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On Flows of Bingham-Type Fluids with Threshold Slippage.

1. Introduction

The statement that a fluid adheres to any solid boundary is one of the main tenets of classical fluid mechanics. However, careful experiments point to various possibilities for the behaviour of fluids at the interphase boundary. In particular, it is known that many non-Newtonian fluids slip over solid surfaces when the shear stresses reach a critical value. In order to describe slip effects, numerous mathematical models have been proposed (see, e.g., the short survey [1]).

In this article, we consider a model describing internal steady-state flows of a viscoplastic fluid of Bingham type [2, 3] in a bounded domain [OMEGA] [subset] [R.sup.3] with locally Lipschitz boundary r under a threshold-slip boundary condition [4] on a fixed subset [[GAMMA].sub.0] [subset] [GAMMA] and the no-slip condition on [GAMMA]\[[GAMMA].sub.0]:

[rho] div (u [cross product] u) - div [sigma] + [nabla][pi] = [rho]f in [OMEGA], (1)

div u = 0 in [OMEGA], (2)

[sigma] = [mu]([absolute value of (D (u))]) D (u) + g D (u)/[absolute value of (D (u))] if [absolute value of (D (u))] [not equal to] 0 in [OMEGA], (3)

[absolute value of ([sigma])] [less than or equal to] g if [absolute value of (D (u))] = 0 in [OMEGA], (4)

u x n = 0 on [GAMMA], (5)

[absolute value of ([([sigma]n).sub.tan])] [less than or equal to] [omega] on [[GAMMA].sub.0], (6)

[absolute value of ([([sigma]n).sub.tan])]< [omega] [??] [u.sub.tan] = 0 on [[GAMMA].sub.0], (7)

[absolute value of ([([sigma]n).sub.tan])] = [omega] [??] [u.sub.tan] [up arrow] [down arrow] [([sigma]n).sub.tan] on [[GAMMA].sub.0], (8)

u = 0 on [GAMMA]\[[GAMMA].sub.0]. (9)

Here u is the velocity, [pi] is the pressure, [sigma] is the deviatoric stress tensor, i is an external body force, D = D(u) is the strain velocity tensor,

[D.sub.ij] = [D.sub.ij] (u) = 1/2 ([partial derivative][u.sub.i]/[partial derivative][x.sub.j] + [partial derivative][u.sub.j]/[partial derivative][x.sub.i]), (10)

[mu]([absolute value of (D)]) > 0 is the viscosity, [rho] is the constant density of the fluid, g denotes the yield stress, g : [OMEGA] [right arrow] [R.sub.+], and [omega] is a critical value for start to slip along the boundary, [omega] : [[GAMMA].sub.0] [subset] [GAMMA] [right arrow] [R.sub.+]. For the sake of simplicity, we put in the sequel [rho] = 1.

The unknowns in systems (1)-(9) are the vector functions u, [sigma] and the function [pi], while all other quantities are assumed to be given.

Let us explain the tensor notation that we use in this article. Given a tensor F, the vector div F is defined by the formula

[(div F).sub.i] = [3.summation over (j=1)] [partial derivative][F.sub.ij]/[partial derivative][x.sub.j]. (11)

Given vectors x and y, the tensor x [cross product] y is the tensor product defined by

[(x [cross product] y).sub.ij] = [x.sub.i][y.sub.j]. (12)

We denote by [absolute value of (v)] the Euclidean norm of a vector v and by [absolute value of (E)] the Frobenius norm of a tensor E:

[mathematical expression not reproducible] (13)

As usual, n denotes the unit outer normal to [GAMMA] and [(x).sub.tan] stands for the tangential component of a vector; that is,

[u.sub.tan] = u - (u x n) n. (14)

The symbol [up arrow] [down arrow] is used to denote oppositely directed vectors.

Remark 1. Obviously, for g [equivalent to] 0 and [mu] [equivalent to] const, we recover the Navier-Stokes system with stick-slip boundary conditions. Such slip problem was studied in [4] (see also [5]). Note that system (6)-(8) is a special case of the following slip boundary condition [1]:

[mathematical expression not reproducible] (15)

where T = -[pi]I + [sigma] is the stress tensor, [(Tn).sub.n] = ((Tn) x n)n, and [psi] : [[GAMMA].sub.0] x [R.sub.+] [right arrow] [R.sub.+] is a given function. Actually, if [psi](x, s) = [omega](x) for any (x, s) [member of] [[GAMMA].sub.0] x [R.sub.+], then it is easily shown that system (6)-(8) is equivalent to (15).

The mathematical models of Bingham-type fluids are used to study the behaviour of materials such as paints, pastes, foams, suspensions, cements, and oils. Starting with the pioneering works by Mosolov and Miasnikov [6] and Duvaut and Lions [7], a large number of mathematicians have worked on the theoretical analysis of Bingham fluids and other similar viscoplastic media (see [8-23] and the references therein).

The novelty of the present paper is that it combines the use ofthe Bingham constitutive equations with threshold-slip boundary conditions and takes into account the dependence of the viscosity on the second invariant of the strain velocity tensor. It should be mentioned at this point that a nonlocal (regularized) friction problem for a class of non-Newtonian fluids has been investigated by Consiglieri [24] (see also [25]).

Let us state the main results of this paper. Following an approach adopted in [4,7], we formulate the boundary-value problem (1)-(9) as a variational inequality for the unknown velocity field. Using some existence results for inequalities with pseudomonotone operators and convex functionals, which naturally arise in this slip problem, and the Krasnoselskii theorem on continuity of the Nemytskii operator [26], we establish sufficient conditions for the existence of weak solutions and derive their energy estimates. Also, it is shown that the set of weak solutions to problem (1)-(9) is sequentially weakly closed in a suitable functional space.

2. Preliminaries

In this section, we describe the necessary functional spaces and the main assumptions used in the paper.

We shall use the classical notation [L.sup.p] ([OMEGA]) and [H.sup.s] ([OMEGA]) = [W.sup.s.sub.2] ([OMEGA]) for the Lebesgue and Sobolev spaces, respectively. Bold face letters will denote functional spaces of vectors or tensors: [L.sup.p] ([OMEGA]) = [L.sup.p] [([OMEGA]).sup.3], [H.sup.s] ([OMEGA]) = [H.sup.s] [([OMEGA]).sup.3], and so forth.

Next, we set

[mathematical expression not reproducible] (16)

We now recall an inequality of Korn's type.

Proposition 2. Let a : [H.sup.1]([OMEGA]) x [H.sup.1] ([OMEGA]) [right arrow] R be a continuous symmetric bilinear form such that a(v, v) [greater than or equal to] 0, for any v [member of] [H.sup.1] ([OMEGA]), and it follows from the conditions

[[integral].sub.[OMEGA]] [absolute value of (D (w))].sup.2] dx = 0, a (w, w) = 0, w [member of] [H.sup.1] ([OMEGA]) (17)

that w = 0. Then there exists a positive constant C such that

[mathematical expression not reproducible] (18)

for all v [member of] [H.sup.1] ([OMEGA]).

The proof of this proposition is given in [27].

Suppose that the 2-dimensional Lebesgue measure of the set [GAMMA]\[[GAMMA].sub.0] is positive, then we can define the scalar product in X([OMEGA]) by the formula

[(v, u).sub.X([OMEGA]) = [[integral].sub.[OMEGA]] D (v) : D (u) dx, (19)

where D(v) : D(u) denotes the scalar product of tensors D(v) and D(u):

D (v) : D (u) = trace (D (v) D [(u).sup.T]). (20)

Setting

[mathematical expression not reproducible] (21)

and applying Proposition 2, we infer that the norm

[[parallel]v[parallel].sub.X([OMEGA])] = [(v, v).sup.1/2.sub.X([OMEGA])] (22)

is equivalent to the norm induced from the Sobolev space [H.sup.1] ([OMEGA]).

Recall that the restriction of a function w [member of] [H.sup.1] ([OMEGA]) to [GAMMA] is defined by the formula w[|.sub.[GAMMA]] = [[gamma].sub.0]w, where [[gama].sub.0] : [H.sup.1]([OMEGA]) [right arrow] [H.sup.1/2]([GAMMA]) is the trace operator (see [7]).

By [M.sup.3x3.sub.sym] denote the space symmetric matrices of size 3 x 3.

In the sequel, we assume that the following conditions hold:

(i) for any matrices A, B [member of] [M.sup.3x3.sub.sym], we have

([mu] ([absolute value of (A)]) A - [mu]([absolute value of (B)]) B) : (A - B) [greater than or equal to] 0; (23)

(ii) the function [mu] is continuous and

0 < [[mu].sub.0] < [mu] (s) < [[mu].sub.1], [for all]s [member of] [R.sub.+]; (24)

(iii) g [member of] [L.sup.2.sub.+] ([OMEGA]), [omega] [member of] [L.sup.2.sub.+]([[GAMMA].sub.0]), and f [member of] [L.sup.2]([OMEGA]);

(iv) the 2-dimensional Lebesgue measure of the set [GAMMA]\[[GAMMA].sub.0] is positive.

Remark 3. We claim that condition (i) holds true if the function [mu] is monotonically increasing. Indeed, using the Cauchy-Schwarz inequality, we obtain

[mathematical expression not reproducible] (25)

for any A, B [member of] [M.sup.3x3.sub.sym].

3. Weak Formulation of Problem (1)-(9)

Definition 4. One shall say that a vector function u : [bar.[OMEGA]] [right arrow] [R.sup.3] is a weak solution to problem (1)-(9) if u [member of] X([OMEGA]) and the following inequality holds:

[mathematical expression not reproducible] (26)

for any vector function v [member of] X([OMEGA]).

Remark 5. Let us explain how variational inequality (26) arises in the definition of weak solutions. Assume that regular functions u, [sigma], [pi] satisfy relations (1)-(9) and v [member of] X([OMEGA]). If we take the scalar product of both sides of (1) by v - u and integrate by parts over the domain [OMEGA], we get

[mathematical expression not reproducible] (27)

where we used the equalities

[[integral].sub.[OMEGA]] (u [cross product] u) : D (u) dx = 0, (u [cross product] u) n[|.sub.[GAMMA]] = 0. (28)

Let us show that under conditions (3) and (4) the following inequality

[mathematical expression not reproducible] (29)

holds true. We set

[[OMEGA].sub.+] := {x [member of] [OMEGA]: [absolute value of (D (u) (x))] > 0},

[[OMEGA].sub.0] := {x [member of] [OMEGA]: [absolute value of (D (u) (x))] = 0}. (30)

Using (3) and the Cauchy-Schwarz inequality, we obtain

[mathematical expression not reproducible] (31)

Besides, taking into account (4), we arrive at the inequality

[mathematical expression not reproducible] (32)

By adding this inequality to (31), we obtain relation (29).

Note also that the system of conditions (6)-(8) is equivalent to the following system:

[absolute value of ([([sigma]n).sub.tan])] [less than or equal to] [omega] on [[GAMMA].sub.0],

[([sigma]n).sub.tan] x [u.sub.[tan] + [omega] [absolute value of ([u.sub.tan])] = 0 on [[GAMMA].sub.0]. (33)

Using these relations, we obtain

[mathematical expression not reproducible] (34)

Finally, combining (27) with (29) and (34), we arrive at inequality (26).

4. Main Results

Our main results are collected in the following theorem.

Theorem 6. Suppose that conditions (i)-(iv) hold. Then

(a) problem (1)-(9) has at least one weak solution;

(b) any weak solution u satisfies the energy equality

[mathematical expression not reproducible] (35)

(c) the set of weak solutions to problem (1)-(9) is sequentially weakly closed in the space X([OMEGA]).

5. Proof of Theorem 6

The proof uses the following two propositions.

Proposition 7 (see [28]). Let V be a reflexive Banach space, [V.sup.*] its the dual space, A : V [right arrow] [V.sup.*] a pseudomonotone operator, and J : V [right arrow] R a lower semicontinuous convex functional. In addition, suppose that

<A (v), v> + J (v)/[[parallel]v[parallel].sub.V] [right arrow] +[infinity] (36)

as [[parallel]v[parallel].sub.V] [right arrow] +[infinity]. Then, for an arbitrary z [member of] [V.sup.*], there exists an element [u.sub.z] [member of] V such that

<A ([u.sub.z]) - z, v - [u.sub.z]> + J (v) - J ([u.sub.z]) [greater than or equal to] 0 [for all]v [member of] V. (37)

Proposition 8 (Krasnoselskii's theorem, see [26]). Let h : [OMEGA]x [R.sup.m] [right arrow] R be a function such that

(a) the function h(x, y) : [OMEGA] [right arrow] R is measurable for every y [member of] [R.sup.m];

(b) the function h(x, x) : [R.sup.m] [right arrow] R is continuous for almost every x [member of] [OMEGA];

(c) for every y [member of] [R.sup.m] and for almost every x [member of] [OMEGA]

[mathematical expression not reproducible] (38)

where [p.sub.k], q [greater than or equal to] 1, [alpha] [member of] [L.sup.q] ([OMEGA]), and C is a positive constant.

Then the Nemytskii operator defined by

[mathematical expression not reproducible] (39)

is a bounded and continuous map.

Proof of Theorem 6. Let us introduce here the following operators:

[mathematical expression not reproducible] (40)

Using these operators, we can rewrite inequality (26) as follows:

<[A.sub.[mu]] (u) + [K.sub.f] (u), v - u> + [J.sub.g,[omega]] (v) - [J.sub.g,[omega]] (u) [greater than or equal to] 0 [for all]v [member of] X ([OMEGA]). (41)

By condition (i), we deduce that

<[A.sub.[mu]] (u) - [A.sub.[mu]] (v), u - v> [greater than or equal to] 0 [for all]u, v [member of] X ([OMEGA]); (42)

that is, the operator [A.sub.[mu]] is monotone. Moreover, applying Proposition 8 and condition (ii), we establish that this operator is continuous. From properties of monotone operators it follows that [A.sub.[mu]] is a pseudomonotone operator.

The embedding [H.sup.1] ([OMEGA]) [??] [L.sup.4] ([OMEGA]) is compact (see, e.g., [29]). This implies that the embedding X([OMEGA]) [??] [L.sup.4] ([OMEGA]) is compact too. Therefore, it is easily shown that the operator [K.sub.f] is completely continuous; that is, if [u.sub.n] [??] [u.sub.0] weakly in the space X([OMEGA]) as n [right arrow] [infinity], then [K.sub.f] ([u.sub.n]) [right arrow] [K.sub.f]([u.sub.0]) strongly in the space [[X([OMEGA])].sup.*] as n [right arrow] [infinity]. This yields that the sum [A.sub.[mu]] + [K.sub.f] is a pseudomonotone operator.

Further, taking into account condition (ii) and the equality

[[integral].sub.[OMEGA]] (u [cross product] u) : D (u)dx = 0, (43)

we obtain

[mathematical expression not reproducible] (44)

as [[parallel]u[parallel].sub.X([OMEGA])] [right arrow] +[infinity].

Then from Proposition 7 we infer that inequality (41) has a solution [u.sub.*] [member of] X([OMEGA]). It is clear that [u.sub.*] is a weak solution to problem (1)-(9).

We claim that energy equality (35) holds true for any weak solution u of problem (1)-(9). Indeed, by setting v = 2u in (26), we find

[mathematical expression not reproducible] (45)

On the other hand, the choice v = 0 in (26) yields that

[mathematical expression not reproducible] (46)

Obviously, if we combine the last inequality with (45), we get (35).

Now we must only prove that the set of weak solutions to problem (1)-(9) is sequentially weakly closed in the space X([OMEGA]). Consider a sequence [{[u.sub.n]}.sup.[infinity].sub.n=1] such that, for any n [member of] N, [u.sub.n] is a weak solution of (1)-(9) and [u.sub.n] [right arrow] [u.sub.0] weakly in X([OMEGA]) as n [right arrow] [infinity]. Let us show that u0 is a weak solution of (1)-(9).

By definition of weak solutions, we have

[mathematical expression not reproducible] (47)

Note that the functional [J.sub.g,[omega]] : X([OMEGA]) [right arrow] R is convex and continuous. Therefore, [J.sub.g,[omega]] is lower semicontinuous with respect to the weak convergence in X([OMEGA]). This implies that

[mathematical expression not reproducible] (48)

Further, we set v = [u.sub.0] in (47) and pass to the lower limit as n [right arrow] [infinity]. Taking into account inequality (48) and the complete continuity of the operator [K.sub.f], we obtain

[mathematical expression not reproducible] (49)

or equivalently,

[mathematical expression not reproducible] (50)

Since [A.sub.[mu]] is a pseudomonotone operator, it follows from the last inequality that

[mathematical expression not reproducible] (51)

Now, we rewrite (47) in the form

[J.sub.g,[omega]] ([u.sub.n]) - [J.sub.g,[omega]] (v) [less than or equal to] - <[A.sub.[mu]] ([u.sub.n]) + [K.sub.f] ([u.sub.n]), [u.sub.n] - v> [for all]v [member of] X ([OMEGA]), n [member of] N (52)

and pass to the upper limit in this inequality:

[mathematical expression not reproducible] (53)

Using (48) and (51), we deduce from (53) that

[mathematical expression not reproducible] (54)

Thus, we have

<[A.sub.[mu]] ([u.sub.0]) + [K.sub.f] ([u.sub.0]), v - [u.sub.o]> + [J.sub.g,[omega]] (v) - [J.sub.g,[omega]] ([u.sub.0]) [greater than or equal to] 0 [for all]v [member of] X ([OMEGA]). (55)

This means that [u.sub.0] is a weak solution of problem (1)-(9).

Theorem 6 is completely proved.

https://doi.org/10.1155/2017/7548328

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, Project no. 16-31-00182 moLa.

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Evgenii S. Baranovskii

Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, Voronezh, Russia

Correspondence should be addressed to Evgenii S. Baranovskii; esbaranovskii@gmail.com

Received 20 September 2017; Accepted 26 November 2017; Published 17 December 2017

Academic Editor: Luigi C. Berselli
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