On Flows of Bingham-Type Fluids with Threshold Slippage

Abstract

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.

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 with locally Lipschitz boundary under a threshold-slip boundary condition [4] on a fixed subset and the no-slip condition on :Here is the velocity, is the pressure, is the deviatoric stress tensor, is an external body force, is the strain velocity tensor, is the viscosity, is the constant density of the fluid, denotes the yield stress, , and is a critical value for start to slip along the boundary, . For the sake of simplicity, we put in the sequel .

The unknowns in systems (1)–(9) are the vector functions , 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 , the vector is defined by the formulaGiven vectors and , the tensor is the tensor product defined by We denote by the Euclidean norm of a vector and by the Frobenius norm of a tensor :

As usual, denotes the unit outer normal to and stands for the tangential component of a vector; that is, The symbol is used to denote oppositely directed vectors.

Remark 1. Obviously, for and , 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]:where is the stress tensor, , and is a given function. Actually, if for any , 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 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.

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 and for the Lebesgue and Sobolev spaces, respectively. Bold face letters will denote functional spaces of vectors or tensors: , , and so forth.

Next, we set

We now recall an inequality of Korn’s type.

Proposition 2. Let be a continuous symmetric bilinear form such that , for any , and it follows from the conditions that . Then there exists a positive constant such that for all .

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

Suppose that the 2-dimensional Lebesgue measure of the set is positive, then we can define the scalar product in by the formula where denotes the scalar product of tensors and :

Setting and applying Proposition 2, we infer that the norm is equivalent to the norm induced from the Sobolev space .

Recall that the restriction of a function to is defined by the formula , where is the trace operator (see [7]).

By denote the space symmetric matrices of size .

In the sequel, we assume that the following conditions hold:(i)for any matrices , we have (ii)the function is continuous and (iii), , and ;(iv)the 2-dimensional Lebesgue measure of the set is positive.

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 .

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

Definition 4. One shall say that a vector function is a weak solution to problem (1)–(9) if and the following inequality holds:for any vector function .

Remark 5. Let us explain how variational inequality (26) arises in the definition of weak solutions. Assume that regular functions , , satisfy relations (1)–(9) and . If we take the scalar product of both sides of (1) by and integrate by parts over the domain , 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 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).

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 satisfies the energy equality(c)the set of weak solutions to problem (1)–(9) is sequentially weakly closed in the space .

5. Proof of Theorem 6

The proof uses the following two propositions.

Proposition 7 (see [28]). Let be a reflexive Banach space, its the dual space, a pseudomonotone operator, and a lower semicontinuous convex functional. In addition, suppose that as . Then, for an arbitrary , there exists an element such that

Proposition 8 (Krasnoselskii’s theorem, see [26]). Let be a function such that(a)the function is measurable for every ;(b)the function is continuous for almost every ;(c)for every and for almost every  where , , , 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 is monotone. Moreover, applying Proposition 8 and condition (ii), we establish that this operator is continuous. From properties of monotone operators it follows that is a pseudomonotone operator.
The embedding is compact (see, e.g., [29]). This implies that the embedding is compact too. Therefore, it is easily shown that the operator is completely continuous; that is, if weakly in the space as , then strongly in the space as . This yields that the sum is a pseudomonotone operator.
Further, taking into account condition (ii) and the equality we obtain as .
Then from Proposition 7 we infer that inequality (41) has a solution . It is clear that is a weak solution to problem (1)–(9).
We claim that energy equality (35) holds true for any weak solution of problem (1)–(9). Indeed, by setting in (26), we findOn the other hand, the choice 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 . Consider a sequence such that, for any , is a weak solution of (1)–(9) and weakly in as . Let us show that is a weak solution of (1)–(9).
By definition of weak solutions, we haveNote that the functional is convex and continuous. Therefore, is lower semicontinuous with respect to the weak convergence in . This implies thatFurther, we set in (47) and pass to the lower limit as . Taking into account inequality (48) and the complete continuity of the operator , we obtain or equivalently, Since is a pseudomonotone operator, it follows from the last inequality thatNow, we rewrite (47) in the form and pass to the upper limit in this inequality:Using (48) and (51), we deduce from (53) that Thus, we have This means that is a weak solution of problem (1)–(9).
Theorem 6 is completely proved.

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 mol_a.