

$\frac{\beta^{2}T_{0}K}{\rho_{0}}=c_{p}-c_{v}\,\,\,,$ (15)

where $c_{v}$ is the specific heat per unit mass at constant volume.[22]
Therefore, equation (14) becomes

$\delta\rho=\frac{\rho_{0}c_{v}}{Kc_{p}}\delta p-\beta\rho_{0}\delta T\,\,\,,$
(16)

or

$\delta p=\frac{Kc_{p}}{\rho_{0}c_{v}}\delta\rho+\frac{\beta
Kc_{p}}{c_{v}}\delta T\,\,.$ (17)

Now we use two other thermodynamic relations

$\frac{c_{p}}{c_{v}}K=K_{ad}\,\,,\,\,\beta K=\alpha\,\,\,,$ (18)

where $K_{ad}$ is the adiabatic modulus of compressibility and
$\alpha=(\partial p/\partial T)_{V}$ is the thermal pressure coefficient.[22]
Finally, we get

$\delta p=\frac{K_{ad}}{\rho_{0}}\delta\rho+\frac{c_{p}}{c_{v}}\alpha\delta
T\,\,\,,$ (19)

which is used in the Navier-Stokes equation. Equations (10) become

$\begin{array}[]{c}\frac{\partial\rho}{\partial
t}+\rho_{0}div\boldsymbol{v}=0\,\,,\\\ \\\
\rho_{0}\frac{\partial\boldsymbol{v}}{\partial
t}=-\frac{K_{ad}}{\rho_{0}}grad\rho-\rho_{0}grad\varphi-\frac{c_{p}}{c_{v}}\alpha
gradT+\\\ \\\
+\eta\Delta\boldsymbol{v}+\left(\frac{1}{3}\eta+\zeta\right)grad\,div\boldsymbol{v}\,\,,\\\
\\\ \rho_{0}c_{p}\frac{\partial T}{\partial t}=\kappa\Delta
T\,\,\,,\end{array}$ (20)

where the second equation (12) is used. We can see that the temperature
equation (20) is independent; it describes the transport of an external
temperature, which may provide a source for the velocity in the Navier-Stokes
equation. We may leave aside this external temperature.

Let us seek a potential-flow solution of the above equations, where the
velocity is derived from a potential $\Phi$, by

$\boldsymbol{v}=grad\Phi\,\,.$ (21)

We notice that $curl\boldsymbol{v}=0$ and $curl\,curl\boldsymbol{v}=0$, _i.e._
$\Delta\boldsymbol{v}=grad\,div\boldsymbol{v}$. Therefore, the Navier-Stokes
equation can be written as

$\begin{array}[]{c}\rho_{0}\frac{\partial\boldsymbol{v}}{\partial
t}=-\frac{K_{ad}}{\rho_{0}}grad\rho-\rho_{0}grad\varphi+\\\ \\\
+\left(\frac{4}{3}\eta+\zeta\right)grad\,div\boldsymbol{v}\,\,.\end{array}$
(22)

By using equation (21), we obtain

$\begin{array}[]{c}\frac{\partial\rho}{\partial
t}+\rho_{0}\Delta\Phi=0\,\,,\\\ \\\ \frac{\partial\Phi}{\partial
t}+\frac{K_{ad}}{\rho_{0}^{2}}\rho+\varphi-\frac{1}{\rho_{0}}\left(\frac{4}{3}\eta+\zeta\right)\Delta\Phi=0\,\,\,,\end{array}$
(23)

up to a function of time, where $\rho$ and $\varphi$ should be viewed as their
corresponding variations. By an additional time differentiation we obtain

$\frac{\partial^{2}\Phi}{\partial
t^{2}}-\frac{K_{ad}}{\rho_{0}}\Delta\Phi+\dot{\varphi}-\frac{1}{\rho_{0}}\left(\frac{4}{3}\eta+\zeta\right)\Delta\dot{\Phi}=0\,\,.$
(24)

This equation provides the potential $\Phi$, therefore the velocity
$\boldsymbol{v}$ through equation (21) and the density $\rho$ through the
first equation (23).

Equation (24) is the wave equation with friction (the term
$\sim\Delta\dot{\Phi}$) and sources ($-\dot{\varphi}$). The ratio
$K_{ad}/\rho_{0}$ is the square of the sound velocity
$c=\sqrt{K_{ad}/\rho_{0}}$. The elementary solutions $e^{-i\omega
t}e^{i\boldsymbol{kr}}$ of this (homogeneous) equation are damped plane waves

$e^{\mp ickt}e^{i\boldsymbol{kr}}e^{-\frac{\sigma k^{2}}{c}t}\,\,\,,$ (25)

for $\sigma k\ll c$, where $\sigma=(4\eta/3+\zeta)/2\rho_{0}$. The relaxation
time is much longer than the wave period. A wave propagating along the
$x$-direction is proportional to $\sim e^{-\gamma x}$, with the attenuation
coefficient $\gamma=\sigma k^{2}/c=\sigma\omega^{2}/c^{3}$. This is the well-
known absorption coefficient for sound (without the $\kappa$-contribution).

##  4 Vorticity

Euler’s equation for an ideal fluid can be written as

$\frac{d\boldsymbol{v}}{dt}=-gradw\,\,\,,$ (26)

where $w$ is the enthalpy ($dw=\frac{1}{\rho}dp$); the pressure $p$ is a
function of density $\rho$. By taking the $curl$, we get

$curl\frac{d\boldsymbol{v}}{dt}=0\,\,.$ (27)

On the other hand,

$\frac{d\boldsymbol{v}}{dt}=\frac{\partial\boldsymbol{v}}{\partial
t}+(\boldsymbol{v}grad)\boldsymbol{v}=\frac{\partial\boldsymbol{v}}{\partial
t}+\left(\frac{\partial\boldsymbol{v}}{\partial t}\right)_{f}\,\,\,,$ (28)

where the suffix $f$ indicates that the derivative is taken along the flow.
Equation (27) becomes

$\frac{\partial}{\partial t}curl\boldsymbol{v}+\left(\frac{\partial}{\partial
t}curl\boldsymbol{v}\right)_{f}=0\,\,;$ (29)

since the two variations of the $curl\boldsymbol{v}$ are independent, we get

$curl\boldsymbol{v}=0\,\,\,,$ (30)

_i.e._ the vorticity $curl\boldsymbol{v}$ is conserved along the flow.
Therefore, we cannot create, or destroy, vorticity $curl\boldsymbol{v}$ in the
flow of an ideal fluid. Equation (30) is valid in the absence of special
external force, which do not derive from a gradient. This is Helmholtz’s
circulation law. As it is well known, an ideal fluid supports only an
irrotational (potential) flow, where the velocity is derived from a scalar
potential ($\boldsymbol{v}=grad\Phi$). By knowing the equation of state of the
fluid, the Euler equation and the continuity equation are fully determined.

For a real, viscid, fluid the Navier-Stokes equation is

$\rho\frac{d\boldsymbol{v}}{dt}=-gradp+\eta\Delta\boldsymbol{v}+\left(\frac{1}{3}\eta+\zeta\right)grad\,div\boldsymbol{v}\,\,\,,$
(31)

where $\eta$, $\zeta$ are the viscosity coefficients. By taking the $curl$, we
get

$\begin{array}[]{c}curl\left(\rho\frac{d\boldsymbol{v}}{dt}\right)=grad\rho\times\frac{d\boldsymbol{v}}{dt}+\rho
curl\frac{d\boldsymbol{v}}{dt}=\\\ \\\ =\eta\Delta
curl\boldsymbol{v}\,\,;\end{array}$ (32)

we can see that the viscosity $\eta$ can generate vorticity
($curl\boldsymbol{v}\neq 0$). In general, the velocity $\boldsymbol{v}$
rotates about the vorticity $curl\boldsymbol{v}$. This is a vortex.

Equation (32) is in conflict with the continuity equation

$\frac{d\rho}{dt}+\rho div\boldsymbol{v}=0\,\,.$ (33)

Indeed, if $curl\boldsymbol{v}\neq 0$, the velocity should be derived from a
$curl$ (not from a $grad$!), _i.e._ we should have
$\boldsymbol{v}=curl\boldsymbol{A}$, where $\boldsymbol{A}$ is a vector
potential. Consequently, the fluid should be incompressible
($div\boldsymbol{v}=div\,curl\boldsymbol{A}=0$). The density should be
constant, both in time and space (along the flow). This indicates that in a
compressible fluid we cannot have vortices. Usually, the variations of the
density are small, such that they may be neglected for the present purpose.

Therefore, we may limit to an incompressible fluid ($div\boldsymbol{v}=0$),
for which equation (32) becomes

$curl\frac{d\boldsymbol{v}}{dt}=\nu\Delta curl\boldsymbol{v}\,\,\,,$ (34)

or

$\frac{\partial}{\partial t}curl\boldsymbol{v}-curl\left(\boldsymbol{v}\times
curl\boldsymbol{v}\right)=\nu\Delta curl\boldsymbol{v}\,\,\,,$ (35)

where $\nu=\eta/\rho$ is the kinematical viscosity and we have used the
identity $(\boldsymbol{v}grad)\boldsymbol{v}=-\boldsymbol{v}\times
curl\boldsymbol{v}+grad(v^{2}/2)$. This is the equation of vorticity; it can
also be written as

$\frac{\partial}{\partial
t}curl\boldsymbol{v}+curl\left[(\boldsymbol{v}grad)\boldsymbol{v}\right]=\nu\Delta
curl\boldsymbol{v}\,\,.$ (36)

This equation gives the velocity. The pressure is obtained from the Navier-
Stokes equation. If we write the Navier-Stokes equation as

$\frac{\partial}{\partial
t}curl\boldsymbol{A}+(\boldsymbol{v}grad)\boldsymbol{v}=-\frac{1}{\rho}gradp+\nu\Delta
curl\boldsymbol{A}\,\,\,,$ (37)

we get

$\begin{array}[]{c}div\left[(\boldsymbol{v}grad)\boldsymbol{v}+\frac{1}{\rho}gradp\right]=0\end{array}$
(38)

and

$div\left(\frac{\partial\boldsymbol{v}}{\partial
t}-\nu\Delta\boldsymbol{v}\right)=0\,\,\,,$ (39)

which is an identity ($div\boldsymbol{v}=0$,
$\boldsymbol{v}=curl\boldsymbol{A}$). In some cases a particular solution of
these equations is provided by

$\begin{array}[]{c}(\boldsymbol{v}grad)\boldsymbol{v}=-grad(p/\rho)\,\,,\\\
\\\ \frac{\partial\boldsymbol{v}}{\partial
t}=\nu\Delta\boldsymbol{v}\,\,.\end{array}$ (40)

We can see that the Navier-Stokes equation is split into (the derivatives of)
a diffusion (heat) equation and an equilibrium equation; the first equation
(40) indicates an equilibrium between the pressure force $-grad(p/\rho)$ and
Euler’s force $(\boldsymbol{v}grad)\boldsymbol{v}$. The diffusion equation
(40) holds also for the vorticity, because the above equations are valid for a
non-vanishing vorticity. It is easy to see that these equations generalize the
equations for the Couette flow.

However, we have also the heat-transfer equation. For an incompressible fluid
it reads

$\rho c_{p}\frac{dT}{dt}=\kappa\Delta
T+\frac{1}{2}\eta\left(\partial_{i}v_{j}+\partial_{j}v_{i}\right)^{2}\,\,\,,$
(41)

or

$\frac{dT}{dt}=\chi\Delta
T+\frac{1}{2}\frac{\nu}{c_{p}}\left(\partial_{i}v_{j}+\partial_{j}v_{i}\right)^{2}\,\,\,,$
(42)

where $c_{p}$ is the specific heat per unit mass at constant pressure,
$\kappa$ is the thermoconductivity, and $\chi=\kappa/\rho c_{p}$ is the
thermometric conductivity. For an incompressible fluid this equation can be
transformed into an equation for the derivatives of the pressure,

$\frac{dp}{dt}=\chi\Delta
p+\frac{1}{2}\frac{\alpha\nu}{c_{p}}\left(\partial_{i}v_{j}+\partial_{j}v_{i}\right)^{2}\,\,\,,$
(43)

where $\alpha=(\partial p/\partial T)_{v}$ is the thermal pressure
coefficient. In general, equation (43) is not compatible with the Navier-
Stokes equation. Therefore, rigorously speaking, we cannot have vorticity in
an incompressible fluid either. Usually, the coefficient $\nu/c_{p}$ is very
small (of the order $10^{-24}-10^{-25}g\cdot cm^{2}/s$), such that, for small
gradients of velocity, we have a low rate of entropy production (though, a
factor of the order $10^{23}K/erg$ should be taken into account). Under these
conditions, we may assume that the flow is isentropic and the heat-transfer
equation may be neglected.

In order to get an idea of how large the variations of the density and the
temperature can be, we may estimate a change $\delta p$ in pressure from
$\delta p\simeq\rho v^{2}$. A velocity $v=100km/h$, which is fairly large,
produces a change $\delta p\simeq 10^{4}dyn/cm^{2}$ in air
($\rho=10^{-3}g/cm^{3}$), whose normal pressure is $10^{6}dyn/cm^{2}$;
therefore, $\delta p/p\simeq 10^{-2}$. Such a velocity ($\simeq 3\times
10^{3}cm/s$) is close to the mean thermal velocity $\simeq 10^{4}cm/s$ (for
normal air), and close to the sound velocity in normal air $c\simeq 3.5\times
10^{4}cm/s$. For this velocity we still expect local thermal equilibrium. The
change in density is given by $\delta p=K(\delta\rho/\rho)$, where
$K=-V(\partial p/\partial V)$ is the (say, isothermal) modulus of compression.
For air $K\simeq 10^{6}dyn/cm^{2}$, for water $K\simeq 10^{10}dyn/cm^{2}$,
such that we get $\delta\rho/\rho\simeq 10^{-2},\,10^{-6}$. The change in
temperature is obtained from $\delta p=\alpha T(\delta T/T)$, where
$\alpha=(\partial p/\partial T)_{V}$ is the thermal pressure coefficient (at
constant volume $V$). For water $\alpha\simeq 10^{22}/cm^{3}$, for gases it is
much higher; for normal temperature $T=300K$ we get $\delta T/T\simeq
10^{-4}$, or much lower. Consequently, we may expect an almost ideal,
incompressible flow. We note that, although we neglect the viscosity in the
heat-transfer equation, we keep it in the Navier-Stokes equation.

Therefore, within these approximations (incompressibility and constant
entropy), we are left with equation (36) and the Navier-Stokes equation for a
vorticial flow. In general, an external pressure which satisfies the Navier-
Stokes equation (or equation (38)) is very special, such that, if they exist,
the vortices might be, in fact, unstable. They develop an Euler’s force, which
is difficult to be compensated by an external force.

We note that, under the conditions stated above, the viscosity may generate
(unstable) vortices. In the next section we show that the viscosity may
generate another type of instabilities.

For small variations of the velocity we may neglect the inertial term in the
vorticity equation (36), which becomes

$\frac{\partial}{\partial t}curl\boldsymbol{v}=\nu\Delta
curl\boldsymbol{v}\,\,.$ (44)

By making use of $\boldsymbol{v}=curl\boldsymbol{A}$, we get

$\left(\Delta-grad\,div\right)\left(\frac{\partial\boldsymbol{A}}{\partial
t}-\nu\Delta\boldsymbol{A}\right)=0\,\,.$ (45)

A solution of this equation is provided by

$\frac{\partial\boldsymbol{A}}{\partial t}-\nu\Delta\boldsymbol{A}=0\,\,\,,$
(46)

which leads to

$\boldsymbol{A}=\boldsymbol{A}_{0}e^{-\lambda\nu t}e^{\pm
i\sqrt{\lambda}r}/r\,\,\,,$ (47)

where $\boldsymbol{A_{0}}$ and $\lambda$ are two constants. The velocity
acquires the form

$\boldsymbol{v}=-\boldsymbol{A}_{0}\times grad\left(e^{-\lambda\nu t}e^{\pm
i\sqrt{\lambda}r}/r\right)\,\,\,,$ (48)

and the pressure is uniform within this approximation. We note that, although
the spatial dependence of the solution does not depend on viscosity, it is
generated by the viscosity term $\nu\Delta\boldsymbol{v}$.

##  5 Instabilities

The equation of energy conservation for an incompressible fluid is

$\begin{array}[]{c}\frac{\partial}{\partial t}\left(\frac{1}{2}\rho
v^{2}\right)+div\left[\boldsymbol{v}\left(\frac{1}{2}\rho
v^{2}+p\right)-\frac{1}{2}\eta grad(v^{2})\right]+\\\ \\\
+\eta\left(\partial_{j}v_{i}\right)^{2}=0\,\,;\end{array}$ (49)

it is obtained by multiplying by $\boldsymbol{v}$ the Navier Stokes equation
(31) for an incompressible fluid ($div\boldsymbol{v}=0$). The $div$-term
represents a transport of energy and mechanical work of the pressure, and an
energy flux associated with collisions (viscosity); the term
$\eta\left(\partial_{j}v_{i}\right)^{2}$ represents the heat produced by
viscosity. We integrate this equation over a volume $V$ enclosed by a surface
$S$,

$\begin{array}[]{c}\frac{\partial}{\partial t}\int dV\left(\frac{1}{2}\rho
v^{2}\right)+\oint dS\left[v_{n}\left(\frac{1}{2}\rho
v^{2}+p\right)-\frac{1}{2}\eta\partial_{n}(v^{2})\right]+\\\ \\\ +\eta\int
dV\left(\partial_{j}v_{i}\right)^{2}=0\,\,\,,\end{array}$ (50)

where $v_{n}$ is the velocity component normal to the surface and
$\partial_{n}$ is the derivative along the normal to the surface.

We compare the orders of magnitude of the surface terms and the $\eta$-volume
term, and get ratios of the form $\frac{Sl}{V}R$, $\frac{Sl}{V}(p/\rho
v^{2})R$, $\frac{Sl}{V}$, where $l$ is the distance over which the velocity
varies and $R=vl/\nu$ is the Reynolds number. For moderate Reynolds numbers
and $Sl/V\ll 1$ we can neglect the surface contributions in comparison with
the heat term. By writing

$\boldsymbol{v}=f(t)\boldsymbol{u}(\boldsymbol{r})\,\,\,,$ (51)

the above equation becomes

$\frac{\partial}{\partial t}f^{2}\cdot\int dV\left(\frac{1}{2}\rho
u^{2}\right)+\eta f^{2}\int dV\left(\partial_{j}u_{i}\right)^{2}=0\,\,.$ (52)

We can see that the time dependence of the velocity is a damped exponential.
The flow is stable, as a consequence of the dissipated heat. The $\eta$-term
in equation (52) gives, in fact, the increase of entropy.

Let us assume that the integration domains are sufficiently small, such that
$Sl/V$ is of the order of unity; the velocity varies over a distance $l$
inside the domains, but we assume that it suffers a large discontinuity across
the surface, over a small distance $\delta\ll l$. Then, it is easy to see that
the dominant term in equation (50) is the collision term, such that equation
(50) becomes

$\frac{\partial}{\partial t}f^{2}\cdot\int dV\left(\frac{1}{2}\rho
u^{2}\right)-\frac{1}{2}\eta f^{2}\oint dS\partial_{n}(u^{2})=0\,\,.$ (53)

We can see that for a positive normal derivative the flow is unstable. The
viscosity is insufficient to disipate the energy as heat, and the energy is
transferred by molecular collisions (viscosity) through surfaces of
discontinuities. The process occurs in small domains, with large
discontinuities of velocity across their surface, and the instabilities imply
returning, swirling and fluctuating, velocities. This is the turbulence
phenomenon. We note that the instabilities are governed by viscosity, which
gives also vorticity (when the entropy production is neglected). Moreover, we
note that the inertial term does not appear in instabilities, though it plays
an important role in turbulence.

The above arguments can be extended to compressible fluids, including the
variations of the temperature. Indeed, the energy conservation in this case
reads

$\frac{\partial}{\partial t}\left(\frac{1}{2}\rho
v^{2}+\rho\varepsilon\right)=-\partial_{j}\left[\rho
v_{j}\left(\frac{1}{2}v^{2}+w\right)-v_{i}\sigma_{ij}^{{}^{\prime}}-\kappa\partial_{j}T\right]\,\,\,,$
(54)

where $\varepsilon$ is the internal energy per unit mass, $w$ is the enthalpy
per unit mass and

$\sigma_{ij}^{{}^{\prime}}=\eta\left(\partial_{i}v_{j}+\partial_{j}v_{i}-\frac{2}{3}\delta_{ij}div\boldsymbol{v}\right)+\zeta\delta_{ij}div\boldsymbol{v}$
(55)

is the viscosity tensor. For a smooth flow and a sufficiently large volume the
surface term in equation (54) can be neglected, and the energy is conserved,
as it is well known. However, in the surface integral we have terms of the
form

$\oint dS\left[\eta
v_{i}\left(\partial_{i}v_{n}+\partial_{n}v_{i}\right)-\left(\frac{2}{3}\eta-\zeta\right)v_{n}div\boldsymbol{v}+\kappa\partial_{n}T\right]\,\,\,,$
(56)

which imply normal derivatives to the surface, both of velocity and
temperature. By collecting these contributions, we get

$\oint
dS\left[\eta\partial_{n}(v^{2}/2)+\left(\frac{1}{3}\eta+\zeta\right)\partial_{n}(v_{n}^{2}/2)+\kappa\partial_{n}T\right]\,\,.$
(57)

We can see that for large normal derivatives across the surface, both for
velocity and temperature, these terms may lead to instabilities.

##  6 Turbulence

As it is well known, for a moderate turbulence, _i.e._ for slowly varying
fluctuations, we may decompose the velocity field into a mean velocity and a
fluctuating part, and limit ourselves to the time averaged Navier-Stokes
equation. This way we get the Reynolds equations, for which the mean energy is
coupled to the fluctuating energy, via model assumptions. A fully developed
turbulence exhibits highly-varying fluctuations, such that we need to consider
the time-dependent Navier-Stokes equation.

The turbulent instabilities ocurring in a fully developed turbulence exhibit
large variations of the velocity over small distances. In this case we may
assume that the velocity is split into a mean-flow velocity
$\boldsymbol{v}_{0}$ and a fluctuating part $\boldsymbol{v}$, where the mean-
flow velocity $\boldsymbol{v}_{0}$ may have a slight time variation, while the
fluctuating velocity $\boldsymbol{v}$ is a rapidly varying velocity. By using
this decomposition for an incompressible fluid, the Navier-Stokes equation
reads

$\begin{array}[]{c}\frac{\partial\boldsymbol{v}_{0}}{\partial
t}+\frac{\partial\boldsymbol{v}}{\partial
t}+\left(\boldsymbol{v}_{0}grad\right)\boldsymbol{v}_{0}+\left(\boldsymbol{v}_{0}grad\right)\boldsymbol{v}+\left(\boldsymbol{v}grad\right)\boldsymbol{v}_{0}+\\\
\\\
+\left(\boldsymbol{v}grad\right)\boldsymbol{v}=-\frac{1}{\rho}gradp_{0}-\frac{1}{\rho}gradp+\nu\Delta\boldsymbol{v}_{0}+\nu\Delta\boldsymbol{v}\,\,\,,\end{array}$
(58)

where $p_{0}$ is the pressure corresponding to the main flow and $p$ is the
fluctuating part of the pressure. In this equation we have three distinct
types of time variations, such that it should be viewed as three equations

$\begin{array}[]{c}\frac{\partial\boldsymbol{v}_{0}}{\partial
t}+\left(\boldsymbol{v}_{0}grad\right)\boldsymbol{v}_{0}=-\frac{1}{\rho}gradp_{0}+\nu\Delta\boldsymbol{v}_{0}\,\,,\\\
\\\ \frac{\partial\boldsymbol{v}}{\partial
t}+\left(\boldsymbol{v}_{0}grad\right)\boldsymbol{v}+\left(\boldsymbol{v}grad\right)\boldsymbol{v}_{0}=-\frac{1}{\rho}gradp+\nu\Delta\boldsymbol{v}\,\,,\\\
\\\ \left(\boldsymbol{v}grad\right)\boldsymbol{v}=0\,\,;\end{array}$ (59)

similarly, the continuity equation should be split into

$div\boldsymbol{v}_{0}=0\,\,,\,\,div\boldsymbol{v}=0\,\,.$ (60)

The first equation (59) is an independent equation, which gives the main flow
velocity $\boldsymbol{v}_{0}$. Since the main part of the velocity is taken by
the fluctuating velocity, we may neglect the quadratic term in this equation.
The energy conservation and the heat transfer for this equation are given by

$\begin{array}[]{c}\frac{\partial}{\partial
t}(v_{0}^{2}/2)=-\partial_{i}\left[v_{0i}\left(p_{0}/\rho+v_{0}^{2}/2\right)-\nu\partial_{i}(v_{0}^{2}/2)\right]-\\\
\\\ -\nu(\partial_{i}v_{0j})^{2}\,\,,\\\ \\\ T_{0}\frac{ds_{0}}{dt}=\chi\Delta
T_{0}+\nu(\partial_{i}v_{0j})^{2}\,\,\,,\end{array}$ (61)

where $T_{0}$ is the temperature of the main flow, $s_{0}$ is the entropy per
unit mass of the main flow and $\chi$ is the thermometric conductivity.

Having solved the mean-flow equation we can pass to solve the second equation
(59) for the fluctuating velocity $\boldsymbol{v}$, with $\boldsymbol{v}_{0}$
as a parameter. This equation has its own energy-conservation and heat-
transfer equations. We note that the temperature and the entropy of the mean
flow are different from the temperature and the entropy of the fluctuating
part of the flow, which means that the two components of the flow (the mean
flow and the fluctuating flow) are not in thermal equilbrium. Indeed, if we
multiply the first equation (59) by $\boldsymbol{v}$ and the second equation
(59) by $\boldsymbol{v}_{0}$, we get cross-terms of the form
$T_{0}\frac{ds}{dt}+T\frac{ds_{0}}{dt}$ in the heat-transfer equation, where
$T$ and $s$ are the temperature and the entropy of the fluctuating flow. This
indicates a heat exchange between the two components of the flow.

We are left with the third equation (59), which, in general, is not satisfied.
We conclude that the fully developed turbulence does not satisfy the Navier-
Stokes equation. The quadratic term of the third equation (59) is equivalent
to a rapidly varying internal force (Euler’s force), which cannot be
compensated by any physical external force. The fully developed turbulence is
unstable.

Under these conditions it is reasonable to be interested in time averaged
quantities. Then, the fluctuating part of the flow is reduced to

$\overline{\left(\boldsymbol{v}grad\right)\boldsymbol{v}}=0\,\,.$ (62)

Since $div\boldsymbol{v}=0$, the components of the velocity $\boldsymbol{v}$
are not independent. In general, equation (62) is not satisfied, which means
that the turbulent motion is unstable even on average. We note that a similar
decomposition is valid for a compressible fluid, as long as the velocity,
density and entropy fluctuations are independent of one another.

Since a fully developed turbulence originates in large variations of the
velocity across small distances, it is reasonable to associate these
variations with a discrete distribution of positions
$\boldsymbol{r}_{i},$which we call centres of turbulence. Further on, we
assume that this is a homogeneous and isotropic distribution, such that we may
write the velocity field as

$\boldsymbol{v}=\sum_{i}\boldsymbol{v}_{i}(t,R_{i})\,\,\,,$ (63)

where $\boldsymbol{R}_{i}=\boldsymbol{r}-\boldsymbol{r}_{i}$. If
$curl\boldsymbol{v}_{i}\neq 0$, this velocity field represents a vorticial
liquid, which is unstable. We assume that the fluctuating velocities are
independent at distinct positions, _i.e._ $\overline{\boldsymbol{v}_{i}}=0$
and $\overline{\boldsymbol{v}_{i}\boldsymbol{v}_{j}}\sim\delta_{ij}$. Equation
(62) becomes

$\overline{\left(\boldsymbol{v}grad\right)\boldsymbol{v}}=\sum_{i}\overline{\left(\boldsymbol{v}_{i}\boldsymbol{R}_{i}/R_{i}\right)\left(d\boldsymbol{v}_{i}/dR_{i}\right)}\,\,.$
(64)

The conditions of homogeneity and isotropy imply that $\boldsymbol{v}_{i}$ in
the above equation may be replaced by the same velociy $\boldsymbol{u}$. For a
sufficiently dense set of positions $\boldsymbol{r}_{i}$ we can define a
density $\rho_{v}$ of such points, which is a constant. Then, equation (64)
can be transformed into the integral

$\overline{\left(\boldsymbol{v}grad\right)\boldsymbol{v}}=\rho_{v}\int dR\cdot
R^{2}\int
do\overline{\left(\boldsymbol{u}\boldsymbol{R}/R\right)\left(d\boldsymbol{u}/dR\right)}\,\,.$
(65)

If the radial integral is finite, the result of integration in the above
equation is zero, due to the integration over the solid angle $o$, such that
the term $\overline{\left(\boldsymbol{v}grad\right)\boldsymbol{v}}$ is zero.
In this case we can say that the Navier-Stokes equation is trivially satisfied
on average, being reduced to the mean-flow equation (first equation (59)).

If the radial integral in equation (65) is singular for $\boldsymbol{R}=0$, as
it may often happen for vortices, we are left with a discrete set of
singularities, extending over a small characteristic distance $a$, where the
singularity is

$\boldsymbol{u}\sim(a/R)^{n}\,\,,\,\,n>1\,\,.$ (66)

In each of these regions there exists a mass $M$ of fluid, which can be
carried by the background fluid and, at the same time, they may have their own
motion.

The averaged energy-conservation equation derived from the second equation
(59),

$\frac{1}{2}\boldsymbol{v}_{0}grad\overline{v^{2}}+\partial_{j}\left(\overline{v_{i}v_{j}}v_{0i}\right)-\frac{1}{2}\nu\Delta\overline{v^{2}}=-\nu\overline{\left(\partial_{j}v_{i}\right)^{2}}\,\,\,,$
(67)

shows that the fluctuating motion produces heat which is partly transported by
the mean flow (the $div$-terms integrated over a volume are irrelevant). This
dissipated heat should be compensated from the outside. Therefore, we are left
with a quasi ideal classical gas of singular vortices, or a solution of
vortices in the background fluid, in thermal quasi equilibrium. This is an
example of emergent dynamics.[23]

##  7 Gas of singularities

Let us assume a homogeneous, isotropic, fluctuating distribution of singular
centres of turbulence localized at $\boldsymbol{r}_{i}$ with mass $M$, as
described above. Their density is

$\rho=M\sum_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})$ (68)

and

$\frac{\partial\rho}{\partial
t}=-M\sum_{i}\boldsymbol{u}_{i}grad\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,$
(69)

where $\boldsymbol{u}_{i}=d\boldsymbol{r}_{i}/dt$ is their velocity. The
velocity field of the singularities is

$\boldsymbol{u}=v\sum_{i}\boldsymbol{u}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,$
(70)

where $v$ is the small volume over which the $\delta$-function is localized,
such that $v=a^{3}$ and $M=\rho v$. Let us compute

$\begin{array}[]{c}div(\rho\boldsymbol{u})=vMdiv\sum_{ij}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\boldsymbol{u}_{j}\delta(\boldsymbol{r}-\boldsymbol{r}_{j})=\\\
\\\
=Mdiv\sum_{i}\boldsymbol{u}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})=\\\
\\\
=M\sum_{i}\boldsymbol{u}_{i}grad\delta(\boldsymbol{r}-\boldsymbol{r}_{i})=-\frac{\partial\rho}{\partial
t}\,\,;\end{array}$ (71)

we can see that the continuity equation is satisfied.

Now, let us focus on the Navier-Stokes equation

$\rho\frac{\partial\boldsymbol{u}}{\partial
t}+\rho(\boldsymbol{u}grad)\boldsymbol{u}=-gradp+\eta\Delta\boldsymbol{u}\,\,,$
(72)

and let us compute each term in this equation for our fluid of singularities.
We have

$\begin{array}[]{c}\frac{\partial\boldsymbol{u}}{\partial
t}=v\sum_{i}\dot{\boldsymbol{u}}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})-v\sum_{i}\boldsymbol{u}_{i}(\boldsymbol{u}_{i}grad)\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,.\end{array}$
(73)

The inertial term is

$\begin{array}[]{c}(\boldsymbol{u}grad)\boldsymbol{u}=v^{2}\sum_{ij}\boldsymbol{u}_{j}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})(\boldsymbol{u}_{i}grad)\delta(\boldsymbol{r}-\boldsymbol{r}_{j})=\\\
\\\
=v\sum_{i}\boldsymbol{u}_{i}(\boldsymbol{u}_{i}grad)\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,.\end{array}$
(74)

On comparing equations (73) and (74), we can see that the inertial (Euler’s)
term disappears from equation. This is expected, since the $\delta$-function
equals the variable $\boldsymbol{r}$ to the function $\boldsymbol{r}_{i}(t)$,
which amounts to Lagrange’s approach. The term on the left in equation (72)
becomes

$\begin{array}[]{c}\rho\frac{\partial\boldsymbol{u}}{\partial
t}+\rho(\boldsymbol{u}grad)\boldsymbol{u}=M\sum_{i}\dot{\boldsymbol{u}}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,\end{array}$
(75)

and equation (72) reads now

$\begin{array}[]{c}M\sum_{i}\dot{\boldsymbol{u}}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})=-v\sum_{i}p_{i}grad\delta(\boldsymbol{r}-\boldsymbol{r}_{i})+\\\
\\\ +\eta
v\sum_{i}\boldsymbol{u}_{i}\Delta\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,\end{array}$
(76)

where $p_{i}$ is the pressure at the position $\boldsymbol{r}_{i}$. The
pressure term in equation (76) is a force per unit volume acting upon the
vortex placed at $\boldsymbol{r}_{i}$; it may arise from the pressure exerted
by the background fluid particles. This term may be written as

$\sum_{i}\boldsymbol{f}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,$
(77)

where $\boldsymbol{f}_{i}$ is the force acting at $\boldsymbol{r}_{i}$. The
factor $\Delta\delta(\boldsymbol{r}-\boldsymbol{r}_{i})$ is of the order
$-\frac{1}{a^{2}}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})$, where $a$ is of
the order of the dimension of the vortex ($v=a^{3})$; consequently, we may
replace the viscosity terms in equation (76) by

$-\frac{\eta
v}{a^{2}}\sum_{i}\boldsymbol{u}_{i}\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,.$
(78)

A similar contribution brings the $\zeta$-term. The equation of motion (76)
describes a set of independent particles with mass $M$, subjected to an
external force $\boldsymbol{f}_{i}$ and a friction force; the equation of
motion of each such particle can be written as

$M\dot{\boldsymbol{u}}_{i}=\boldsymbol{f}_{i}-\eta a\boldsymbol{u}_{i}\,\,\,,$
(79)

which is Newton’s law of motion. The damping coefficient caused by the
friction force is very small, such that we can consider the ensemble of
singularities as a (quasi) ideal classical gas of independent, identical,
pointlike particles. Therefore, a (singular) fully developed turbulence may be
viewed as the (quasi) thermodynamic equilibrium of such a gas (or a solution
of singularities in the background fluid). We can define a temperature of
turbulence, which is approximately the mean kinetic energy of the
translational motion of a singularity. Also, we can estimate a chemical
potential by evaluating $\overline{v^{2}}/2$. We note that the density
$\rho_{v}$ and the dimension $a$ of the singularities remain undetermined;
these parameters can be estimated from experiment.

Also, it is worth noting that we may consider the equations of the fluid
mechanics for this new gas of singularities, viewed as a continuous medium, at
a higher scale.

##  8 Vorticial liquids

###  8.1 Vortex

For an incompressible fluid with an isentropic flow we consider a velocity
field given by

$\boldsymbol{v}=\boldsymbol{\omega}\times gradf(r)\,\,\,,$ (80)

where $\boldsymbol{\omega}$ is a vector which may depend only on the time and
the function $f(r)$ is smooth everywhere, except, possibly, at the origin
$\boldsymbol{r}\neq 0$, and vanishing rapidly at infinity. The velocity
$\boldsymbol{v}$ rotates about $\boldsymbol{\omega}$; such a velocity field
defines a vortex.

We can ckeck that $div\boldsymbol{v}=0$ and

$\boldsymbol{v}=-curl\left[\boldsymbol{\omega}f(r)\right]\,\,\,,$ (81)

such that we can define a vector potential
$\boldsymbol{A}=-\boldsymbol{\omega}f(r)$
($\boldsymbol{v}=curl\boldsymbol{A}$). We note that $div\boldsymbol{A}\neq 0$
and the vorticity differs from $\Delta\boldsymbol{A}$, in general,

$curl\boldsymbol{v}=\Delta\left[\boldsymbol{\omega}f(r)\right]-grad\,div\left[\boldsymbol{\omega}f(r)\right]\neq-\Delta\boldsymbol{A}\,\,.$
(82)

If the velocity given by equation (81) satisfies the second equation (40), we
should have $\boldsymbol{\omega}\sim e^{-\nu\lambda t}$ and $\Delta f+\lambda
f=0$, $i.e.$ $f\sim e^{\pm i\sqrt{\lambda}r}/r$, where $\lambda$ is, in
general, complex. In two dimensions $f$ is a Bessel function. The Navier-
Stokes equation gives the pressure.

The most common example of a velocity field given by equation (80) is the
filamentary vortex (the "cyclon"), with $\boldsymbol{\omega}=const$,
($\lambda=0$), $f(r)=-\ln r$ and the two-dimensional position vector
$\boldsymbol{r}$ perpendicular to $\boldsymbol{\omega}$. In this case
$\boldsymbol{v}=\boldsymbol{r}\times\boldsymbol{\omega}/r^{2}$,
$\boldsymbol{A}=\boldsymbol{\omega}\ln r$ and
$curl\boldsymbol{v}=-2\pi\boldsymbol{\omega}\delta(\boldsymbol{r})$. The
velocity is singular at $\boldsymbol{r}=0$. For the filamentary vortex the
Navier-Stokes equation can be satisfied for $p=-\rho\omega^{2}/2r^{2}$, _i.e._
for an external potential $\varphi=p/\rho=-\omega^{2}/2r^{2}$. This can be
provided by the gravitational field for a fluid with a free surface.

For convenience, we give the expression of the various terms in the Navier-
Stokes equation for the vortex given by equation (80),

$\begin{array}[]{c}\frac{\partial\boldsymbol{v}}{\partial
t}=\left(\dot{\boldsymbol{\omega}}\times\boldsymbol{r}\right)\frac{f^{{}^{\prime}}}{r}\,\,,\\\
\\\
(\boldsymbol{v}grad)\boldsymbol{v}=\left[\boldsymbol{\omega}(\boldsymbol{\omega}\boldsymbol{r})-\omega^{2}\boldsymbol{r}\right]\frac{f^{{}^{\prime}2}}{r^{2}}\,\,,\\\
\\\
\Delta\boldsymbol{v}=(\boldsymbol{\omega}\times\boldsymbol{r})\frac{(f^{{}^{\prime\prime}}+2f^{\prime}/r)^{{}^{\prime}}}{r}\,\,\,,\end{array}$
(83)

where the primes denote the derivatives of the function $f;$ the second
equation (83) is derived by using the identity

$(\boldsymbol{v}grad)\boldsymbol{v}=-\boldsymbol{v}\times
curl\boldsymbol{v}+grad(v^{2}/2)\,\,.$ (84)

For a filamentary vortex the above expressions become

$\begin{array}[]{c}(\boldsymbol{v}grad)\boldsymbol{v}=-\omega^{2}\boldsymbol{r}\frac{f^{{}^{\prime}2}}{r^{2}}\,\,,\\\
\\\
\Delta\boldsymbol{v}=(\boldsymbol{\omega}\times\boldsymbol{r})\frac{(f^{{}^{\prime\prime}}+f^{\prime}/r)^{{}^{\prime}}}{r}\,\,.\end{array}$
(85)

In general, the vortex given by equation (80) does not satisfy the Navier-
Stokes equation; it develops internal (Euler’s) forces which cannot be
compensated; the vortex is unstable. Such vortices are examples of singular
velocities.

###  8.2 Vorticial liquid

A set of vectors $\boldsymbol{\omega}_{i}$ placed at $\boldsymbol{r}_{i}$ form
a vorticial liquid. The velocity field is

$\boldsymbol{v}=\sum_{i}\boldsymbol{\omega}_{i}\times gradf_{i}(R_{i})\,\,\,,$
(86)

where $\boldsymbol{R}_{i}=\boldsymbol{r}-\boldsymbol{r}_{i}$. This velocity
field is a superpositon of independent vortices. Each $i$-th vortex develops
an internal force, while the inertial term generates an interaction which
brings an additional force; this additional force depends on all the other
$j$-th vortices, $j\neq i$. Therefore, the Navier-Stokes equation is not
satisfied by the velocity field given by equation (86), in the sense that
there is no physical external pressure to compensate the Euler force.

We assume randomly fluctuating vectors $\boldsymbol{\omega}_{i}$, as a
distinctive feature of a fully developed vorticial turbulence. It is likely
that the fluid develops fluctuations as a reaction to its uncompensated
internal forces. Specifically, we assume
$\overline{\boldsymbol{\omega}}_{i}=0$ and
$\overline{\omega_{i}^{\alpha}\omega_{j}^{\beta}}=\frac{1}{3}\overline{\omega_{i}^{2}}\delta_{ij}\delta^{\alpha\beta}$,
where $\alpha,\,\beta=1,2,3$ are the cartesian labels of the components of the
vectors $\boldsymbol{\omega}_{i}$, and the overbar indicates the average over
time. Then, the average velocity is zero ($\overline{\boldsymbol{v}}=0$) and
we are left with the inertial term

$\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=\overline{v_{j}\partial_{j}v_{i}}\,\,.$
(87)

The calculation of this term is straightforward; we get

$\begin{array}[]{c}\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=\frac{1}{3}\sum_{i}\omega_{i}^{2}[\frac{1}{2}grad\left(gradf_{i}(R_{i})\right)^{2}-\\\
\\\ -gradf_{i}(R_{i})\cdot\Delta f_{i}(R_{i})]\,\,.\end{array}$ (88)

The averaged Euler forces are

$\begin{array}[]{c}-\overline{\boldsymbol{v}\times
curl\boldsymbol{v}}=-\frac{1}{3}\sum_{i}\omega_{i}^{2}[\frac{1}{2}grad\left(gradf_{i}(R_{i})\right)^{2}+\\\
\\\ +gradf_{i}(R_{i})\cdot\Delta f_{i}(R_{i})]\,\,,\\\ \\\
\overline{grad(v^{2}/2)}=\frac{1}{3}\sum_{i}\omega_{i}^{2}grad\left(gradf_{i}(R_{i})\right)^{2}\,\,.\end{array}$
(89)

By using the spherical symmetry of the function $f_{i}(R_{i})$, these
expressions can be cast in the form

$\begin{array}[]{c}\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=-\frac{2}{3}\sum_{i}\omega_{i}^{2}f_{i}^{{}^{\prime}2}\frac{\boldsymbol{R}_{i}}{R_{i}^{2}}\,\,,\\\
\\\ -\overline{\boldsymbol{v}\times
curl\boldsymbol{v}}=-\frac{2}{3}\sum_{i}\omega_{i}^{2}f_{i}^{{}^{\prime}}\left(f_{i}^{{}^{\prime\prime}}+\frac{1}{R_{i}}f_{i}^{{}^{\prime}}\right)\frac{\boldsymbol{R}_{i}}{R_{i}}\,\,,\\\
\\\
\overline{grad(v^{2}/2)}=\frac{2}{3}\sum_{i}\omega_{i}^{2}f_{i}^{{}^{\prime}}f_{i}^{{}^{\prime\prime}}\frac{\boldsymbol{R}_{i}}{R_{i}}\,\,.\end{array}$
(90)

We can see that even on average the Navier-Stokes equation is not satisfied,
in the sense discussed above. Even on average the vorticial liquid develops
internal forces which are not equilibrated; it is unstable. We shall give
specific examples of such an instability below.

Now, let us assume that the vorticial liquid is sufficiently dense, _i.e._ if
we can define a density $\rho_{v}$ of points $\boldsymbol{r}_{i}$; further, we
assume that the liquid is homogeneous and isotropic, _i.e._ this density is
constant and the $\omega_{i}$ and the functions $f_{i}$ can be replaced in the
above equations by uniform functions $\omega_{i}=\omega$ and $f_{i}=f$. We
assume that this is another distinctive feature of a fully developed
turbulence. Then, we may transform the summation over $i$ in equations (90)
into an integral, like in equation (65). By choosing the origin at
$\boldsymbol{r}=\boldsymbol{r}_{i}$ for a fixed $\boldsymbol{r}_{i}$, and
using the notation $\boldsymbol{R}=\boldsymbol{r}-\boldsymbol{r}_{i}$, we get

$\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=-\frac{2}{3}\rho_{v}\omega^{2}\int_{0}^{\infty}dR\cdot
Rf^{{}^{\prime}2}(R)\int do(\boldsymbol{R}/R)\,\,;$ (91)

the result of integration in this equation is zero, due to the integration
over the solid angle $o$, providing the radial integration is finite. In this
case the Navier-Stokes equation is satisfied trivially.

Let us assume that the integral over $R$ is singular at $R=0$, as another
distinctive feature of a fully developed turbulence (the function $f$ is
assumed to decrease sufficiently rapid at infinity to have a finite integral
in this limit). The integration outside a small region around the $i$-th point
is zero, while the integration over such a small region is indefinite. This
singularity implies

$f(R)\sim(a/R)^{n}\,\,,\,\,n>0$ (92)

for $R\ll a$, where $a$ is a small characteristic distance (compare with
equation (66)). Therefore, we are left with a discrete set of points
$\boldsymbol{r}_{i}$, where the function $f(R_{i})$ and the velocity ($v\sim
1/R_{i}^{n+1}$) are singular. We have now a small region of dimension $a$,
around each point $\boldsymbol{r}_{i}$, which includes a mass of fluid, say,
$M$, where the inertial term given by equation (91) is not defined. Since the
positions $\boldsymbol{r}_{i}$ may change in time, we are left with a
classical gas of particle-like vortices (or a solution of vortices in the
background fluid), as discussed above. Of course, we may have also a mixture
of vorticial gases, each characterized by a dimension $a$ and a mass $M$.

The equation of energy conservation (equation (67))

$\frac{\partial}{\partial
t}\left(\frac{1}{2}v^{2}\right)+\boldsymbol{v}\cdot(\boldsymbol{v}grad)\boldsymbol{v}=\nu\boldsymbol{v}\Delta\boldsymbol{v}\,\,,$
(93)

averaged over the fluctuating vectors $\boldsymbol{\omega}_{i}$, is reduced to
the viscosity term

$\nu\overline{\boldsymbol{v}\Delta\boldsymbol{v}}=\frac{2\nu}{3}\sum_{i}\omega_{i}^{2}f_{i}^{{}^{\prime}}\left(f_{i}^{{}^{\prime\prime}}+2f_{i}^{{}^{\prime}}/R_{i}\right)^{{}^{\prime}}\,\,.$
(94)

This term should be computed outside the regions with dimension $a$, where the
motion is defined. The non-vanishing value of this term indicates an energy
loss, which should be compensated from the outside. The vorticial gas is in
quasi equilibrium.

The above considerations are valid for spherical-symmetric functions
$f_{i}(R_{i})$; if the vortices have a lower (internal) symmetry, the unit
vector $\boldsymbol{R}/R$ in equation (91) is replaced by functions which do
not have a spherical symmetry, and the inertial term is not vanishing, in
general. The vortices are unstable, and, likely, they could tend to acquire a
spherical symmetry, which ensures a (quasi)-equilibrium.

###  8.3 Filamentary liquid

Let us consider a set of rectilinear, parallel filaments, directed along the
$z$-axis, placed at positions $\boldsymbol{r}_{i}$ in the $(x,y)$-plane, with
vorticities $\boldsymbol{\omega}_{i}$. This is a two-dimensional vorticial
liquid of "cyclons". The velocity field is given by

$\boldsymbol{v}=-\sum_{i}\boldsymbol{\omega}_{i}\times grad\ln
R_{i}=\sum_{i}\frac{\boldsymbol{R}_{i}\times\boldsymbol{\omega}_{i}}{R_{i}^{2}}\,\,\,,$
(95)

where $\boldsymbol{R}_{i}=\boldsymbol{r}-\boldsymbol{r}_{i}$.

Equation (95) shows that the velocity is derived from a vector potential
$\boldsymbol{A}$, through $\boldsymbol{v}=curl\boldsymbol{A}$. By taking the
$curl$ in this equation, we get

$\Delta\boldsymbol{A}=-2\boldsymbol{\omega}$ (96)

(providing $div\boldsymbol{A}=0$ and $div\boldsymbol{\omega}=0$), where we
introduce the notation $curl\boldsymbol{v}=2\boldsymbol{\omega}$;
$\boldsymbol{\omega}$ is called vorticity. The vorticity distribution
corresponding to equation (95)

$\boldsymbol{\omega}(\boldsymbol{r})=\frac{1}{2}curl\boldsymbol{v}=-\pi\sum_{i}\boldsymbol{\omega}_{i}\delta(\boldsymbol{R}_{i})\,\,\,,$
(97)

gives the vector potential

$\boldsymbol{A}=\sum_{i}\boldsymbol{\omega}_{i}\ln R_{i}\,\,.$ (98)

According to equation (84), the inertial term has the components

$\boldsymbol{f}=-\boldsymbol{v}\times curl\boldsymbol{v}=-2\pi\sum_{i\neq
j}\omega_{i}\omega_{j}grad_{i}\ln
R_{ij}\cdot\delta(\boldsymbol{R}_{i})\,\,\,,$ (99)

where $\boldsymbol{R}_{ij}=\boldsymbol{r}_{i}-\boldsymbol{r}_{j}$, and
$grade$, where

$e=\frac{1}{2}v^{2}=\sum_{i\neq
j}\omega_{i}\omega_{j}\frac{\boldsymbol{R}_{i}\boldsymbol{R}_{j}}{2R_{i}^{2}R_{j}^{2}}$
(100)

is a density of kinetic energy (per unit mass). Apart from the force
$\boldsymbol{f}$, which acts at the positions of the vortices, there exist
internal forces given by $grade$, which make the liquid unstable. The motion
and the statistics of parallel, rectilinear filaments have been extensively
investigated,[24]-[31] the instability being associated with a negative
temperature in an attempt of a statistical theory.[25, 27]

The total force given by equation (99)

$\boldsymbol{F}=\int d\boldsymbol{r}\boldsymbol{f}=-2\pi\sum_{i\neq
j}\omega_{i}\omega_{j}grad_{i}\ln R_{ij}$ (101)

is zero. We can see that a force

$\boldsymbol{f}_{ij}=-\boldsymbol{f}_{ji}=-2\pi\omega_{i}\omega_{j}grad_{i}\ln
R_{ij}$ (102)

acts between any pair $(ij)$ of vortices. This force derives from a potential

$U_{ij}=2\pi\omega_{i}\omega_{j}\ln R_{ij}\,\,\,,$ (103)

such that

$\boldsymbol{F}=-\sum_{i\neq j}grad_{i}U_{ij}$ (104)

The density of kinetic energy $e$ can be written as

$e=\frac{1}{2}v^{2}=\frac{1}{2}\boldsymbol{v}curl\boldsymbol{A}=-\frac{1}{2}div(\boldsymbol{v}\times\boldsymbol{A})+\boldsymbol{A}\boldsymbol{\omega}\,\,.$
(105)

The first term in equation (105) is singular at
$\boldsymbol{r}=\boldsymbol{r}_{i}$; we integrate this term over the whole
space, transform it into surface integrals, both at infinity and over small
circles around each filament, and neglect their contributions. The result of
such integrations is a self-energy (or a self-force), which may be left aside.
We call this procedure a "renormalization".[32] By doing so, we are left with
a total kinetic energy

$\begin{array}[]{c}E=\int d\boldsymbol{r}e=\int
d\boldsymbol{r}\boldsymbol{A}\boldsymbol{\omega}=-\pi\sum_{i\neq
j}\omega_{i}\omega_{j}\ln R_{ij}=\\\ \\\ =-\frac{1}{2}\sum_{i\neq
j}U_{ij}=-U\,\,\,,\end{array}$ (106)

where $U$ is the total potential energy. We can see that the total energy is
conserved, _i.e._ $E+U=const$. Also, the total force $\boldsymbol{F}=0$, such
that the total momentum is conserved. The total angular momentum is $-2\int
d\boldsymbol{r}\boldsymbol{A}$; it is proportional to
$\sum_{i}\boldsymbol{\omega}_{i}$. The total torque

$2\pi\sum_{i\neq
j}\omega_{i}\omega_{j}\frac{\boldsymbol{r}_{i}\times\boldsymbol{r}_{j}}{R_{ij}^{2}}$
(107)

is zero. By this "renormalization" procedure the points $\boldsymbol{r_{i}}$
are completely decoupled from the fluid, and they may have their own motion.

The average over fluctuating vorticities can be computed straightforwardly, by
using $\overline{\boldsymbol{\omega}_{i}^{2}}=\frac{1}{2}\omega_{i}^{2}$; it
is given by

$\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=-\sum_{i}\omega_{i}^{2}\frac{\boldsymbol{R}_{i}}{2R_{i}^{4}}\,\,.$
(108)

We can see that for a sufficiently dense, homogeneous and isotropic liquid we
get a gas of (singular) vortices, as discussed above.

###  8.4 Coulombian liquid

For $f_{i}(R_{i})=-1/R_{i}$ in equation (86) we get a coulombian vorticial
liquid with the velocity field

$\boldsymbol{v}=-\sum_{i}\boldsymbol{\omega}_{i}\times
grad(1/R_{i})=\sum_{i}\frac{\boldsymbol{\omega}_{i}\times\boldsymbol{R}_{i}}{R_{i}^{3}}$
(109)

and the vector potential

$\boldsymbol{A}=\sum_{i}\frac{\boldsymbol{\omega}_{i}}{R_{i}}\,\,.$ (110)

The equation $\Delta\boldsymbol{A}=-2\boldsymbol{\omega}$ is satisfied for

$\boldsymbol{\omega}(\boldsymbol{r})=2\pi\sum_{i}\boldsymbol{\omega}_{i}\delta(\boldsymbol{R}_{i})\,\,\,,$
(111)

but this vorticity differs from

$\frac{1}{2}curl\boldsymbol{v}=2\pi\sum_{i}\boldsymbol{\omega}_{i}\delta(\boldsymbol{R}_{i})-\frac{1}{2}grad\,div\sum_{i}(\boldsymbol{\omega}_{i}/R_{i})\,\,.$
(112)

By applying the "renormalization" procedure we get the force

$\boldsymbol{F}=\int d\boldsymbol{r}\boldsymbol{f}=-\int
dr\boldsymbol{v}\times curl\boldsymbol{v}=4\pi\sum_{i\neq
j}grad_{i}\frac{\boldsymbol{\omega}_{i}\boldsymbol{\omega}_{j}}{R_{ij}}$ (113)

and the energy

$E=\int d\boldsymbol{r}e=\frac{1}{2}\int drv^{2}=2\pi\sum_{i\neq
j}\frac{\boldsymbol{\omega}_{i}\boldsymbol{\omega}_{j}}{R_{ij}}\,\,\,,$ (114)

where, in both cases $\boldsymbol{v}=curl\boldsymbol{A}$ is used. The
potential from equation (103) is now
$U_{ij}=-4\pi\frac{\boldsymbol{\omega}_{i}\boldsymbol{\omega}_{j}}{R_{ij}}$.
The total energy is conserved, the total force is zero, the total angular
momentum is proportional to $\sum_{i}\boldsymbol{\omega}_{i}$ and the total
torque (zero) is

$4\pi\sum_{i\neq
j}\boldsymbol{\omega}_{i}\boldsymbol{\omega}_{j}\frac{\boldsymbol{r}_{i}\times\boldsymbol{r}_{j}}{R_{ij}^{3}}\,\,.$
(115)

The average of the inertial term over fluctuating vortices is obtained from
equation (90)

$\overline{(\boldsymbol{v}grad)\boldsymbol{v}}=-\frac{2}{3}\sum_{i}\omega_{i}^{2}\frac{\boldsymbol{R}_{i}}{R_{i}^{6}}\,\,\,,$
(116)

such that we may get a gas of singular vortices.

###  8.5 Dipolar liquid

A dipolar liquid is defined by the vorticity

$\boldsymbol{\omega}=-2\pi\sum_{i}\boldsymbol{m}_{i}\times
grad\delta(\boldsymbol{r}-\boldsymbol{r}_{i})\,\,\,,$ (117)

where the vectors $\boldsymbol{m}_{i}$ may depend on the time, at most. We get
the vector potential

$\boldsymbol{A}=\sum_{i}\frac{\boldsymbol{m}_{i}\times\boldsymbol{R}_{i}}{R_{i}^{3}}=-\sum_{i}\boldsymbol{m}_{i}\times
grad(1/R_{i})$ (118)

and the velocity field

$\begin{array}[]{c}\boldsymbol{v}(\boldsymbol{r})=\sum_{i}\left[-\boldsymbol{m}_{i}/R_{i}^{3}+3\boldsymbol{R}_{i}(\boldsymbol{m}_{i}\boldsymbol{R}_{i})/R_{i}^{5}\right]=\\\
\\\ =\sum_{i}grad\left[\boldsymbol{m}_{i}grad(1/R_{i})\right]\,\,.\end{array}$
(119)

We recognize in these equations magnetic (dipole) moments
$\boldsymbol{m}_{i}$, a dipolar vector potential $\boldsymbol{A}$ and a
magnetic field $\boldsymbol{v}$.

The inertial term has the components

$\begin{array}[]{c}\boldsymbol{f}=-\boldsymbol{v}\times
curl\boldsymbol{v}=2\boldsymbol{\omega}\times\boldsymbol{v}=\\\ \\\
=4\pi\sum_{i\neq
j}grad\left[\boldsymbol{m}_{j}grad(1/R_{j})\right]\times\left[\boldsymbol{m}_{i}\times
grad\delta(\boldsymbol{R}_{i})\right]\end{array}$ (120)

and $grade$, where

$\begin{array}[]{c}e=\frac{1}{2}v^{2}=\frac{1}{2}\sum_{i\neq
j}[\frac{\boldsymbol{m}_{i}\boldsymbol{m}_{j}}{R_{i}^{3}R_{j}^{3}}-\frac{3(\boldsymbol{m}_{i}\boldsymbol{R}_{j})(\boldsymbol{m}_{j}\boldsymbol{R}_{j})}{R_{i}^{3}R_{j}^{5}}-\\\
\\\
-\frac{3(\boldsymbol{m}_{j}\boldsymbol{R}_{i})(\boldsymbol{m}_{i}\boldsymbol{R}_{i})}{R_{i}^{5}R_{j}^{3}}+\\\
\\\
+\frac{9(\boldsymbol{R}_{i}\boldsymbol{R}_{j})(\boldsymbol{m}_{i}\boldsymbol{R}_{i})(\boldsymbol{m}_{j}\boldsymbol{R}_{j})}{R_{i}^{5}R_{j}^{5}}]\,\,.\end{array}$
(121)

We can see that the dipolar liquid is unstable.

The total force and the total kinetic energy are

$\begin{array}[]{c}\boldsymbol{F}=\int d\boldsymbol{r}\boldsymbol{f}=2\int
d\boldsymbol{r}\boldsymbol{\omega}\times\boldsymbol{v}=-\sum_{i\neq
j}grad_{i}U_{ij}\,\,\,,\\\ \\\ E=\int d\boldsymbol{r}e=\int
d\boldsymbol{r}\boldsymbol{\omega}\boldsymbol{A}=-\frac{1}{2}\sum_{i\neq
j}U_{ij}\,\,\,,\end{array}$ (122)

where

$U_{ij}=-4\pi\boldsymbol{m}_{i}grad_{i}\left[\boldsymbol{m}_{j}grad_{i}(1/R_{ij})\right]\,\,.$
(123)

By this "renormalization" procedure, the liquid is reduced to a set of
interacting particle-like vortices. We can see that the energy is conserved,
the total force is zero, the total angular momentum and the total torque are
zero.

The time average over vorticities in equation (121) leads to

$\overline{e}=\frac{1}{2}\sum_{i}\left[\frac{m_{i}^{2}}{R_{i}^{6}}+\frac{3\overline{(\boldsymbol{m}_{i}\boldsymbol{R}_{i})^{2}}}{R_{i}^{8}}\right]=\sum_{i}\frac{m_{i}^{2}}{R_{i}^{6}}\,\,\,,$
(124)

which gives a force

$grad\overline{e}=-6\sum_{i}\frac{m_{i}^{2}\boldsymbol{R}_{i}}{R_{i}^{8}}\,\,.$
(125)

For a dense, homogeneous and isotropic liquid the force given by equation
(125) is zero (for $\boldsymbol{R}_{i}\neq 0$); also, the force
$\boldsymbol{f}$ is zero for $\boldsymbol{R}_{i}\neq 0$, and the Navier-Stokes
equation is satisfied trivially (on average). We note that
$\Delta\boldsymbol{v}$ is zero for $\boldsymbol{R}_{i}\neq 0$ (equation
(119)), such that the viscosity contribution is zero. We are left with a set
of positions $\boldsymbol{r}_{i}$, each surrounded by a small region, where
the motion is not defined. According to the above discussion, such a structure
may be viewed as a (quasi) ideal classical gas of vortices (or a solution of
vortices in the background fluid).

##  9 Concluding remarks

In fairly general conditions we have given in this paper an explicit (smooth)
solution for the potential flow. We have shown that, rigorously speaking, the
equations of the fluid mechanics have not rotational solutions. However,
usually we may neglect the variations of the density and the temperature, such
that, in these conditions, the Navier-Stokes equation may exhibit
(approximate) vorticial solutions, governed by the viscosity. We give
arguments that the vortices are unstable. On the other hand, for large
variations of the velocity over small distances, the fluid velocity exhibits
turbulent, highly fluctuating instabilities, controlled by viscosity. Such a
fully developed turbulence occurs as a consequence of the insufficiency of the
viscosity to dissipate heat. We represent the fully developed turbulence as a
superposition of highly fluctuating velocities, associated to a discrete
distrbution of turbulence centres, and are interested in the temporal average
of this velocity field. A regular mean flow may be added. It is shown that the
Navier-Stokes equation is not satisfied on average. However, for a homogeneous
and isotropic distribution of (non-singular) turbulence centres (as another
distinctive feature of a fully developed turbulence), the temporal average of
both the fluctuating velocity and the inertial term is zero, such that the
Navier-Stokes equation is satisfied trivially. If the velocity is singular at
the turbulence centres we are left with a quasi ideal classical gas of
singularities (or a solution of singularities in the background fluid), in
thermal quasi equilibrium, as an example of emergent dynamics. The Navier-
Stokes equation for this fluid of singularities is reduced to Newton’s law of
motion (with a small friction). At a higher scale, equations of fluid
mechanics can be considerd for this gas, as a continuous medium. We have
illustrated all the above considerations with three examples of (singular)
vorticial liquids.

Acknowledgements

The author is indebted to the members of the Department of Theoretical
Physics, the Institute of Physics and Nuclear Engineering, the Institute of
Atomic Physics, Magurele, for many enlightening discussions. A helpful
analysis by dr. F. Buzatu is particularly acknowledged. This work was carried
out within the Program Nucleu, funded by the Romanian Ministry of Research,
Innovation and Digitization, projects no. PN23210101/2023.

Conflict of interests: The author declare no conflict of interest.

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