International Journal of Fluid Mechanics Research, Vol. 29, No. 1, 2002
On the Relationship between Fluid Velocity
and de Broglie’s Wave Function and the Implications
to the Navier – Stokes Equation†
V. V. Kulish
School of Mechanical & Production Engineering, Nanyang
Technological University, 50 Nanyang Ave., Singapore, 639798
Tel.: +(65) 790 4950; Fax: +(65) 791 1859; E-mail: [email protected]
J. L. Lage
Mechanical Engineering Department, Box 750337,
School of Engineering, Southern Methodist University, Dallas, TX, USA, 75275-0337
Tel.: +(1 214) 768 2361; Fax: +(1 214) 768 1473; E-mail: [email protected]
By exploring the relationship between the group velocity of the de Broglie’s
waves and a particle velocity we can demonstrate the existence of a close relationship between the continuity equation and the Schrödinger’s equation. This
relationship leads to the proportionality between the fluid velocity v and the
corresponding de Broglie’s wave’s phase at the same location. That is, the
existence of a scalar function θ̃ proportional to the phase of the de Broglie’s
wave, such that v = ∇θ̃ can be proven without reference to the flow being
inviscid. We then proceed to show that the Navier – Stokes equation in the
case of constant viscosity incompressible fluid is equivalent to a reaction –
diffusion equation for the wave function of the de Broglie’s wave associated
to the moving fluid element. A general solution to this equation, written in
terms of the Green’s functions, and the criterion for the solution to be stable is presented. Finally, in order to provide an example, the procedure is
applied to obtain the solution for the simplest case of the Burgers’ equation.
* * *
Introduction
The present article is an attempt to look at the problem of solving the governing equation of fluid
motion in the most general way, considering, whenever possible, only the fundamental principles of
scientific knowledge.
The Navier – Stokes equation, together with the continuity equation, is the governing equation
†
Received 22.03.2001
ISSN 1064-2277
c
°
2002 Begell House, Inc.
40
of fluid motion valid as long as the continuum hypothesis holds. For a wide variety of flow phenomena encountered in engineering, the assumption of constant properties is adequate. However, even
when simplified by this assumption the Navier – Stokes equation still presents a challenge, namely
the nonlinear convective term, which makes the solving procedure very cumbersome, if possible at
all.
Closed form solutions of the Navier – Stokes equation become possible after much more significant simplifications and are few, e. g., Landau & Lifschiz (1988) [1] and White (1991) [2]. Even
from the point-of-view of numerical flow simulation, the task of finding an exact solution to the
Navier – Stokes equation is often impossible to be accomplished because of the necessary computer
resources (to take into account the very large range of time and spatial scales) to avoid spurious
effects caused by the non-linear nature of the equation, Anderson (1995) [3].
Surprisingly enough, it becomes possible to transform the Navier – Stokes equation, written for
the case of constant and uniform dynamic viscosity and uniform density, into the linear so called
reaction – diffusion equation. The transformation is accomplished by rewriting the convective and
diffusion terms in such a way as to eliminate the convective part.
This work underlines the belief that only very few basic principles (such as conservation of
energy or the least action) are necessary to model any physical process and, therefore, differential
equations describing different processes must be kin through following these principles.
Another idea underlying this work is the concept according to which any fundamental equation must comprise all the fundamental principles in itself. It is not possible, for example, for a
fundamental equation to satisfy the energy conservation principle and to not satisfy the least action
one.
It is commonly believed that the Navier – Stokes equation indeed governs any fluid motion,
including turbulence. If it is so, no additional hypothesis should be made in order to describe turbulence by means of the Navier – Stokes equation. However, since no general method for solving
the Navier – Stokes equation exists nowadays, it has become usual to resort to various models that
often are inconsistent with the phenomenon in question. Thus, for example, the turbulence concept
of eddy viscosity emerges, although it is independent of physical properties of the fluid. The advantage of splitting the velocity components into time averaged and fluctuating parts is doubtful, for
it leads to the closure problem of the Navier – Stokes equation. Besides, it is not completely clear
how the averaging should take place in reality, for depending on the scale of the time step used for
averaging the result can differ significantly. This is so because any turbulent process is by definition
a fractional Brownian process, Koulich (1999) [4]. In fact this problem is similar, if not the same, to
the problem of measuring the length of the coast of Britain, Mandelbrot (1982) [5], where the result
depends on the measuring scale.
All such practices in studying turbulence became so common that they shadow the real problem
of tackling the Navier – Stokes equation, trying to recover the information of turbulence encoded in
it.
In what follows we provide a fresh look at the Navier – Stokes equation without having recourse
to any additional model or hypothesis but considering this famous equation as it is.
1. Wave Function
Any freely moving particle, which has energy E and momentum p, can be associated with a
flat wave
(1)
ψ(r, t) = C exp[i(ωt − kr)],
41
where r is the radius-vector of an arbitrary point in the space, C is the wave amplitude, t is the
time, de Broglie (1923) [6]. The frequency ω of this wave and its wave number k are related to the
particle energy E and momentum p by the same equations which are valid for a photon, i. e.,
E = h̄ω,
(2)
p = h̄k,
(3)
where h̄ is the Planck’s constant. Eqs. (2) and (3) are called the de Broglie’s equations. By substituting Eqs. (2) and (3) into Eq. (1), the latter equation can be written as
·µ
¶¸
Et pr
ψ(r, t) = C exp i
−
(4)
.
h̄
h̄
Such a wave is called the de Broglie’s wave. The function ψ defined by Eq. (4) is called the
wave function. The wave function determines the probability with which the particle can be found
in a given point in space at a given moment of time. In fact, the probability to find the particle in
an infinitesimally small volume d3 r around the point r is proportional to |ψ(r, t)|2 d3 r. Thus, the
probability density is proportional to the squared modulus of the wave function.
A given wave function corresponds to a certain motion of the particle. It is worth to emphasize
that if ψ(r, t) is possible for a certain state of the particle, then the wave function ψ1 (r, t) =
ψ(r, t) exp(iθ) is also possible provided θ is a real quantity, Blokhintsev (1976) [7]. It is necessary
that the probability densities determined by both functions ψ and ψ1 be equal, so that these two
functions describe the same state of the particle motion.
2. Group Velocity
In principle, there seems to be no relationship between the wave movement described by Eq. (1)
and the mechanical laws of the particle motion. However, this is not so. Consider, for simplicity, a
one-dimensional de Broglie’s wave propagating in the direction that coincides with the x-axis of a
Cartesian co-ordinate system, i. e.,
ψ(x, t) = C exp[i(ωt − kx)].
(5)
The quantity (ωt − kx) is the phase of the wave. Consider now a certain point x, where the
phase has a given value α. The location of this point can be found from the equation α = ωt − kx.
One can see that the phase value α moves in the course of time along the x axis with the velocity u
(the phase velocity) whose value can be determined by differentiating α = ωt − kx with respect to
time, and equating the result to zero. Hence,
dα
= ω − ku = 0
dt
⇒
u=
ω
.
k
(6)
Consider now a group of the de Broglie’s waves, i. e., a superposition of waves whose wave
lengths belong to a very narrow interval. The wave function of this group of waves is
k0Z+∆k
ψ(x, t) =
C(k) exp[i(ωt − kx)]dk,
k0 −∆k
42
(7)
where k0 is the wave number, around which the wave numbers of other waves, forming the group,
are located. The value ∆k is very small.
Now, for a non-relativistic particle, the particle energy can be written as (see Blokhintsev
(1976) [7])
£
¤ 1/2
p2
E = m20 c4 + p2 c2
+ ...,
(8)
= m0 c2 +
2m0
where m0 is the rest-mass of the particle, and c is the speed of light. By substituting Eq. (8) into
Eq. (2) and expressing p2 through k 2 , using Eq. (3), one obtains
ω=
m0 c2
h̄k 2
+
+ ...
h̄
2m0
Expanding the frequency ω as a function of k in series around k0 :
µ ¶
dω
(k − k0 ) + . . . ,
k = k0 + (k − k0 ).
ω = ω0 +
dk 0
(9)
(10)
Now, introducing a new integration variable ξ = k − k0 and assuming c(k) a slowly varying
function of k, one finds:
Z∆k
ψ(x, t) = C(k0 ) exp[i(ω0 t − k0 x)]
exp {i[(dω/dk)0 t − x]ξ} dξ,
(11)
−∆k
where C(k0 ) is the mean between C(k0 − ∆k) and C(k0 + ∆k) and, therefore, a constant. The
integration yields
ψ(x, t) = 2C(k0 )
sin {[(dω/dk)0 t − x]∆k}
exp[i(ω0 t − k0 x)]
(dω/dk)0 t − x
(12)
= C(x, t) exp[i(ω0 t − k0 x)].
Observe that C(k0 ) and C(x, t) both denote the same quantity – the wave amplitude – however,
expressed in two of the three possible co-ordinate systems (k, ω), (x, t), or (E, p). Since ∆k in the
sine term of Eq. (12) is small, then C(x, t) changes slowly with time and x. Therefore, one can
consider C(x, t) as the amplitude of an almost monochromatic wave whose phase is (ωt − k0 x).
It is obvious that the amplitude has its maximum value at x = (dω/dk)0 t. This point is called
the center of the wave group. This center moves with the velocity U = (dω/dk)0 obtained by
differentiating the expression for x with respect to time. The velocity U is called the group velocity.
The group velocity U can be found by using Eq. (9) as U = dω/dk = h̄k/m0 . On the other
hand, according to Eq. (3), h̄k = p = m0 v, where v is the particle velocity. Therefore, one comes
to an important conclusion: the group velocity of the de Broglie’s waves U equals to the velocity v
of the particle corresponding to this group of waves, i. e., U = v.
The above conclusion can be easily drawn in a multi-dimensional case, Blokhintsev (1976) [7],
having
∂ω
∂E
∂ω
∂E
∂ω
∂E
Ux =
=
,
Uy =
=
,
Uz =
=
(13)
∂kx
∂px
∂ky
∂py
∂kz
∂pz
or, in a vector form,
U = ∇k ω = ∇p E = v.
43
(14)
3. Schrödinger’s Equation and Continuity Equation
The Schrödinger’s equation forms the basis of quantum mechanics. This equation determines
the wave function of a particle of mass m moving in a potential field Φ(x, t). It can be written in
the form, Blokhintsev (1976) [7],
ih̄
∂ψ
h̄2 2
=−
∇ ψ + Φ(x, t)ψ.
∂t
2m
(15)
The most important particularity of the Schrödinger’s equation is the presence of the imaginary
unity i in front of the time derivative. In classical physics, equations of first order in time have no
periodic solutions and, therefore, describe irreversible processes, e. g., mass diffusion, heat conduction, etc. Due to the imaginary coefficient, the Schrödinger’s equation, although first order in time,
allows periodic solutions.
It will be shown now that the Schrödinger’s equation also allows to derive the continuity equation in its most general form. In order to do so, consider the Schrödinger’s equation written for the
conjugate wave function ψ ∗ = C exp(−iθ), namely:
−ih̄
∂ψ ∗
h̄2 2 ∗
=−
∇ ψ + Φ(x, t)ψ ∗ .
∂t
2m
(16)
By multiplying Eq. (15) by ψ ∗ and Eq. (16) by ψ, and subtracting the resulting equations, one
obtains:
µ
¶
∂ψ
∂ψ ∗
h̄2 ∗ 2
ih̄ ψ ∗
+ψ
=−
(ψ ∇ ψ − ψ∇2 ψ ∗ ).
(17)
∂t
∂t
2m
Eq. (17) can be re-written as
∂w
+ divj = 0,
∂t
where
divj = i
(18)
¢
h̄ ¡ ∗ 2
ψ ∇ ψ − ψ∇2 ψ ∗ .
2m
Observe also that ψψ ∗ = w by definition. Indeed, ψ = C exp(iθ) and its conjugate ψ ∗ = C exp(−iθ)
lead to ψψ ∗ = C 2 = w, i. e., the probability density to find the particle in the state described by
ψ. This is stated in, for instance, Blatt (1992) [8]. Since mass itself is a form of energy and the
wave function ψ represents the wave field of a particle of mass m, then ψψ ∗ may be thought of
as an energy density associated with that mass. Accordingly, the product ψψ ∗ = |ψ|2 = w is to be
interpreted as a probability density.
If one notes that w can be also considered as the density of the particles, then Eq. (18) is the
equation describing conservation of the amount of particles. Hence, j should be interpreted simply
as the flux of particles.
It is worth to emphasize that after multiplying Eq. (18) by the mass of a particle m, one obtains
∂ρm
+ divj m = 0,
∂t
(19)
where ρm is the particle density and j m is the density flux. Therefore, Eq. (19) is nothing else but
the continuity equation. A more elaborate procedure to reach the same conclusion is presented in
the Appendix.
44
Observe that the form Eq. (18), derived from the Schrödinger’s equation, is the most general
form of the continuity (conservation) equation. Thus, after multiplying Eq. (18) by the electric
charge of the particle, one obtains the equation describing conservation of the electric charge, i. e.,
∂ρe
+ divj e = 0,
∂t
(20)
where ρe is the density of the charge and j e is the density of the electric current.
Recall now the representation of the wave function given by Eq. (5). This equation can be
written in a more compact form
ψ(x, t) = C exp(iθ),
(21)
where C is the real amplitude and θ is the real phase of the wave function. By substituting Eq. (21)
into the explicit expression for j,
j=−
ih̄
(ψ∇ψ ∗ − ψ ∗ ∇ψ),
2m
(22)
one obtains
h̄
∇θ.
m
Since C 2 = w, then the quantity (h̄/m)∇θ must be the velocity at the point x.
j = C2
(23)
The conclusion, drawn above, is of great importance and is hard to overestimate. It will be
extensively used in the course of further derivations. Indeed, velocity at a certain location x is
proportional to the gradient of the corresponding de Broglie’s wave’s phase in the same location,
i. e.,
h̄
ih̄ ∇ψ
v(x) = ∇θ̃ = ∇θ = −
,
(24)
m
m ψ
where the value (h̄/m)θ = θ̃ is the velocity potential.
4. Navier – Stokes and Diffusion Equations
Consider the Navier – Stokes equation written for the case of a constant viscosity fluid, Landau
& Lifschiz (1988) [1]:
·
¸
³
∂v
µ´
ρ
+ (v∇)v = −∇p + µ∇2 v + λ +
∇(∇ · v),
(25)
∂t
3
where ρ, λ, and µ are density, second viscosity, and dynamic viscosity of the fluid respectively, p is
the pressure, and v is the fluid velocity.
In the case of an incompressible fluid, Eq. (25) can be written in a simpler form, since ∇v = 0,
i. e.,
∂v
1
+ (v∇)v = − ∇p + ν∇2 v,
∂t
ρ
(26)
where ν = µ/ρ is called kinematic viscosity.
Now, using the result expressed by Eq. (24), the Navier – Stokes equation can be written in
terms of the phase of the de Broglie’s wave corresponding to the moving fluid particle, (taking into
account that
(v · ∇)v = (∇θ̃ · ∇)∇θ̃ =
1
1
∇(∇θ̃ · ∇θ̃) − ∇θ̃ × (∇ × ∇θ̃) = ∇(∇θ̃ · ∇θ̃),
2
2
45
and
∇2 v = ∇(∇ · v) − ∇ × (∇ × v) = ∇(∇ · ∇θ̃) − ∇ × (∇ × ∇θ̃) = ∇(∇2 θ̃),
since ∇ × ∇ ≡ 0:
#
·
¸
p
∂ θ̃ 1
2
∇
+ (∇θ̃ · ∇θ̃) = ∇ − + ν∇ θ̃ ,
∂t
2
ρ
"
(27)
or
∂ θ̃ 1
∆p
+ (∇θ̃ · ∇θ̃) = −
+ ν∇2 θ̃,
(28)
∂t
2
ρ
where ∆p is the difference between the actual pressure p and a certain reference pressure p0 .
Now, substituting θ̃ = −2ν ln ψ̃, the Navier – Stokes equation becomes
∆p
∂ ψ̃
− ν∇2 ψ̃ =
ψ̃,
∂t
2µ
(29)
which is the reaction – diffusion equation written in terms of the scalar function ψ̃.
From Eq. (24), it follows that θ̃ = −(ih̄/m) ln ψ. On the other hand, θ̃ = −2ν ln ψ̃ was
used in order to transform the Navier – Stokes equation. It is obvious that ψ and ψ̃ are related
as ψ̃ = ψ exp(ih̄/2mν). It was already mentioned in Section 1 that both ψ and ψ̃ represent the
same state of the particle, for their amplitudes are equal and, therefore, the probability densities
determined by them are the same. Thus, it is no wonder that ψ̃ = ψ exp(ih̄/2mν) being substituted
into Eq. (29) leads to
∂ψ
∆p
− ν∇2 ψ =
ψ
(30)
∂t
2µ
which is the reaction – diffusion equation for the wave function corresponding to the de Broglie’s
wave of a fluid particle moving with velocity v.
Thus, it was so far shown that the Navier – Stokes equation for an incompressible fluid of constant viscosity yields the reaction – diffusion equation for the wave function of the de Broglie’s wave
associated with the moving fluid element. That is, the Navier – Stokes equation allows only an irrotational solution (because of Eq. (24)). Otherwise, in case of a rotational flow, the solution becomes
a set of chaotic values and, in fact, does not exist in the sense understood outside the chaos theory.
Eq. (30) is a particular case of the Einstein – Kolmogorov differential equation whose derivation is given in Tikhonov & Samarskii (1963) [9]. The general case of the equation describes a
random motion, so called Brownian motion, of a microscopic particle, existing in free suspension
in a medium.
5. Initial and Boundary Conditions
From the introduction of the wave function ψ, it is seen that this function and velocity are
related as follows:
∇ψ
v = −2ν
(31)
.
ψ
In order to be able to solve Eq. (30), one has to re-write the initial and boundary conditions,
which are given in terms of velocity, into conditions written in terms of the wave function. However,
from Eq. (31) the way for converting these conditions cannot be found just as:
¶
µ
Z
1
vdx
(32)
ψ = exp −
2ν
46
for it is impossible to determine the wave function precisely but to a certain constant coefficient
only.
From Eq. (24), it follows that
and, therefore,
m
θ=
h̄
Z
vdx
(33)
µ
¶
Z
m
ψ(x, t) = C exp −i
vdx .
h̄
(34)
In particular, one can immediately see that the no-slip condition becomes equivalent to the wave
function being a certain non-zero constant value (in fact, the wave function value can never be zero
to avoid degenerating the solution into a trivial one). One should not be concerned with the fact
that the wave function can only be determined to a constant multiplier, for, looking at Eq. (31), it
becomes clear that the velocity is uniquely determined even though the wave function contains an
arbitrary non-zero coefficient.
According to Eq. (34), the initial condition for the Navier – Stokes equation can be written in
terms of ψ-function as
µ
¶
Z
m
(35)
ψ(x, 0) = C exp −i
v(x, 0)dx .
h̄
6. General Solution
The general solution of Eq. (30) can now be written in terms of the Green’s function of the
non-homogeneous diffusion equation, Haberman (1987) [10, p. 409]:
Zt Z Z Z
ψ(x, t) =
0
℘(x, t; ξ, τ ) ∆p
ψ(x, t)d3 ξdτ +
2µ
ZZZ
℘(x, t; ξ, 0)ψ(ξ, 0)d3 ξ
Zt ZZ
+ν
° [℘(x, t; ξ, τ )∇ξ ψ − ψ(ξ, τ )∇ξ ℘(x, t; ξ, τ )] n dS dτ,
(36)
0
where ℘(x, t; ξ, τ ) is the Green’s function of the diffusion equation for the domain of interest.
Eq. (36) is a linear integral equation for ψ(x, t). One can show that ψ(x, t) is a solution of
Eq. (30) if, and only if, ψ(x, t) is a solution of Eq. (36), Logan (1994) [11].
Observe that Eq. (36) has the form ψ = Υ(ψ), where Υ is the mapping defined on the set of
bounded continuous functions. Thus one can define a fixed-point, iterative process by ψm+1 =
Υ(ψm ) with
ZZZ
ψ0 (x, t) =
℘(x, t; ξ, 0)ψ(ξ, 0)d3 ξ.
The initial approximation ψ0 (x, t) is just the second term on the right side of Eq. (36), and it is
the solution to the linear, homogeneous diffusion equation with initial condition ψ(x, 0) = ψ0 (x).
Therefore, the solution of Eq. (30) can be found from
v = −2ν
where ψ(x, t) is determined by Eq. (36).
47
∇ψ
,
ψ
(37)
Consider now the influence of the domain boundary on the solution, i. e., the third term in
Eq. (36). Both ψ and its normal derivative seem to be needed on the boundary. In most fluid
mechanics problems, however, the velocity value, but not its normal derivative, is given as the
boundary condition. Hence, ψB (x, t) can be found by using Eq. (34). The Green’s function satisfies
the related homogeneous boundary conditions, in this case ℘(x, t; ξ, τ ) = 0 along the boundary.
Thus, the effect of this imposed wave function distribution is
Zt ZZ
−ν
° [ψ(ξ, τ )∇ξ ℘(x, t; ξ, τ )] n dS dτ
0
for those problems in fluid mechanics in which the velocity value is given as the only boundary
condition.
It may be helpful to illustrate the modifications necessary for one-dimensional problems. Volume integrals in Eq. (36) become one-dimensional integrals. Boundary contributions on the closed
surface become contributions at the two ends x = x1 and x = x2 .
Further simplifications can be made in the case of zero pressure gradient, i. e., if ∆p is equal to
a constant. In this special case, Eq. (30), by the substitution ψ = ψ̂ exp[(∆p/(2µ))t], reduces to
∂ ψ̂
= ν∇2 ψ̂
∂t
(38)
with the same initial condition and with the boundary condition ψ̂ = ψ exp[−(∆p/(2µ))t]. The
solution of Eq. (38) is not a mapping anymore and can be written as:
ZZZ
ψ̂(x, t) =
℘(x, t; ξ, 0)ψ̂(ξ, 0)d3 ξ
+ν
Zt ZZ h
i
° ℘(x, t; ξ, τ )∇ξ ψ̂ − ψ̂(ξ, τ )∇ξ ℘(x, t; ξ, τ ) n dS dτ.
(39)
0
7. Stability of the Solution. Transition to Chaos
Consider now the mapping ψm+1 = Υ(ψm ), in general non-linear, to draw a conclusion about
stability of the solution to Eq. (30).
It was pointed out by Landau & Lifschiz (1988) [1] that transition to turbulence is related to the
flow losing stability and can be treated as a bifurcation process, i. e., period doubling which follows
the Feigenbaum’s scenario, Glendinning (1995) [12]. Under a certain condition the solution (36)
becomes non-periodic (“chaotic”) but bounded and nearby solutions separate rapidly in time. This
latter property, called sensitive dependence upon the initial condition, can be thought as a loss of
memory of the flow of the past history. It implies that long term predictions of the flow are almost
impossible despite the deterministic nature of the Navier – Stokes equation.
The solution of the mapping ψm+1 = Υ(ψm ) becomes chaotic, if the following condition
holds:
ψm − ψm−1
(40)
lim
= λ̄ = 4.6692 . . .
m→∞ ψm−1 − ψm−2
The constant λ̄ = 4.6692 . . ., called Feigenbaum’s constant, is universal and marks the transition to
chaos, Landau & Lifschiz (1988) [1], Peitgen, Jügens & Saupe (1992) [13].
48
Turbulence becomes fully developed, if the solution of Eq. (36) satisfies condition (40). In the
flow region, for which the condition in question is not satisfied, either transient or laminar regimes
take place depending on the limit value given by the left-hand side of Eq. (40).
8. Analytic Solution of the Burgers’ Equation
In order to illustrate feasibility of the described way of solving the Navier – Stokes equation,
consider a simple case of the Burgers’ equation.
The Burgers’ equation is a simplified form of the Navier – Stokes equation in the one-dimensional, Cartesian, time-dependent, compressible, Newtonian case. This equation has been used in
studying the decay of free turbulence. The Burgers’ equation is a simple equation showing the
complicated interplay between the non-linear growth and decay of a wave. Consider the initial
value problem for the Burgers’ equation, Landau & Lifschiz (1988) [1]:
∂u
∂2u
∂u
+u
=ν 2,
∂t
∂x
∂x
(41)
where v is uniform throughout the domain, with the initial condition u(x, 0) = u0 (x).
By using the substitution
u = −2νψ
∂ψ
,
∂x
(42)
Eq. (41) reduces to the diffusion equation
∂ψ
∂2ψ
− ν 2 = 0.
∂t
∂x
(43)
Also, via Eq. (42), the initial condition on u transforms into an initial condition on ψ, i. e.,
u(x, 0) = u0 (x) = −2ν
ψx (x, 0)
.
ψ(x, 0)
(44)
Integrating both sides of Eq. (44) yields

1
ψ(x, 0) = ψ0 (x) = exp −
2ν
Zx

u0 (ξ)dξ  .
Now, the solution to the initial value problem for the diffusion Eq. (43) is
·
¸
Z
1
(x − ξ)2
ψ(x, t) =
ψ
(x)
exp
−
dξ.
0
4νt
(4πνt)1/2
Therefore,
∂ψ
1
=−
∂x
(4πνt)1/2
Z
ψ0 (x)
·
¸
x−ξ
(x − ξ)2
exp −
dξ.
2νt
4νt
Consequently, from Eq. (44), we obtain
Z
[(x − ξ)/t] ψ0 (ξ) exp[−(x − ξ)2 /(4νt)] dξ
Z
,
u(x, t) =
ψ0 (ξ) exp[−(x − ξ)2 /(4νt)] dξ
49
(45)
0
(46)
(47)
(48)
where ψ0 (ξ) is given by Eq. (45). It is now straightforward to see that the solution (48) can be
written as
Z
[(x − ξ)/t] exp[−G(ξ, x, t)/(2ν)] dξ
Z
u(x, t) =
.
(49)
exp[−G(ξ, x, t)/(2ν)] dξ
Here,
(x − ξ)2
G(ξ, x, t) =
+
2t
Zξ
˜ ξ.
˜
u0 (ξ)d
(50)
0
Observe that Eq. (49) can be derived from the general solution listed in Eq. (36), by realizing
that ∆p is constant (because ∇p = 0 in the Burgers’ equation), so one can re-define ψ to transform
Eq. (30) into a homogeneous equation – this is equivalent of setting the ∆p of Eq. (36) as equal to
zero. Moreover, in infinite unidirectional space the surface integral in Eq. (36) disappears, and the
last term reduces to a single integral in x.
Concluding Remarks
It is important to point out a distinction between the approach described so far and the one
following from the inviscid flow assumption. It is usual in fluid mechanics to invoke the inviscid
condition and from it to demonstrate that the flow is irrotational, i. e., $ = ∇ × v = 0, where $
is the vorticity function. As a consequence, a potential function φ must exist, such that ∇φ = v
because ∇ × ∇φ = 0 for any scalar function θ̃. In the present case, the existence of the scalar
function θ̃, proportional to the phase of the de Broglie’s wave, such that v = ∇θ̃, was shown without
requiring the flow to be inviscid.
Moreover, turbulence is not precluded in this case, even though the flow is irrotational, because
the viscous effect necessary for turbulence is still present in the momentum transport equation.
The similarity between the diffusion form of the Navier – Stokes equation and the Schrödinger’s
equation is by no means accidental but reflects the fundamental connection of the diffusion process to that one of random walks. The transformation performed for the Navier – Stokes equation
stretches a bridge between quantum mechanics of the micro-world and the continuous world of fluid
motion.
Appendix
Consider the Schrödinger’s equation
ih̄
∂ψ
h̄2 2
=−
∇ ψ + U ψ,
∂t
2m
where the wave function ψ can be defined as
·
µ
¶¸
µ ¶
i
S
ψ = C exp(iθ) = C exp −
Et − pr
= C exp i
,
h̄
h̄
(A1)
(A2)
where S is called action, and ∇S/m = p/m = v is the classical velocity of the particle. Since
µ
¶
µ ¶
∂ψ
∂C
i ∂S
S
=
+ C
exp i
(A3)
∂t
∂t
h̄ ∂t
h̄
50
µ ¶
C
i 2
S
2
∇ ψ = [∇ C − 2 (∇S) + (∇ S + 2∇S · ∇C)] exp i
.
h̄
h̄
h̄
and
2
2
On substituting Eqs. (A3) and (A4) into Eq. (A1), the latter becomes
·
¸
∂S
h̄2 2
C
∂C
1
C 2
C
−
∇ C+
(∇S)2 + U C − ih̄
+ ∇S · ∇C +
∇ S = 0.
∂t
2m
2m
∂t
m
2m
(A4)
(A5)
Eq. (A5) consists of the pure real and the pure imaginery parts and, thus,
C
and
h̄2 2
C
∂S
−
∇ C+
(∇S)2 + U C = 0
∂t
2m
2m
1
C 2
∂C
+ ∇S · ∇C +
∇ S=0
∂t
m
2m
(A6)
(A7)
must hold simultaneously.
If the term containing h̄2 in Eq. (A6) is neglected, this equation becomes nothing else but the
classical Hamilton – Jacobi equation for the particle’s action. This is, by the way, a good illustration
of the fact that as h̄ → 0 the classical mechanics is valid up to the values of the first (and not zeroth –
as is the common belief) order of h̄ inclusive.
Now, Eq. (A7), on being multiplied by 2C, becomes
2C
2C
2C 2 2
∂C
+
∇S · ∇C +
∇ S=0
∂t
m
m
which can be re-written as
µ
¶
∂C 2
∇S
+ ∇ C2
= 0.
∂t
m
(A8)
(A9)
Since C 2 = w = ψψ ∗ and
C2
where
∇S
= wv = j = ψ∇ψ ∗ − ψ ∗ ∇ψ,
m
µ
¶
µ ¶
µ
¶
i
S
ψ ∗ = C exp −iθ = C exp
(Et − pr) = C exp −i
h̄
h̄
denotes the complex conjugate wave function, Eq. (A9) can be written as
∂w
+∇·j =0
∂t
(A10)
which is the continuity equation written in its most general form. The vector j in Eq. (A10) can be
called the probability flux.
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51
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