On Flows of Bingham-Type Fluids with Threshold Slippage

We investigate a mathematical model describing 3D steady-state flows of Bingham-type fluids in a bounded domain under threshold-slip boundary conditions, which state that flows can slip over solid surfaces when the shear stresses reach a certain critical value. Using a variational inequalities approach, we suggest the weak formulation to this problem. We establish sufficient conditions for the existence of weak solutions and provide their energy estimates.Moreover, it is shown that the set of weak solutions is sequentially weakly closed in a suitable functional space.


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 Ω ⊂ R 3 with locally Lipschitz boundary Γ under a threshold-slip boundary condition [4] on a fixed subset Γ 0 ⊂ Γ and the no-slip condition on Γ\Γ 0 : u ⋅ n = 0 on Γ, (5)     (n) tan     ≤  on Γ 0 , (n) tan     <  ⇒ u tan = 0 on Γ 0 , (n) tan     =  ⇒ u tan ↑↓ (n) tan on Γ 0 , Here u is the velocity,  is the pressure,  is the deviatoric stress tensor, f is an external body force, D = D(u) is the strain velocity tensor, (|D|) > 0 is the viscosity,  is the constant density of the fluid,  denotes the yield stress,  : Ω → R + , and  is a critical value for start to slip along the boundary,  : For the sake of simplicity, we put in the sequel  = 1.
The unknowns in systems (1)-( 9) are the vector functions u,  and the function , 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

Advances in Mathematical Physics
Given vectors x and y, the tensor x ⊗ y is the tensor product defined by We denote by |k| the Euclidean norm of a vector k and by |E| the Frobenius norm of a tensor E: As usual, n denotes the unit outer normal to Γ and (⋅) tan stands for the tangential component of a vector; that is, The symbol ↑↓ is used to denote oppositely directed vectors.
The novelty of the present paper is that it combines the use of the 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.

Preliminaries
In this section, we describe the necessary functional spaces and the main assumptions used in the paper.
We shall use the classical notation   (Ω) and   (Ω) =   2 (Ω) for the Lebesgue and Sobolev spaces, respectively.Bold face letters will denote functional spaces of vectors or tensors: 3 , and so forth.
Next, we set fl the closure of the set Q (Ω) in the space H 1 (Ω) . ( We now recall an inequality of Korn's type.
, and it follows from the conditions that w = 0. Then there exists a positive constant  such that for all k ∈ H 1 (Ω).
The proof of this proposition is given in [27].Suppose that the 2-dimensional Lebesgue measure of the set Γ\Γ 0 is positive, then we can define the scalar product in X(Ω) by the formula where D(k) : D(u) denotes the scalar product of tensors D(k) and D(u): Setting and applying Proposition 2, we infer that the norm is equivalent to the norm induced from the Sobolev space H 1 (Ω).
By M 3×3 sym denote the space symmetric matrices of size 3 × 3.
Remark 3. We claim that condition (i) holds true if the function  is monotonically increasing.Indeed, using the Cauchy-Schwarz inequality, we obtain for any A, B ∈ M 3×3 sym .

Weak Formulation of Problem (1)-(9)
Definition 4. One shall say that a vector function u : Ω → R 3 is a weak solution to problem (1)-( 9) if u ∈ X(Ω) and the following inequality holds: for any vector function k ∈ X(Ω).
Remark 5. Let us explain how variational inequality (26) arises in the definition of weak solutions.Assume that regular functions u, ,  satisfy relations (1)-( 9) and k ∈ X(Ω).If we take the scalar product of both sides of (1) by k − u and integrate by parts over the domain Ω, we get where we used the equalities Let us show that under conditions ( 3) and ( 4) the following inequality holds true.We set Using (3) and the Cauchy-Schwarz inequality, we obtain

Advances in Mathematical Physics
Besides, taking into account (4), we arrive at the inequality 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: Using these relations, we obtain Finally, combining ( 27) with ( 29) and (34), we arrive at inequality (26).

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

Proof of Theorem 6
The proof uses the following two propositions.
Proposition 7 (see [28]).Let V be a reflexive Banach space, V * its the dual space, A : V → V * a pseudomonotone operator, and  : V → R a lower semicontinuous convex functional.In addition, suppose that as ‖k‖ V → +∞.Then, for an arbitrary z ∈ V * , there exists an element u z ∈ V such that Proposition 8 (Krasnoselskii's theorem, see [26]).Let ℎ : Ω× R  → R be a function such that (a) the function ℎ(⋅, y) : Ω → R is measurable for every y ∈ R  ; (b) the function ℎ(x, ⋅) : R  → R is continuous for almost every x ∈ Ω; (c) for every y ∈ R  and for almost every where   ,  ≥ 1,  ∈   (Ω), and  is a positive constant.
Then the Nemytskii operator defined by is a bounded and continuous map.
Proof of Theorem 6.Let us introduce here the following operators: Using these operators, we can rewrite inequality (26) as follows: By condition (i), we deduce that that is, the operator A  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  is a pseudomonotone operator.
The embedding H 1 (Ω) → L 4 (Ω) is compact (see, e.g., [29]).This implies that the embedding X(Ω) → L 4 (Ω) is compact too.Therefore, it is easily shown that the operator K  is completely continuous; that is, if u  ⇀ u 0 weakly in the space X(Ω) as  → ∞, then K  (u  ) → K  (u 0 ) strongly in the space [X(Ω)] * as  → ∞.This yields that the sum A  + K  is a pseudomonotone operator.
Further, taking into account condition (ii) and the equality we obtain as ‖u‖ X(Ω) → +∞.
Then from Proposition 7 we infer that inequality (41) has a solution u * ∈ X(Ω).It is clear that u * 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 k = 2u in (26), we find On the other hand, the choice k = 0 in (26) yields that 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(Ω).Consider a sequence {u  } ∞ =1 such that, for any  ∈ N, u  is a weak solution of (1)-( 9) and u  ⇀ u 0 weakly in X(Ω) as  → ∞.Let us show that u 0 is a weak solution of (1)- (9).
By definition of weak solutions, we have Note that the functional  , : X(Ω) → R is convex and continuous.Therefore,  , is lower semicontinuous with respect to the weak convergence in X(Ω).This implies that Further, we set k = u 0 in (47) and pass to the lower limit as  → ∞.Taking into account inequality (48) and the complete continuity of the operator K  , we obtain    ( This means that u 0 is a weak solution of problem (1)-( 9).Theorem 6 is completely proved.