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O. Macchi, C.-M. Marle, Y. Meyer, H. Moffatt, R. Moreau, M. Nivat, L. Pandolfi, M. Pélegrin, P. Perrier, H. Petryk, O. Pironneau, 
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031
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Publication information: Comptes Rendus Mecanique (ISSN 1631-0721). For 2018, volume 346 is scheduled for publication. From 2016 all sub-
scription is electronic only, prices are available upon request from the Publisher or from the Regional Sales Office nearest you or from this 
journal’s website (http://www.elsevier.com/locate/issn/1631-0721). Further information is available on this journal and other Elsevier products 
through Elsevier’s website: (http://www.elsevier.com). Subscriptions are accepted on a prepaid basis only and are entered on a calendar year 
basis.
Desk Editor – Jean-Michel Blengino. Tel.: (+33) (0)1 71 16 54 90 – Fax: (+33) (0)1 71 16 51 84. E-mail: j.blengino@elsevier.com
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Editorial Board

Peng Du

Abdellatif Ouahsine

Philippe Sergent

Influences of the separation distance, ship speed and channel dimension on ship maneuverability in a confined waterway

Editorial board: Académie des sciences, 23, quai de Conti, 75006 Paris, France
Tel.: (+33) (0)1 44 41 43 61.
fatima.messadi@academie-sciences.fr
http://www.academie-sciences.fr
Objective of the journal: the Comptes rendus Mecanique are open to all scientists, whatever their position or nationality. Through a policy of rapid 
publication of original work, the journal is a highly advantageous means of making advances in fundamental research known to the international 
scientific community. Such communication is achieved in the form of research articles, review articles or articles forming part of a thematic issue.
The journal guarantees a rapid publication and an international diffusion.
Editorial director: Catherine Bréchignac.
Editor-in-chief: Jean-Baptiste Leblond.
Associate editors: Z. Bažant, F. Chinesta, F. Darve, J. Etay, R. Friedrich, P. Johnson, E. Mastorakos, J. Rice, H. Stone, L. Vervisch, E. Villermaux, 
J. Willis.
Editorial board:
A. Bensoussan, G. Berry, S. Candel, Y. Choquet-Bruhat, Ph. G. Ciarlet, P. Clavin, M. Combarnous, G. Comte-Bellot, R. Dautray,
P. Deheuvels, G. Duvaut, O. Faugeras, P. Flandrin, R. Glowinski, P. Huerre, G. Huet, G. Iooss, A. Jaumotte, D. Knuth, P. Lax, A. Linan, 
O. Macchi, C.-M. Marle, Y. Meyer, H. Moffatt, R. Moreau, M. Nivat, L. Pandolfi, M. Pélegrin, P. Perrier, H. Petryk, O. Pironneau, 
Y. Pomeau, P.-A. Raviart, J. Salençon, É. Sanchez-Palencia, J. Sifakis, P. Suquet, E. Van der Giessen, M. G. Velarde, A. Zaoui.
Indexation: Current Contents, Inspec, ISI, Pascal, Scopus, Zentralblatt MATH.
© 2018 Académie des sciences. All rights reserved.
Published by Elsevier Masson SAS, Société par actions simplifiée à associé unique, au capital social de 47 275 384 € - RCS Nanterre B 542 037 
031
Office: 65, rue Camille-Desmoulins, 92442 Issy-les-Moulineaux cedex, France.
Shareholder: Elsevier Holding France
Publication information: Comptes Rendus Mecanique (ISSN 1631-0721). For 2018, volume 346 is scheduled for publication. From 2016 all sub-
scription is electronic only, prices are available upon request from the Publisher or from the Regional Sales Office nearest you or from this 
journal’s website (http://www.elsevier.com/locate/issn/1631-0721). Further information is available on this journal and other Elsevier products 
through Elsevier’s website: (http://www.elsevier.com). Subscriptions are accepted on a prepaid basis only and are entered on a calendar year 
basis.
Desk Editor – Jean-Michel Blengino. Tel.: (+33) (0)1 71 16 54 90 – Fax: (+33) (0)1 71 16 51 84. E-mail: j.blengino@elsevier.com
Subscriptions – Tel.: (+33) (0)1 71 16 55 99 – Fax: (+33) (0)1 71 16 55 77. E-mail: infos@elsevier-masson.fr
Publisher – Giovanni Raneri
President – Daniel Rodriguez
Author enquiries
For enquiries relating to the submission of articles please visit Elsevier’s Authors Home at http://www.elsevier.com/authors. This also provides 
the facility to track accepted articles and set up e-mail alerts to inform you of when an article’s status has changed, as well as detailed artwork 
guidelines, copyright information, frequently asked questions and more. Contact details for questions arising after acceptance of an article, 
especially those relating to proofs, are provided after registration of an article for publication.
Imprimé en France par JOUVE Commission paritaire : 0517 T 81774
1, rue du Dr-Sauvé, Mayenne, France Périodicité : 12 numéros par an
Dépôt légal à parution 


Christian Licht

Thibaut Weller

An asymptotic Reissner–Mindlin plate model

Jianqiu Tian

Enlong Liu

Lian Jiang

Xiaoqiong Jiang

Yi Sun

Ran Xu

Influence of particle shape on the microstructure evolution and the mechanical properties of granular materials

Nicolas Bur

Pierre Joyot

Pierre Villon

Reduced-order model of optimal temperature control for the automated fibre placement process

Houssem Badreddine

Khemais Saanouni

Carl Labergère

Jean-Louis Duval

Effect of the kinematic hardening on the plastic anisotropy parameters for metallic sheets

Regina R. Bulatova

Gregory A. Chechkin

Tatiana P. Chechkina

Vyacheslav N. Samokhin

C. R. Mecanique 346 (2018) 807–814Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
On the influence of a magnetic field on the separation of the 
boundary layer of a non-Newtonian MHD medium ✩
Sur l’influence d’un champ magnétique sur la séparation de la couche limite 
d’un milieu MHD non newtonien
Regina R. Bulatova a, Gregory A. Chechkin a, Tatiana P. Chechkina b, 
Vyacheslav N. Samokhin c
a M.V. Lomonosov Moscow State University, Moscow 119991, Russia
b National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russia
c Moscow Polytechnical University, Pryanishnikova ul., 2A, Moscow 127550, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received 15 June 2018
Accepted 18 June 2018
Available online 27 June 2018
Keywords:
Magnetohydrodynamics
Boundary layer of a fluid
Separation point
O.A. Ladyzhenskaya fluid
Mots-clés :
Magnétohydrodynamique
Couche limite d’un fluide
Point de séparation
O.A. Ladyzhenskaya fluide
We study the behavior of a magnetohydrodynamic (MHD) stationary boundary layer in a 
framework modified according to O.A. Ladyzhenskaya. We estimate the shift of a separation 
point under the influence of a magnetic field.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
Nous étudions le comportement d’une couche limite stationnaire magnétohydrodynamique 
(MHD) dans le cadre modifié par O.A. Ladyzhenskaya. Nous estimons le déplacement du 
point de séparation sous l’influence d’un champ magnétique.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
To describe boundary layers of fluids, mathematicians use the Prandtl system of equations, which is obtained from 
Navier–Stokes system by rescaling. The solvability of boundary value problems as well as initial boundary value problems 
for boundary layers of Newtonian fluids, and separation of boundary layers were investigated in [1], [2] and [3]. Similar 
topics for non-Newtonian fluids and liquid crystals were studied in [4], [5], [6], [7], [8]. Nonuniqueness of solutions to 
the Navier–Stokes system leads to several modifications of this system. We consider the O.A. Ladyzhenskaya modification, 
✩ The paper was partially supported by RFBR grant 18-01-00046.
E-mail addresses: regina.bulatova@mech.math.msu.su (R.R. Bulatova), chechkin@mech.math.msu.su (G.A. Chechkin), chechkina@mail.ru (T.P. Chechkina), 
vnsamokhin@mtu-net.ru (V.N. Samokhin).https://doi.org/10.1016/j.crme.2018.06.010
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
808 R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814Fig. 1. The separation of a boundary layer.
where a nonlinear viscosity was suggested also for Newtonian fluids [9], [10]. The theory of boundary layer for the O.A. 
Ladyzhenskaya fluids was developed in [11].
In this paper, we study a boundary layer of magnetic fluid of O.A. Ladyzhenskaya. We show the influence of magnetic 
field on the separation of the boundary layer (Fig. 1).
2. Settings
In the two-dimensional case, the stationary system of modified magnetic hydrodynamical boundary layer has the form⎧⎪⎪⎨
⎪⎪⎩
ν
∂
∂ y
((
1 + k
( ∂u
∂ y
)2) ∂u
∂ y
)
− u ∂u
∂x
− v ∂u
∂ y
+ B2(U − u) = −U dU
dx
,
∂u
∂x
+ ∂v
∂ y
= 0,
(1)
where ν denotes the kinematic viscosity, k is a small positive constant, the density ρ and the medium conduction σ are 
equal to one, B(x), U (x) are given functions. The function U (x), the pressure p(x), and the electromagnetic field B(x) satisfy
U
dU
dx
= −dp
dx
− E B
σ
− B2U .
We consider equations (1) in D = {0 < x < X, 0 < y < ∞} with the boundary conditions
u(0, y) = u0(y), u(x,0) = 0, v(x,0) = v0(x),
u(x, y) → U (x) as y → +∞. (2)
Here the functions u0(x), v0(x) are given.
Definition 2.1. It is said that the functions u(x, y) and v(x, y) are a classical solution to problem (1), (2), if they satisfy 
the following properties: u and v are continuous in D , have in D continuous derivatives appearing in Eq. (1); and satisfy 
pointwise equations (1) and conditions (2).
To prove the existence and the uniqueness results, we need some more restrictions to given functions. Assume that the 
following conditions hold true.
Conditions I:
• U (x) > 0; u0(y) > 0 as y > 0;
• u0(0) = 0, u′0 > 0, u0(y) → U (0) as y → ∞;
• dU
dx
, v0(x), B(x) are differentiable on [0, X];
• u0(y), u′0(y), u′′0(y) are bounded as 0 ≤ y < ∞ and satisfy the Hölder condition;• in (0, 0) the following compatibility condition holds true as y → 0:
ν
∂
∂ y
((
1 + k
(∂u0
∂ y
)2)∂u0
∂ y
)
− v0(0) ∂u0
∂ y
+ B2(0)(U (0) − u0(y))+
+ U (0)dU (0) = O (y2).
dx
R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814 809The system (1) with conditions (2) can be rewritten as a quasilinear differential equation by means of the Mises trans-
form. To this end, we introduce new independent variables x = x, ψ = ψ(x, y), where
u = ∂ψ
∂ y
, v − v0(x) = −∂ψ
∂x
, ψ(x,0) = 0,
and a new unknown function w(x, ψ) = u2(x, y). Then
u = √w; y =
ψ∫
0
dψ√
w(x,ψ)
;
∂u
∂x
= 1
2
√
w
∂ w
∂x
+ 1
2
√
w
∂ w
∂ψ
∂ψ
∂x
;
∂u
∂ y
= 1
2
√
w
∂ w
∂ y
= 1
2
√
w
∂ w
∂ψ
∂ψ
∂ y
= 1
2
∂ w
∂ψ
;
∂2u
∂ y2
= ∂
∂ y
(1
2
∂ w
∂ψ
)
= 1
2
∂
∂ψ
(∂ w
∂ψ
)∂ψ
∂ y
=
√
w
2
∂2 w
∂ψ2
.
Substituting this expressions into the first equation of the system (1), we get
ν
√
w
2
∂2 w
∂ψ2
(
1 + 3k
(1
2
∂ w
∂ψ
)2) − √w( 1
2
√
w
∂ w
∂x
+ 1
2
√
w
∂ w
∂ψ
∂ψ
∂x
)
−
−1
2
(
v0(x) − ∂ψ
∂x
)∂ w
∂ψ
+ B2(U − √w) = −U ∂U
∂x
.
Finally, the new equation has the form
ν
√
w
(
1 + 3
4
k
(∂ w
∂ψ
)2)∂2 w
∂ψ2
− ∂ w
∂x
− v0 ∂ w
∂ψ
+ 2 B2(U − √w) = −2 U ∂U
∂x
(3)
in domain G = {0 < x < X, 0 < ψ < ∞} with conditions
w(0,ψ) = w0(ψ), w(x,0) = 0, w(x,ψ) → U 2(x) as ψ → ∞. (4)
The function w0(ψ) is defined from the equation
w0
( y∫
0
u0(η)dη
)
≡ u20(y).
The above conditions in terms of the new unknown function become the following ones.
Conditions II:
• w0(ψ) > 0, as ψ > 0;
• w0(0) = 0, w ′0(0) > 0;
• w0(ψ) → U 2(x) as ψ → ∞;
• dU
dx
, v0(x), B(x) are continuously differentiable as x ∈ [0, X];
• w0(ψ), w ′0(ψ), w ′′0(ψ) are bounded as 0 ≤ ψ < ∞ and Hölder continuous;
• in (0, 0), the compatibility condition holds true as ψ → 0:
μ(ψ) =ν√w0(ψ)((1 + 3
4
k
(∂ w0(ψ)
∂ψ
)2)∂2 w0(ψ)
∂ψ2
)
− v0(0) ∂ w0(ψ)
∂ψ
+
+ 2B2(0)(U (0) − √w0(ψ)) + 2U (0)dU (0)
dx
= O (ψ).
(5)
810 R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–8143. Existence and uniqueness
The following assertions hold true.
Theorem 3.1 (Existence). Suppose that the functions U (x), u0(y), v0(x) satisfy the conditions I. In the domain D for some X > 0, the 
problem (1), (2) has a solution u(x), v(x), satisfying the following properties: u(x) is continuous and bounded in D; u > 0 as y > 0; 
∂u
∂ y
> m > 0 as 0 ≤ y ≤ y0; m, y0 = const; ∂u
∂ y
,
∂2u
∂ y2
are bounded and continuous in any finite subdomain of D. The solution to 
problem (1), (2) does exist for any X > 0 if
U
(
dU
dx
+ B2
)
≥ 0 and v0 ≤ 0 or U
(
dU
dx
+ B2
)
≥ α = const > 0.
Theorem 3.2 (Existence). Suppose that the functions U (x), w0(ψ), v0(x) satisfy the conditions II. Then problem (3), (4) has a solution 
w(x, ψ) in domain G = {0 < x < X, 0 < ψ < ∞} for some X. This solution satisfies the following properties: w(x, ψ) is bounded in 
G, w(x, ψ) > 0 as ψ > 0, ∣∣∣∣∂ w∂ψ
∣∣∣∣ ≤ M,
∣∣∣∣√w ∂2 w∂ψ2
∣∣∣∣ ≤ M in G,∣∣∣∣∂ w∂x
∣∣∣∣ ≤ Mψ1−β, ∂ w∂ψ ≥ m > 0 as 0 ≤ ψ ≤ ψ1, 0 < β < 12 ,
where the positive constants M, m, ψ1 depend only on X, U , w0 , v0 .
If U
(
dU
dx
+ B2
)
≥ 0 and v0(x) ≤ 0 or U
(
dU
dx
+ B2
)
> 0, then such a solution does exist in G for any X > 0.
Theorem 3.3 (Uniqueness). Assume that the functions u(x, y), v(x, y) satisfy the system (1) in the domain D = {0 < x < X,
0 < y < ∞}, are continuous in D and satisfy the conditions (2); moreover, the inequalities 0 < u < C as y > 0,
k1 y ≤ u ≤ k2 y as 0 < y < y0, (6)
∂2u
∂ y2
≤ k3 in D (7)
hold true, where C , k j , y0 are positive constants. Then the solution u(x, y), v(x, y) to the problem (1), (2), satisfying the conditions 
above, is unique. If U
(
dU
dx
+ B2
)
≥ 0 and v0 ≤ 0, then the conditions (6), (7) can be omitted. If U
(
dU
dx
+ B2
)
≥ 0, and v0 is an 
arbitrary function, then the condition (6) can be omitted (i.e. we only need the condition (7)).
Theorem 3.4 (Uniqueness). The solution w(x, ψ) to the problem (3), (4), continuous and bounded in G, satisfying the inequalities 
k1ψ ≤ w(x, ψ) ≤ k2ψ as ψ ≤ ψ1; w(x, ψ) ≥ a > 0 as ψ ≥ ψ1; √w ∂2 w∂ψ2 ≤ M, is unique in G, where k1 , k2 , ψ1 , M are positive 
constants.
The proof of the theorems is based on the following statements.
Lemma 3.5. Assume that in G = {0 < x < X, 0 < ψ < ∞}, there exists a solution w(x, ψ) to the problem (3), (4), with the following 
properties: the function w(x, ψ) is bounded in G; w(x, ψ) > 0 as ψ > 0. There exist constants M, m, ψ1, only dependent on X, u0, 
v0 , and p(x), such that∣∣∣∣∂ w∂ψ
∣∣∣∣ ≤ M,
∣∣∣∣√w ∂2 w∂ψ2
∣∣∣∣ ≤ M, (x,ψ) ∈ G; (8)
moreover,∣∣∣∣∂ w∂x
∣∣∣∣ ≤ Mψ1−β, ∂ w∂ψ ≥ m > 0 as 0 ≤ ψ ≤ ψ1, 0 < β < 12 .
Then in D = {0 < x < X, 0 < y < ∞}, there exists a solution u(x, y), v(x, y) to the problem (1), (2), satisfying the following prop-
erties: the function u(x, y) is continuous and bounded in D, u > 0 as y > 0; ∂u
∂ y > m1 > 0 as 0 < y < y0 (m1 and y0 are some 
constants); ∂u and ∂
2u
2 are continuous and bounded in D; ∂u , v, ∂v are continuous and bounded in any finite subdomain of D.∂ y ∂ y ∂x ∂ y
R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814 811Consider the domain Gε =
{
0 < x < X, 0 < ψ <
1
ε
}
. Let
ε = {0 < x < X, ψ = 0} ∪
{
x = 0, 0 < ψ < 1
ε
}
∪
{
0 < x < X, ψ = 1
ε
}
.
In the domain Gε , we consider Eq. (3) with conditions:
w(x,0) = w0(ε)exp
(
μ(ε)x
w0(ε)
)
, w(0,ψ) = w0(ε + ψ),
w
(
x,
1
ε
)
= w0
(
ε + 1
ε
)
exp
(
μ(ε + 1/ε)x
w0(ε + 1/ε)
)
.
(9)
Lemma 3.6. If the problem (3), (9) has a positive solution wε(x, ψ) in the domain Gε , then there exist numbers X > 0 and ε0 > 0
such that, for any ε > ε0 , the inequality
wε(x,ψ) ≥ wε(x,0) + f (ψ)(1 + e−αx), (x,ψ) ∈ Gε (10)
holds true, where α > 0, and the function f (ψ) satisfies f (ψ) = A1ψ4/3 + A2ψ as ψ ≤ 1 and f (1) ≤ f (ψ) ≤ A3 as ψ > 1; in 
addition, 
∣∣ f ′(ψ)∣∣ ≤ A4, ∣∣ f ′′(ψ)∣∣ ≤ A5 as ψ > 1. Here Ai are positive constants.
If U
(
dU
dx
+ B2
)
≥ 0 and v0(x) ≤ 0, then in the domain Gε , the a priori estimate
wε(x,ψ) ≥ wε(x,0) + f (ψ)e−αx (11)
holds for any X > 0 and sufficiently small ε0 > 0.
If U
(
dU
dx
+ B2
)
≥ β0 > 0, then in the domain Gε , the inequality
wε(x,ψ) ≥ wε(x,0) + f (ψ) (12)
holds true.
Lemma 3.7. Let wε(x, ψ) be a solution to the problem (3), (9). Then there exist positive constants M1, M2 , M3 , independent of ε, such 
that
0 < wε(x,ψ) < M1, (13)
M2 <
∂ w
∂ψ
∣∣∣
ψ=0 < M3. (14)
Lemma 3.8. There exists a constant M4 such that∣∣∣∣∂ w∂ψ
∣∣∣∣ ≤ M4, (x,ψ) ∈ Gε.
Lemma 3.9. There exist positive constants C1, C2 independent of ε such that in Gε the estimates
∂ wε
∂x
≥ −C1, √wε ∂
2 wε
∂ψ2
≥ −C2
hold.
Lemma 3.10. There exist positive constants C3, C4 , C5 independent of ε such that in domain Gε the estimates
√
wε
∂2 wε
∂ψ2
≤ C3, ∂ wε
∂x
≤ C4; (15)
∂ wε
∂ψ
≤ C5, 0 ≤ ψ ≤ ψ̃ (16)
hold true for some ψ̃ ≥ 0.
Lemma 3.11. In domain Gε , the estimate∣∣∣∣wβ−1ε ∂ wε∂x
∣∣∣∣ ≤ C6 (17)
takes place, where C6 does not depend on ε, 0 < β < 1/2.
812 R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814Fig. 2. Separation phenomenon.
4. Auxilliary propositions
Consider the domain
ω = {0 < x < X, ψ1 < ψ < ψ2}
and denote by γ the part of its boundary
γ = {x,ψ : 0 < x < X,ψ = ψ1}∪{x,ψ : 0 < x < X,ψ = ψ2}∪
∪{x,ψ : x = 0,ψ1 ≤ ψ ≤ ψ2}.
Let us consider a differential operator
A(w) ≡ a(w, ∂ w
∂ψ
)∂2 w
∂ψ2
− ∂ w
∂x
+ b(x,ψ)∂ w
∂ψ
+ c(x,ψ)w, (18)
where a(w, p) ≥ 0.
Lemma 4.1. Assume that the functions a(w, p), b(x, ψ), c(x, ψ) are bounded as (x, ψ) ∈ ω and −∞ < p < ∞; w(x, ψ) is continuous 
in ω and has in ω continuous derivatives appearing in (18). If A(w) ≤ 0 in ω \ γ and w ≥ 0 on γ , then w ≥ 0 in ω. If A(w) ≥ 0 in 
ω \ γ and w ≤ 0 on γ , then w ≤ 0 in ω.
Lemma 4.2. Suppose that the conditions of Lemma 4.1 are fulfilled and moreover that a(w, p) has bounded partial derivatives on w
and p in ω. Suppose also that (x, ψ) is a continuous function in ω, which has in ω \ γ continuous partial derivatives appearing in 
(18). If A(w) − (w) ≤ 0 in ω \ γ and w ≥  on γ , then w ≥  in ω. If A(w) − (w) ≥ 0 in ω \ γ and w ≤  on γ , then w ≤ 
in ω.
The proof of these Lemmas can be found in [3, Ch. 2, §2.1].
5. On the separation of the boundary layer
The boundary layer with large Reynolds numbers exhibits a phenomenon called separation of boundary layer. The mean-
ing of this phenomenon is that a backward flow appearing for some movement of viscous medium separates (detach) the 
boundary layer from the streamline surface and takes it in the outer flow (see Fig. 2).
Definition 5.1. The separation point of a boundary layer is a point x0 such that u y(x, 0) > 0 for 0 < x < x0, and u y(x0, 0) = 0. 
Then x0 is the supremum of x > 0 such that, in domain D , the problem (1), (2) has a solution u(x, y), v(x, y), for which 
u y(x, 0) > 0.
In the point x0 as y = 0, the derivative of the velocity on y equals to zero. Hence an inflection point appears, and the 
stability of the flow is violated. Then, it appears a domain with negative derivative on y, i.e. the back flow inducing vorticity.
It is possible to avoid this phenomenon imposing a magnetic field perpendicular to the flow. Under this influence, the 
separation point moves to the right (see Fig. 3).
R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814 813Fig. 3. Influence of a magnetic field on the separation of boundary layer.
Theorem 3.1 says that, for B2 + dU
dx
≥ α, where α is a positive constant, there is no separation. The following statement 
is valid.
Theorem 5.1. If there exists a solution to the problem (1), (2) in domain D = {0 < x < X, 0 < y < ∞}, then X < x0 , where x0 is the 
separation point that can be found from the conditions
max
y
u20(y) −
x0∫
0
(
− 2U (x)dU
dx
− 2B2(x)U (x)
)
dx = 0 and dU (x0)
dx
< 0.
Proof. Consider the problem (3), (4) under the condition v0(x) ≡ 0. Suppose that dU
dx
< 0, B2 <
∣∣∣dU
dx
∣∣∣.
Let us denote p0 := max
ψ
w0(ψ) and define
w̃(x) = p0 −
x∫
0
(
− 2U (x)dU
dx
− 2B2(x)U (x)
)
dx.
Define the operator
L(V ) := ν√w
(
1 + 3
4
k
(∂ w
∂ψ
)2)∂2 V
∂ψ2
− ∂V
∂x
− 2B2√V .
Assume that, in the domain G = {0 < x < X, 0 < ψ < ∞}, there exists a solution w(x, ψ) to the problem (3), (4) and 
there exists x0 such that w̃(x0, ψ) = 0, w̃(x, ψ) > 0 as x < x0, dU
dx
< 0.
Consider the difference L(w̃) ≡ −2 U dU
dx
− 2B2U − 2B2√w̃ and L(w) ≡ −2 U dU
dx
− 2B2U . We have
L(w̃) − L(w) = −2B2√w̃ ≤ 0 as w̃ ≥ 0, 0 < x ≤ x0.
Since
max
ψ
w0(ψ) + 2
x∫
0
(
U
dU
dx
+ B2U
)
dx > 0 = w(x,0)
and w(0, ψ) ≤ max
ψ
w0(ψ) ≤ w̃(0, ψ), then by the maximum principle (see Lemma 4.2), we get the inequality w(x, ψ) ≤
w̃(x, ψ) for x  x0.
Hence, if w̃(x0) = 0 and ∂ w(x0,ψ)
∂x
=
x∫
0
2
(
U
dU
dx
+ B2U
)
dx < 0, then there is no positive solution w(x, ψ) as x > x0. It 
means that X < x0, i.e. for 0 < x < X there is no separation of the boundary layer. Theorem 5.1 is proved. 
814 R.R. Bulatova et al. / C. R. Mecanique 346 (2018) 807–814Remark 1. Due to Theorem 5.1 there is no separation of the boundary layer provided that:
B2(x) >
∣∣∣dU
dx
∣∣∣.
References
[1] O.A. Oleinik, The Prandtl system of equation in boundary layer theory, Sov. Math. Dokl. 4 (1963) 583–586.
[2] O.A. Olejnik, On a system of equations in boundary layer theory, USSR Comput. Math. Math. Phys. 3 (1963) 650–673.
[3] O.A. Oleinik, V.N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman & Hall, CRC, Boca Raton, FL, USA, 1999.
[4] Z.P. Shul’man, B.M. Berkovskii, Boundary Layer of Non-Newtonian Fluids, Nauka Tekhnika, Minsk, 1966, in Russian.
[5] A.Yu. Linkevich, T.S. Ratiu, S.V. Spiridonov, G.A. Chechkin, Thin layer of a non-Newtonian fluid flowing on a rough surface and percolating through a 
perforated obstacle, J. Math. Sci. (N.Y.) 189 (3) (2013) 525–535.
[6] W.L. Wilkinson, Non-Newtonian Fluids, Pergamon Press, London, 1960.
[7] M. Reiner, Lectures on Theoretical Rheology, North-Holland, Amsterdam, 1960.
[8] G.A. Chechkin, T.S. Ratiu, M.S. Romanov, V.N. Samokhin, Existence and uniqueness theorems for the full three-dimensional Ericksen–Leslie system, 
Math. Models Methods Appl. Sci. 27 (5) (2017) 807–843.
[9] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, etc., 1969.
[10] O.A. Ladyzhenskaya, On modifications of Navier–Stokes equations for large gradients of velocities, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 
7 (1968) 126–154, in Russian.
[11] V.N. Samokhin, G.M. Fadeeva, G.A. Chechkin, Equations of the boundary layer for a modified Navier–Stokes system, J. Math. Sci. (N.Y.) 179 (4) (2011) 
557–577.


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