Steady state non-Newtonian flow with strain rate dependent viscosity in domains with cylindrical outlets to infinity

Abstract. The paper deals with a stationary non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets to infinity. The viscosity is assumed to be smoothly dependent on the gradient of the velocity. Applying the generalized Banach fixed point theorem, we prove the existence, uniqueness and high order regularity of solutions stabilizing in the outlets to the prescribed quasi-Poiseuille flows. Varying the limit quasi-Poiseuille flows, we prove the stability of the solution.


Introduction
Asymptotic behaviour of solutions of elliptic and parabolic equations in domains with noncompact boundaries was considered in [12], where the first theorems on stabilization of solutions were proved. They were called Phrägmen-Lindelöf theorems. The stationary elasticity equations in unbounded domains are studied in [15], and the stabilization theorems were associated there with the Saint-Venant principle. For the stationary and nonstationary Stokes and Navier-Stokes equations with no-slip condition at the boundary of the outlets, these questions were studied in [1, 9-11, 21-24, 28], and for the viscoelastic flows, in [25]. For the non-Newtonian flows with viscosity depending on the gradient of the velocity, the existence, uniqueness and asymptotic behaviour in the outlets were studied in [13,20]. Note that this non-Newtonian rheology governs the blood circulation in vessels (see [2, pp. 84-89, 196-200]).
A part of theoretical interest for partial differential equations, this set of questions is important for construction of asymptotic expansions of solutions in thin domains. Namely, matching of the asymptotic solutions via the boundary layer method leads exactly to the scaled partial differential equations in unbounded domains with cylindrical outlets (see, e.g., [16][17][18][19] for Newtonian flows and [14] for the power law fluids). In particular, results of the present paper are used for the construction of an asymptotic expansion of a non-Newtonian flow in a network of thin cylinders, modeling blood vessels.
In the present paper the results obtained in [20] will be extended and generalized. First, we reconstruct the pressure, while in [20], only the weak formulation of the problem without pressure was studied. Second, in order to reconstruct the pressure, we need to have more regularity for the solution, so we will prove the third-order regularity of the velocity and second-order regularity of the pressure in weighted spaces with exponential decay at infinity. Of course, we need more regularity (C 3 ) for the viscosity ν, depending on the shear rate y. However, we will rid of a restrictive condition of boundedness of ∇(ν(y)y), which was assumed in [20]. Finally, we will focus on the questions of stability of solutions with respect to the quasi-Poiseuille flows to which they stabilize in the outlets. These new theorems are important for the construction of boundary layers of non-Newtonian flows.
The paper has the following structure. In Section 2, we give the definition of the domain with outlets. In Section 3, we cite and prove some auxiliary results: embedding inequalities in domains with cylindrical outlets and a lemma on the stabilization to a constant for functions with exponentially decaying gradient. In the same Section 3, we recall some results for the stationary Stokes equation and prove the weak Banach contraction principle. This theorem generalizes the classical Banach fixed point theorem. This result is well known in the mathematical community and is widely used. However, we could not find the proof in literature. Therefore, for the reader's convenience, we present a proof. This generalization is used in the proofs of the regularity of the solutions. The main problem for the stationary non-Newtonian flow in unbounded domains with outlets is formulated in Section 4. In Section 5 the quasi-Poiseuille flow for the stationary non-Newtonian equations in an infinite tube is studied. A Poiseuille flow is an exact solution to the equations of the fluid motion (Stokes, Navier-Stokes) in an infinite cylinder with the no-slip condition at the boundary, with a linear pressure with respect to the longitudinal variable, and with the velocity vector having only longitudinal component (called normal velocity) different from zero; this normal velocity depends only on the transversal variables. A quasi-Poiseille (or Hägen-Poiseuille) flow is an exact solution having the same structure and corresponding to some non-Newtonian rheology. Such flows for various rheologies were studied in [2,6,7,26,27]. Contrary to [20], where also the quasi-Poiseuille flow was studied, we focus on the regularity issues. Finally, Section 6 contains the main results of the paper: existence and uniqueness of a regular classical solution (velocity and pressure) and continuity of the solution with respect to the data of problem (stability). The proof of continuity of the solution in the norm W 2,2 for the velocity and L 2 norm for the gradient of the pressure needs the regularity "plus one" of the solution. It explains the difference of norms in Theorems 5 and 6.

Definitions of domains
Consider the domain Ω ⊂ R n , n = 2, 3, with J cylindrical outlets to infinity: where Ω 0 is a bounded domain, Ω 0 ∩Ω j = ∅ for j ∈ {1, . . . , J}, Ω j ∩Ω l = ∅ for j = l, j, l ∈ {1, . . . , J}, and the outlets to infinity Ω j in some coordinate systems 1 , x (j) ), having the origins within the boundary of domain Ω 0 , are given by the relations where σ j are some bounded domains in R n−1 , cross-sections of the cylinders (see Fig. 1). Assume that for any k ∈ {1, . . . , J}, there exists a δ j > 0 such that the cylinder Denote d σ the maximal diameter of the cross-sections σ j . We assume that the boundary ∂Ω is C 3 -regular and that ∂Ω ∩ ∂Ω 0 = ∅ has a positive measure. Evidently, there exists a positive real number R > d σ such that the ball B R = {x ∈ R n : |x| < R} contains Ω 0 . We introduce the following notation: where j = 1, . . . , J, and k 0 is an integer.

Lemma 2.
(i) For any function u ∈ W 2,2 β β β (Ω), the inequality holds: (ii) For any function u ∈ W 2,2 β β β (Ω), the inequality holds: Proof. (i) Let us represent the domain Ω as a union of bounded domains: In every ω jk , we have the inequalities in ω jk , we obtain Here the constants depend on β only. Summing these inequalities over k and j and adding the inequality u 4 c u 4 W 1,2 (Ω0) , we obtain (1). (ii) By same token, using the inequalities Lemma 3. Let us define the half-cylinder Π + = {x = (z, x ) ∈ R n : x ∈ σ, z ∈ (0, +∞)}, where σ is a bounded domain in R n−1 with Lipschitz boundary. Suppose that p ∈ W 1,2 loc (Π + ) and Then there exists a constant p 0 such that the following estimate holds: Proof. First, we prove that the mean valuep(z) = σ p(z, x ) dx is a bounded function.
It is easy to prove that there exists a constant p 0 such that lim z→+∞p (z) = p 0 . Indeed, sincep(z) is bounded, there is a sequence {z k } such that lim k→+∞p (z k ) = p 0 for some constant p 0 < +∞. Consider Here we used that ξ ∈ [z, 2z], and so, ξ → +∞ implies z → +∞. By the triangle inequality we get lim z→+∞ σ |p(z, x ) − p 0 | 2 dx = 0. In order to finish the proof of the lemma, we need an auxiliary inequality Remark 1. From the last lemma it follows that if Ω E β β β (x)|∇p(x)| 2 dx < +∞, then there exist constants p j , j = 1, 2, . . . , J, such that

Weak Banach contraction principle
Then T admits exactly one fixed point x * ∈ M : T x * = x * .
Proof. Let us define a sequence {x n } by the recurrent formulas Since T maps the bounded set M to itself, there exists a positive constant c 0 such that x n X c 0 and T x n X c 0 . Since the space X is reflexive, there exists a subsequence {x n k } such that For simplicity, we will not distinguish in notation the subsequence {x n k } and the sequence {x n }. From (7) it follows that Therefore, {T x n } is strongly convergent in Y and T x n Y → y * . From (8) we obtain Relations (10) and (11) yield T x * = x * . The uniqueness of the fixed point is obvious.

Formulation of the problem
Let n = 2, 3, ν 0 , λ be positive constants. Let ν be a bounded C 3 -smooth function where A is a positive constant independent of y.
Consider the steady state boundary value problem for the non-Newtonian fluid motion equations in the domain Ω where D(v) is the strain rate matrix with the elements We look for the solution v having prescribed fluxes F j over the cross sections σ j of outlets to infinity: where Here and below an integral over σ j is understood as an integral over any orthogonal crosssection of Ω j . Note that this integral for a divergence-free vector function is independent of the position of this cross-section.
Since div v = 0, equations (13) can be written in the form The non-Newtonian Poiseuille flow with the strain rate dependent viscosity was studied in the book [2] and recently in [20]. We will need below some extended versions of theorems proved there.
Let us recall the definition of a quasi-Poiseuille flow for equations (13). Let σ be a bounded domain with Lipschitz boundary in R n−1 . Consider in the infinite cylinder Π = R × σ the Dirichlet boundary value problem where v Pα is the solution of the following problem: and α is the given pressure slope.
Then for any λ ∈ (0, λ 0 ), operator L is a contraction inW 1,2 (σ) with the contraction factor and by Theorem 2 there exists a unique fixed point v Pα of the operator L, which is a solution of problem (16).
From estimates (20), (21) applied to the fixed point v Pα it follows that

Operator relating the pressure slope and the flux
Define F (α) = σ v Pα (x ) dx the flux corresponding to the pressure slope −α. Note that in the case of the steady Newtonian flow (the steady form of Navier-Stokes or Stokes equations), F (α) is proportional to α. This case corresponds to the value λ = 0, and so, We consider as well the operator (function) corrector of the non-Newtonian flux with respect to the Newtonian one: G(α) = F (α) − κα, and prove that for sufficiently small The next lemma is an extension of Lemma 2.3 and Corollary 2.4 [20].

Continuity of the non-Newtonian Poiseuille flow
Theorem 4.

Therefore, the classical estimate for the Poisson equation (29) yields
Thus, inequality (27) is proved.
6 The non-Newtonian flow equations in domain with cylindrical outlets to infinity 6

.1 Existence and uniqueness of a solution
Consider the domain Ω ⊂ R n with J cylindrical outlets to infinity. We assume that the boundary ∂Ω is C 4 -regular. Consider in Ω problem (13)- (15). Denote F = J j=1 F 2 j . Let F 0 be a nonnegative number. By Lemma 7 there exists a number λ 00 depending on F 0 such that for every λ ∈ (0, λ 00 ) and for any set of fluxes (F 1 , . . . , F J ) such that F F 0 , there exist J pressure slopes α j and corresponding J quasi-Poiseuille flows 1 ∈ R}, j = 1, . . . , J, such that F (α j ) = F j . We define cut-off functions χ j associated to each outlet Ω j as C (3) -smooth functions vanishing everywhere in Ω except for the outlet Ω j , where they depend on the local longitudinal variable x
Extend the functions W and V χ by zero into the whole Ω and set and for x ∈ Ω j \Ω j3 , the vector-field V χ (x) coincides with the velocity part V Pα j (x (j) ) of the corresponding Poiseuille flow. Note that the vector field W has zero flux. By denoting in (13) where P χ = J j=1 χ j α j x j 1 , we obtain the following problem: The following estimate holds: (37) Moreover, there exist constants q 1 , q 2 , . . . , q J such that (38) where After subtracting and adding the expression λ div(ν(γ( V χ ))D( V χ )), we write H in the form The function g has compact support, supp g ⊂ Ω (3) , and Note that where ∇γ is the Jacobian matrix ofγ, and by using (12) we obtain the estimate Using the embedding inequalities (1), (2) and estimates (26), (33), we obtain Let us estimate the integrals containing the terms |∇U| 2 |∇ V χ | 2 | ∇ 2 V χ | 2 and |∇ 2 V χ | 2 |∇U| 2 . Inequalities (1), (2) and (3) yield β β β (Ω) . Similar considerations give us also the estimates β β β (Ω) . Collecting the above inequalities and adding (40), we derive Similarly, https://www.journals.vu.lt/nonlinear-analysis The L 2 β β β -norm of this expression is evaluated according to the following scheme: in each product of gradients, the first-order terms |∇U| and |∇ V χ | are evaluated by sup x∈Ω |∇U(x)| and sup x∈Ω |∇ V χ (x)|, the second-order terms |∇ 2 U| and |∇ 2 V χ | are evaluated in the L 4 β β β -norm, finally, the third-order terms |∇ 3 U| and |∇ 3 V χ | are evaluated in the L 2 β β β -norm. Then we apply the embedding inequalities of Lemma 2. So, for the gradient of H, we obtain the estimate From (41) and (42) it follows that the right-hand side R = f + H(U + V χ ) of system (39) satisfies the estimate Then, by Theorem 1, for sufficiently small β > 0, the solution (KU, q) of the Stokes problem (39) is subject to Assume that U 2 Then from (43) it follows and if λ satisfies we obtain the estimate Thus, by (44), the operator K maps the ball Let us show that K is a contraction in the space H(Ω). Multiplying equations (39) by arbitrary η η η ∈ H(Ω) and integrating by parts, we get From (45) it follows that for any U 1 , U 2 ∈ B M , the following equality holds: Applying Young's inequality, we have Since, by (12), we get Taking in (46) η η η = KU 1 − KU 2 , we derive the inequality Therefore, Let .
Then for any λ ∈ (0, Λ 0 ), the operator K is a contraction with the contraction factor and, by Theorem 2, there exists a unique fixed point U of the operator K, which is a solution (together with the corresponding pressure function q) of problem (36). Estimate (37) for the fixed point u and the pressure q follows from the fact that u ∈ B M (see inequality (44)). The existence of the constants q 1 , . . . , q J and estimate (38) follows from Lemma 3 and Remark 1.