# Squire’s theorem for nonlinear monotone energy stability

Giuseppe Mulone111Università degli Studi di Catania (retired), Dipartimento di
Matematica e Informatica, Viale Andrea Doria 6, 95125 Catania, Italy,
<EMAIL_ADDRESS>
###### Abstract

The Squire’s theorem holds for parallel shear flows governed by the linearized
Navier-Stokes equations. Squire writes “For the study of the stability of flow
between parallel walls it is sufficient to confine attention to disturbances
of two-dimensional type”, Squire [2, p. 627]. Instead, for nonlinear Navier-
Stokes system it is supposed that the theorem does not hold in general (see
Drazin and Reid [3, pp. 429–430]).

Here we prove that the Squire theorem holds also for nonlinear monotone energy
stability of parallel shear flows that include Couette and Poiseuille flows
between parallel walls.

MSC Code  76E05

Keywords: Squire theorem, plane shear flows, nonlinear stability, Couette
flow, Poiseuille flow

##  1 Introduction

Squire, [2], considers the steady two-dimensional flow of an incompressible
viscous fluid upon which a small disturbance is superimposed (linearized
Navier-Stokes equations). In the summary of his paper, he writes: “The
stability of the flow of a viscous fluid between parallel walls for three-
dimensional disturbances is discussed. A fourth-order differential equation is
derived and it is shown that, if any velocity profile is unstable for a
particular value of Reynolds’ number, it will be unstable at a lower value of
Reynolds’ number for two-dimensional disturbances”. This is now called Squire
theorem, see Drazin and Reid [3]: “to each unstable three-dimensional
disturbance there corresponds a more unstable two-dimensional one”.

Squire’s theorem has also been proved for many other fluids including flows in
magnetohydrodynamics and in viscoelastic fluids, see [4] \- [13].

Drazin and Reid in their monograph, [3, pp. 429–430], write in the case of
nonlinear theory:

“Orr (1907) further assumed that two-dimensional perturbations, i.e. the least
eigenvalue of equations (53.21) can be obtained as the least eigenvalue of
equation (53.22). Although Squire’s theorem provides a justification for this
assumption in the case of linear theory, it is not applicable to equations
(53.21) and the assumption is false in general.
The determination of the least eigenvalue of equations (53.21) is clearly a
formidable problem in general and results are known for only a few flows. In
the case of Couette flow with $U(z)=z$ $(-1\leq z\leq 1)$, however, Joseph
(1966) has proved that the least eigenvalue of equations (53.21) is still
associated with a two-dimensional disturbance but one which varies only in the
$yz$-plane, i.e. the perturbed flow consists of rolls whose axes are in the
direction of the basic flow.”

In the nonlinear case, studying the nonlinear stability with the Lyapunov
method and choosing the energy of the perturbation as the Lyapunov function,
the critical energy Reynolds number is obtained by determining the maximum of
a functional ratio obtained from the Reynolds-Orr equation, Orr [14]. Orr
assumed (without proving it) that the maximum is obtained on the perturbations
to the velocity field which satisfy no-slip boundary conditions, are periodic,
divergence-free, and they are two-dimensional spanwise. Joseph, [15], [16]
instead assumed that the maximum is attained on streamwise perturbations
(2.5-dimensional perturbations).

Busse, [17] writes “SQUIRE’S Theorem does not hold for the Euler equations
determining the energy stability limit ${\rm Re}_{E}$”.

Despite these results, this problem was taken up by Falsaperla et al. [18] \-
[19]. In particular, in [19], the authors conjectured that the search for the
maximum should be performed on a subspace of the space of kinematically
admissible perturbations, the spanwise perturbations, as Orr assumed.

Recently, Mulone [20] proved that the conjecture of [19] is true. A
consequence of this proof is that Squire’s theorem holds also for the
nonlinear monotone energy stability: to obtain the minimum critical nonlinear-
energy Reynolds number it is sufficient to consider only two-dimensional
spanwise disturbances.

The plan of the paper is as follows. In Sec. 2 we write the non-dimensional
perturbation equations of basic laminar flows between two horizontal planes
with no-slip boundary conditions. In Section 3, we recall the Squire theorem
for linear Navier-Stokes system. In Sec. 4 we study monotone nonlinear energy
stability and prove that the Squire theorem holds for nonlinear monotone
energy stability. In Sec. 5 we make a conclusion.

##  2 Laminar flows between two parallel planes

Given a reference frame $Oxyz$, with unit vectors ${\bf i},{\bf j},{\bf k}$,
consider the (non-dimensional) layer
$\mathcal{D}={\mathbb{R}}^{2}\times[-1,1]$ of thickness $2$ with horizontal
coordinates $x,y$ and vertical coordinate (orthogonal to the layer) $z$.

The basic flows which are solutions of the stationary Navier-Stokes equations,
[3], are characterized by the functional form

$\displaystyle{\bf U}=(f(z),0,0)=f(z){\bf i},$ (1)

where ${\bf U}$ is the velocity field. The function $f(z):[-1,1]\to\mathbb{R}$
is assumed to be sufficiently smooth and it is called the shear profile (base
or mean flow). The velocity filed is written in a non-dimensional form. To
non-dimensionalize the Navier-Stokes equations and the gap of the layer we use
a Reynolds number ${\rm Re}$ based on the average shear and half gap $d$ (see
[18]).

In particular, for fixed velocity at the boundaries $z=\pm 1$, we consider two
well known profiles: Couette flow $f(z)=z$, and Poiseuille flow
$f(z)=1-z^{2}$.

###  2.1 Perturbation equations

The perturbation equations to the plane parallel shear flows, in non-
dimensional form, are

$\displaystyle\left\\{\begin{array}[]{l}u_{t}=-{\bf u}\\!\cdot\\!\nabla u+{\rm
Re}^{-1}\Delta u-(fu_{x}+f^{\prime}w)-p_{x}\\\\[5.0pt] v_{t}=-{\bf
u}\\!\cdot\\!\nabla v+{\rm Re}^{-1}\Delta v-fv_{x}-p_{y}\\\\[5.0pt]
w_{t}=-{\bf u}\\!\cdot\\!\nabla w+{\rm Re}^{-1}\Delta
w-fw_{x}-p_{z}\\\\[5.0pt] u_{x}+v_{y}+w_{z}=0.\\\ \end{array}\right.$ (6)

In (6) ${\bf u}$ is the perturbation velocity field and $p$ the perturbation
to the pressure field. The velocity field u has components $(u,v,w)$ in the
directions $x,y,z$, respectively. Here we use the symbols $g_{x}$ as
$\frac{\partial g}{\partial x}$, $g_{t}$ as $\frac{\partial g}{\partial t}$,
etc., for any function $g$. To system (6) we append the rigid (no-slip)
boundary conditions

${\bf u}(x,y,\pm 1,t)=0,{\quad}(x,y,t)\in{\mathbb{R}}^{2}\times(0,+\infty),$

and the initial condition

${\bf u}(x,y,z,0)={\bf u_{0}}(x,y,z),{\quad}{\rm in}{\quad}\mathcal{D},$

with ${\bf u_{0}}(x,y,z)$ solenoidal vector which vanishes at the boundaries.

We recall that the streamwise (or longitudinal) perturbations $(u.v.w),p$ are
those fields which do not depend on $x$; the spanwise (or transverse)
perturbations are those fields which do not depend on $y$. In particular, we
have two-dimensional spanwise perturbations if also $v=0$.

###  2.2 Linear stability/instability, the Squire’s theorem

Assume that both ${\bf u}$ and $\nabla p$ are $x,y$-periodic with periods
$2\pi/a$ and $2\pi/b$ in the $x$ and $y$ directions, respectively, with wave
numbers $(a,b)\in{\mathbb{R}}^{2}_{+}$ . In the following, it suffices
therefore to consider functions over the periodicity cell

$\Omega=[0,\frac{2\pi}{a}]\times[0,\frac{2\pi}{b}]\times[-1,1].$

Linear stability/instability is obtained by studying the linearized system
neglecting the nonlinear terms in (6).

We recall that, for laminar flows between parallel planes, Romanov [21] proves
that Couette flow is linearly stable for any Reynolds number. Orszag [22]
proves that Poiseuille flow is unstable for any Reynolds number bigger that
$5772$. In the last case, Orszag assumes two-dimensional spanwise disturbances
(in agreement with the results of Orr and Squire) and solves the Orr-
Sommerfeld equation, see Drazin and Reid [3, p. 156].

The Squire’s theorem, [2], holds for the linearized system: the most
destabilizing perturbations are two-dimensional spanwise perturbations.

##  3 Nonlinear monotone energy stability

As the basic function space, we take $L_{2}(\Omega)$, which is the space of
square-summable functions in $\Omega$ with the scalar product denoted by

$(g,h)=\int_{0}^{\frac{2\pi}{a}}\int_{0}^{\frac{2\pi}{b}}\int_{-1}^{1}g(x,y,z)h(x,y,z)dxdydz,$

and the corresponding norm $\|g\|=(g,g)^{1/2}.$

The  nonlinear monotone energy stability is based on the Reynolds-Orr energy
equation, [23]

$\displaystyle\dot{E}=-(f^{\prime}w,u)-{\rm Re}^{-1}[\|\nabla u\|^{2}+\|\nabla
v\|^{2}+\|\nabla w\|^{2}],$ (7)

where

$E(t)=\dfrac{1}{2}[\|u\|^{2}+\|v\|^{2}+\|w\|^{2}]$ (8)

is the (density)  energy of disturbance.

We first note that Lorentz [24] made this observation (see Lamb [25] , p. 640)
: “One or two consequences of the energy equation may be noted. In the first
place, the relative magnitude of the two terms on the right-hand side is
unaffected if we reverse the signs of $u,v,w$, or if we multiply them by any
constant factor. The stability of a given state of mean motion should not
therefore depend on the scale of the disturbance”. This scale invariance
property in particular means that the $-(f^{\prime}u,w)$ term has a positive
or negative sign and cannot change by exchanging the sign of only one of the
two components $u$ and $w$.

For perturbations with $-(f^{\prime}w,u)\leq 0$ and $\|\nabla{\bf u}\|>0$,
then $\dot{E}<-C_{0}E$, with $C_{0}$ a positive (Poincaré) constant. If
instead, $-(f^{\prime}u,w)>0$, from (7), we easily have

$\displaystyle\begin{array}[]{l}\dot{E}=\left(\dfrac{-(f^{\prime}w,u)}{\|\nabla
u\|^{2}+\|\nabla v\|^{2}+\|\nabla w\|^{2}}-\dfrac{1}{{\rm
Re}}\right)\|\nabla{\bf u}\|^{2}\leq\left(m-\dfrac{1}{{\rm
Re}}\right)\|\nabla{\bf u}\|^{2},\end{array}$ (10)

where

$\displaystyle\dfrac{1}{{\rm Re}_{E}}=m=\max_{\cal
S}\dfrac{-(f^{\prime}w,u)}{\|\nabla u\|^{2}+\|\nabla v\|^{2}+\|\nabla
w\|^{2}},$ (11)

$\cal S$ is the space of the kinematically admissible fields

$\displaystyle\begin{array}[]{l}{\cal S}=\\{u,v,w\in
H^{1}(\Omega),\;u=v=w=0{\quad}\hbox{on the boundaries,}\\\\[8.53581pt] \hbox{
periodic in {x}, and {y},}{\quad}u_{x}+v_{y}+w_{z}=0,{\quad}\|\nabla{\bf
u}\|>0\\},\end{array}$ (14)

with $H^{1}(\Omega)=W^{1,2}(\Omega)$ the Sobolev space: the subset of
functions ${\displaystyle h}\in{\displaystyle L^{2}(\Omega)}$ such that
${\displaystyle h}$ and its weak derivatives up to order ${\displaystyle 1}$
have a finite $L^{2}$-norm.

Assuming $-(f^{\prime}w,u)>0$, consider the functional ratio

${\cal F}(u,v,w)=\frac{-(f^{\prime}w,u)}{\|\nabla u\|^{2}+\|\nabla
v\|^{2}+\|\nabla w\|^{2}}$

in ${\cal S}$. For any $(u,v,w)$ in $\cal S$ satisfying the scale invariance
property, we have

$\displaystyle{\cal F}(u,v,w)\leq\frac{-(f^{\prime}w,u)}{\|\nabla
u\|^{2}+\|\nabla w\|^{2}}\leq\max_{\cal S}\frac{-(f^{\prime}w,u)}{\|\nabla
u\|^{2}+\|\nabla w\|^{2}}.$

In [20] it has been proved that

$\displaystyle\max_{\cal S}\frac{-(f^{\prime}w,u)}{\|\nabla u\|^{2}+\|\nabla
w\|^{2}}=\max_{\cal S}{\cal F}(u,0,w),$ (15)

with ${\bf u}=(u(x,y,z),0,w(x,,y,z))$ which do not depend on $y$ and satisfy
the scale invariance property (note that the scale invariance property is not
verified for streamwise perturbations used by Joseph [15] \- [16]). To obtain
(15) it is sufficient to solve the Euler-Lagrange equations of the maximum
problem and take into account the boundary conditions, the solenoidality of
the velocity field, the periodicity, and the invariance of scale of the energy
equation (cf. [20]). The Euler-Lagrange equations of the maximum (15) are:
$-f^{\prime}w+2\bar{m}\Delta
u=\lambda_{x},0=\lambda_{y},-f^{\prime}u+2\bar{m}\Delta w=\lambda_{z}$, where
$\bar{m}$ is the maximum, and $\lambda$ is a Lagrange multiplier. Searching
solutions $(u,v,w,\lambda)=\bar{H}(z)e^{i(ax+by)}$, with $a$ and $b$ wave
numbers and $\bar{H}(z)=\bar{u}(z),\bar{v}(z),\bar{w}(z),\lambda(z)$
respectively, we have: $b=0$. Therefore, $u,v,w,\lambda$ are functions of $x$
and $z$, and $v_{y}=0$, $u_{x}+w_{z}=0$. Since the space of fields
$(u(x,z),v(x,z),w(x,z))$ in $\cal S$, with $u_{x}+w_{z}=0$, is a subspace of
$\cal S$, the two maxima: the maximum (11) and the maximum on the first side
of equation (15) are equal. Hence, $v_{x}=v_{z}=0$. These equations and
$v_{y}=0$ imply $v=0$ for the maximizing field. Note that if $\lambda=0$, then
from the Euler-Lagrange equations and the boundary conditions we would have
$u=w=0$. In this case it should be noted that the perturbation $(0,v(x,z),0)$
is always stabilizing.

Therefore, we have

$\displaystyle\max_{\cal S}{\cal F}(u,v,w)=\max_{\cal S}{\cal
F}(u(x,z),0,w(x.z)),$ (16)

because the space of fields $(u,0,w)$ in $\cal S$, with $u_{x}+w_{z}=0$, is a
subspace of $\cal S$.

The Euler-Lagrange equations of the two maxima problems (11) and (15), after
eliminating the Lagrange multipliers, become

\- for the $\max_{\cal S}{\cal F}(u,v,w)$ :

$\displaystyle\begin{cases}f^{\prime}(\zeta_{y}+2w_{xz})+f^{\prime\prime}w_{x}+2m\Delta\Delta
w=0\\\ f^{\prime}w_{y}+2m\Delta\zeta=0,\end{cases}$ (17)

where $\zeta=v_{x}-u_{y}$ is the third component of vorticity. The boundary
conditions are $w=w_{z}=\zeta=0$ on the boundaries.

\- for the $\max_{{\cal S}}{\cal F}(u,0,w)$:

$\displaystyle 2f^{\prime}w_{xz}+f^{\prime\prime}w_{x}+2m\Delta\Delta w=0,$
(18)

with boundary conditions $w=w_{z}=0$.

It is easy to verify that the last equation and the boundary conditions may be
derived immediately from system (17) and the boundary conditions by putting
$v=0$ and $\partial/\partial y\equiv 0$. This is exactly what happens for the
equation (26) of Orr [14].

Therefore, we have

######  Theorem 3.1.

The critical nonlinear energy Reynolds number ${\rm Re}_{E}$ is given by
solving the Orr’s equation

$\displaystyle{\rm
Re}_{E}(f^{\prime\prime}w_{x}+2f^{\prime}w_{xz})+2\Delta\Delta w=0.$ (19)

with b.c. $w=w_{z}=0$, and $\Delta=\dfrac{\partial^{2}}{\partial
x^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}$.

######  Theorem 3.2.

Assuming ${\rm Re}<{{\rm Re}_{E}}$, the basic shear flow (1) is nonlinear
monotone exponentially stable according to the energy:

$E(t)\leq E(0)\exp\left\\{{\dfrac{\pi^{2}}{2}\left(\dfrac{1}{{\rm
Re}_{E}}-\dfrac{1}{{\rm Re}}\right)t}\right\\},{\quad}\forall t\geq 0.$ (20)

######  Theorem 3.3.

(Squire Theorem for nonlinear energy stability).

The least stabilizing perturbations in the energy norm (8) are two-dimensional
perturbations $(u,0,w)$ which do not depend on $y$, the spanwise
perturbations.

##  4 Conclusion

We have proved that, in the case of the classical flows of Couette and
Poiseuille between parallel walls, Squire’s theorem holds also in the
nonlinear energy stability: for the study of the nonlinear energy stability of
flow between parallel walls it is sufficient to confine attention to
disturbances of two-dimensional spanwise perturbations. This result was
obtained by comparing two maximum functional problems.

We note that Busse [17] uses the correct Reynolds-Orr energy equation, but
defines an incorrect functional ratio (see [17], formula (2)), precisely
replacing $-(f^{\prime}u,w)$ with its absolute value. This leads to the
erroneous result of his manuscript and of Joseph’s paper: they only get
sufficient nonlinear stability conditions and underestimated the critical
Reynolds number. For instance, in the case of Couette, the nonlinear Reynolds
number they obtain is $20.6$ instead of the exact critical Reynolds value
$44.3$ obtained by Orr.

Finally, we note that the Squire’s theorem may be also proved to other fluid
flows including nonlinear flows in magnetohydrodynamics and to viscoelastic
fluids, and possibly to flows between parallel planes with other boundary
conditions.

Statements and Declarations

I declare that I have no competing financial and / or non-financial interests.

Acknowledgements
The research that led to the present paper was partially supported by the
following Grants: 2017YBKNCE of national project PRIN of Italian Ministry for
University and Research, PTR DMI-53722122146 ”ASDeA” of the University of
Catania. We also thank the group GNFM of INdAM for financial support.

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