Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0089 Acta Polytechnica 61(SI):89–98, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES Stanislav Kračmara, Jiří Neustupab,a,∗ a Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha, Czech Republic b Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha, Czech Republic ∗ corresponding author: neustupa@math.cas.cz Abstract. We deal with a mathematical model of a flow of an incompressible Newtonian fluid through a channel with an artificial boundary condition on the outflow. We explain how several artificial boundary conditions formally follow from appropriate variational formulations and the way one expresses the dynamic stress tensor. As the boundary condition of the “do nothing”-type, that is predominantly considered to be the most appropriate from the physical point of view, does not enable one to derive an energy inequality, we explain how this problem can be overcome by using variational inequalities. We derive a priori estimates, which are the core of the proofs, and present theorems on the existence of solutions in the unsteady and steady cases. Keywords: Variational inequality, Navier-Stokes equation, “do nothing” outflow boundary condition. 1. Introduction 1.1. The considered initial–boundary value problem We denote by Ω a Lipschitzian domain in R3, which represents a channel. An incompressible Newtonian fluid is supposed to flow into the channel through the part Γ1 of the boundary ∂Ω and to flow essentially out of the channel through the part Γ2 of ∂Ω. (See Fig. 1.) By “essentially” we mean that we do not exclude possible backward flows on Γ2. A fixed wall of the channel is denoted by Γ0. The flow is described by the equations of motion ∂tv + v ·∇v− div Sd + ∇p = f, (1) div v = 0, (2) 6 � � � � � ��3 � � � � �� Q Q Q Q QQs Q Q Q Q Q QQ ���: ���: ���: - - - Ω Γ0 Γ1 Γ2 x1 x2 x3 Fig. 1 The channel. where v denotes the velocity, p is the pressure, Sd is the dynamic stress tensor and f represents an external body force. For simplicity, we assume that the density of the fluid is equal to one. We use the homogeneous Dirichlet boundary condition v = 0 on Γ0 × (0,T), (3) where (0,T ) is a time interval. The velocity on Γ1 can be naturally assumed to be known, which yields the inhomogeneous Dirichlet boundary condition v = v∗ on Γ1 × (0,T). (4) On the other hand, since the velocity profile on Γ2 cannot be predicted in advance, it is logical to apply some “artificial” boundary condition. There appear various artificial boundary conditions in the litera- ture, see e.g. [1–8]. Boundary conditions, that follow automatically from an appropriate weak formulation of the considered problem if one a priori assumes a sufficient regularity of asolution, are usually called the “do nothing” conditions. (See e.g. [1, 6, 9] for more details.) An example, and probably the most often used artificial boundary condition is −pn + ν ∂v ∂n = g on Γ2 × (0,T), (5) where n denotes the outer normal vector field on ∂Ω, ν is the coefficient of viscosity and g is a given function. The non–steady problem also contains the initial condition v = v0 in Ω ×{0}. (6) 1.2. On some previous related existential results The existential theory for the system (1)–(6) is based on appropriate a priori estimates. As the boundary condition (5) admits a possible reverse flow on Γ2, which may bring to Ω an arbitrarily large amount of kinetic energy from the outside, the usual energy inequality does not hold. This does not matter if the given data of the problem are in an appropriate sense 89 https://doi.org/10.14311/AP.2021.61.0089 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Stanislav Kračmar, Jiří Neustupa Acta Polytechnica “sufficiently small” or if the number T is “sufficiently small” in the non–steady case. (See [1, 9, 10].) The existence of a weak solution of the problem (1)–(6) on an arbitrarily large time interval for “large” data (which is well known for the Navier–Stokes equations with the Dirichlet or Navier boundary conditions on ∂Ω), is an open problem. A similar problem arises if one studies a flow through a 2D turbine cascade, see [3, 4]. Some authors use boundary conditions on Γ2, modified in such a way that it enables one to estimate the kinetic energy of the fluid flowing to Ω through Γ2. Examples of such modifications can be found in [2, 5, 7, 11, 12]. Another modification, used in connection with the heat transfer, can be found in [8]. A different approach has been suggested in papers [13–15]. There the authors have considered the sta- tionary and non-stationary problems and impose an additional condition on Γ2, that enables one to es- timate the kinetic energy of a possible reverse flow and obtain an a priori estimate of the energy type. However, the new additional condition implies that the solution cannot lie in the whole Sobolev space W 1,2(Ω), but only in a certain closed convex subset of this space. Since one does not know in advance whether the solution of the problem (1)–(5) falls into this convex set, one must consider a variational in- equality instead of the momentum equation. Note that artificial boundary conditions on a part of the boundary are also being used, if one approximates a problem in an exterior domain D by a problem in a bounded domain D ∩ BR(0) (for “large” R) and prescribes an artificial boundary condition on ∂BR(0). (See e.g. [16] and [17].) 2. Several boundary conditions of the “do nothing” type 2.1. Three equivalent forms of the dynamic stress tensor in equation (1) Since the difficulties, caused by the artificial boundary conditions on Γ2, are of the same nature in stationary and non-stationary problems, here we consider, for simplicity, only the stationary problem. The dynamic stress tensor Sd, in an incompressible Newtonian fluid, equals 2νD, where D is the rate of the deformation tensor. (It coincides with the symmetrized gradient of velocity.) The term div Sd, which appears in equa- tion (1), can be expressed by any of these formulas: a) div Sd = ν∆v, b) div Sd = ν div [ ∇v + (∇v)T ] , c) div Sd = −ν curl2v.   (7) 2.2. Variational formulations of the initial–boundary value problem A variational formulation of the system (1), (2) with the boundary conditions (3), (4) formally follows from the classical formulation if one multiplies equation (1) by a “smooth” test function φ, such that div φ = 0, and integrates in Ω. As v should satisfy the Dirich- let boundary conditions (3) and (4) on Γ0 and Γ1, respectively, it is logical to assume that φ = 0 on Γ0 ∪Γ1. On the other hand, one imposes no boundary condition on φ on Γ2. Applying the integration by parts, using the forms a) – c) of the dynamic stress tensor, we successively obtain the equations a) ∫ Ω [ v ·∇v ·φ + ν∇v : ∇φ ] dx = ∫ Γ2 [−pn + ν∇v ·n] ·φ dS + ∫ Ω f ·φ dx, b) ∫ Ω [ v ·∇v ·φ + ν(∇v + (∇v)T ) : ∇φ dx = ∫ Γ2 [ −pn + ν ( ∇v + (∇v)T ) ·n ] ·φ dS + ∫ Ω f ·φ dx, c) ∫ Ω [ v ·∇v ·φ + ν curl v · curl φ ] dx = ∫ Γ2 [−pn−ν curl v×n] ·φ dS + ∫ Ω f ·φ dx. However, the integrals on Γ2 cannot be involved by the weak formulation because the integrands are generally not integrable if one stays in the usual level of weak solutions. This is why it is logical to neglect these integrals or to replace them just by ∫ Γ2 g·φ dS, where g is an arbitrarily given function on Γ2. Then the variational forms of the system (1), (2) are a) ∫ Ω [ v ·∇v ·φ + ν∇v : ∇φ ] dx = ∫ Γ2 g ·φ dS + ∫ Ω f ·φ dx, b) ∫ Ω [ v ·∇v ·φ + ν ( ∇v + (∇v)T ) : ∇φ ] dx = ∫ Γ2 g ·φ dS + ∫ Ω f ·φ dx, c) ∫ Ω [ v ·∇v ·φ + ν curl v · curl φ ] dx = ∫ Γ2 g ·φ dS + ∫ Ω f ·φ dx.   (8) The equations should satisfy all functions φ with the aforementioned properties and v should also satisfy the boundary conditions (3) and (4). If a weak solution v exists and is sufficiently smooth then, by a reverse integration by parts, one can prove that there exists an appropriate associated pressure p and show that v and p satisfy the boundary conditions a) −pn + ν∇v ·n = g, b) −pn + ν [∇v + (∇v)T ] ·n = g, c) −pn−ν curl v×n = g,   (9) 90 vol. 61 Special Issue/2021 Modeling of flows through a channel respectively, on Γ2. It is well known that the pressure p in equation (1) is not unique, because p + c, where c is an arbitrary additional constant (or a function of time), also satisfies the same equation. However, the same consideration is not possible in the boundary conditions a) – c) in (9). Here, it only follows from the variational formulation that , one can choose only one pressure p from all associated pressures, that satisfies the boundary condition. 2.3. The momentum equation with the Bernoulli pressure None of the conditions a) – c) in (9) excludes a possible reverse flow on Γ2 that could hypothetically bring an arbitrarily large amount of the kinetic energy back to Ω. One usually derives an a priori energy inequality so that the momentum equation (1) is multiplied by v and integrated over Ω. Then the flow of the kinetic energy through Γ2 comes from the integral of (v·∇v)·v . Applying the integration by parts, we can express this integral as follows:∫ Ω (v ·∇v) ·v dx = ∫ ∂Ω (v ·n) 12 |v| 2 dS = ∫ Γ1 (v∗ ·n) 12 |v ∗|2 dS + ∫ Γ2 (v ·n) 12 |v| 2 dS. (10) The last integral can hypothetically take an arbitrarily large value if v ·n < 0 on the part of Γ2, i.e. in the case of a reverse flow. The situation is different if the nonlinear term in equation (1) is considered in the form curl v ×v + ∇12 |v| 2 and one involves ∇12 |v| 2 and p to the so called Bernoulli pressure q := p + 12 |v| 2. Then, instead of (9), one obtains from the integral equations (8) the boundary conditions a) −qn + ν∇v ·n = g, b) −qn + ν [∇v + (∇v)T ] ·n = g, c) −qn−ν curl v×n = g.   (11) Now, the nonlinear term in equation (1) is just curl v×v (instead of v ·∇v), which yields the term∫ Ω curl v × v · φ dx in the variational formulation. If one formally multiplies equation (1) by the ve- locity v then the nonlinear term vanishes, because (curlv × v) · v = 0. This has the following conse- quences: 1) the nonlinear term curl v×v generates no back- ward inflow of kinetic energy to Ω through the surface Γ2, 2) the usual energy–type inequality can be derived, 3) the existence of a weak solution on an arbitrarily long time interval can be proven by similar methods, as if one considers the homogeneous or inhomogeneous Dirichlet boundary condition on the whole boundary ∂Ω (see e.g. [18]). 2.4. Which artificial boundary condition is the best? There arises a natural question: which of the formu- lated boundary conditions a) – c) in (9) and a) – c) in (11) on Γ2 is most appropriate? The advantage of all the conditions in (11) is that, in contrast to the conditions from (9), they enable one to prove the existence of a weak solution. On the other hand, the condition a) from (9) is fulfilled (with g = 0) by the Poiseuille flow in a circular pipe. This is mainly why this condition is usually considered as the best from a physical point of view. On the other hand, in any of the formulated boundary conditions, one can al- ways calculate an appropriate function g so that the Poiseuille flow satisfies the considered condition with this concrete function g. Thus, the suitability of the chosen boundary condition probably depends only on a particular situation. Moreover, in our opinion, it would be very useful to perform numerical calculations with various boundary conditions so that one could compare the results among themselves and also with physical measurements. 3. The Navier–Stokes inequality – the non-steady case In this section, we deal with the Navier–Stokes prob- lem (1)–(4) in Ω with the boundary condition (5) on Γ2. Due to the reasons, explained in Sections 1 and 2, we study the problem in the form of a variational inequality. 3.1. Notation (i) We use the usual notation of the norms in the Lebesgue spaces: ‖ .‖r is the norm in Lr(Ω) or in Lr(Ω) (the space of vector functions) or in Lr(Ω)3×3 (the space of tensorial functions). By analogy, ‖ .‖k,r denotes the norm in the Sobolev space Wk,r(Ω) or Wk,r(Ω) or Wk,r(Ω)3×3. If the norm is related to another set than Ω then we denote it e.g. by ‖ .‖r; Γ2 , etc. (ii) We assume that v∗ is a given function on Γ1 × (0,T), such that 12 ∫ Γ1 [v∗ · (−n)] |v∗|2 dS (the inflow of the kinetic energy to Ω through Γ1) is bounded, as a function of t, for t ∈ (0,T). (iii) Furthermore, we assume that v∗ can be extended to Ω×(0,T ) so that the extended function (which is denoted by v∗ext) satisfies the boundary condi- tion (3) on Γ0 × (0,T) and a) v∗ext ∈ L∞ ( 0,T; W 1,2(Ω) ) and ∂tv∗ext ∈ L2 ( 0,T; W−1,2(Ω) ) , b) v∗ext is divergence–free. (Here, we denote by W−1,2(Ω) the dual space to W 1,2(Ω). The du- ality pairing between W−1,2(Ω) and W 1,2(Ω) is denoted by 〈 . , .〉.) Due to [19, Theorem I.3.1] function v∗ext belongs to C0 ( [0,T]; L2(Ω) ) . (iv) We denote by V the linear space of all divergence– free vector functions φ ∈ W 1,2(Ω), such that φ = 0 on Γ0∪Γ1. Then v∗ext(t)+V (for a.a. t ∈ (0,T)) 91 Stanislav Kračmar, Jiří Neustupa Acta Polytechnica is the set of all functions φ from W 1,2(Ω), such that div φ = 0, φ = 0 on Γ0 and φ = v∗ext(t) on Γ1. (v) Let �1 > 0 and ζ ∈ L∞(0,T ) satisfy the inequality∥∥(v∗ext(t) ·n)− |v∗ext(t)|2∥∥1; Γ2 + �1 < ζ(t) (12) for a.a. t ∈ (0,T). (The subscript “−” denotes the negative part.) Such number �1 and function ζ exist, because the trace of v∗ on Γ2×(0,T) is in L∞ ( 0,T; W 1/2,2(Γ2) ) and this space is continu- ously imbedded to L∞ ( 0,T ; L4(Γ2) ) . We denote by Kt the set of all functions φ ∈ v∗ext(t) + V such that∥∥(φ ·n)− |φ|2‖1; Γ2 ≤ ζ(t) for a.a. t ∈ (0,T), (13) by Kct the convex hull of Kt and define K c t to be the closure of Kct. Set K c t is the so called closed convex hull of Kt. (See [20] for more properties of the convex hull and the closed convex hull. Note that Kct can also be defined as the intersection of all closed convex sets in v∗ext(t) + V , containing Kt.) We assume that the number �1, and the func- tions v∗ext and ζ are fixed throughout the whole paper. Using the presence of �1 > 0 in inequal- ity (12), one can also show that there exists �2 > 0 such that Kct contains the �2–neighborhood of v∗ext(t), independently of t. (vi) Denote by W (0,T) the space { w ∈ L2(0,T; W 1,2(Ω)); ∂tw ∈ L2(0,T ; W−1,2(Ω)) } with the norm |||w||| := (∫ T 0 ‖w‖21,2 dt+ ∫ T 0 ‖∂tw‖2−1,2 dt )1/2 . Using [19, Theorem I.3.1], one can show that W (0,T) ⊂ C0 ( [0,T]; L2(Ω) ) . (vii) Put K c(0,T) := { w ∈ W (0,T); w(t) ∈ Kct for a.a. t ∈ (0,T) } . 3.2. A formal derivation of the variational inequality Suppose that v, p is a sufficiently smooth solu- tion of the problem (1)–(6) and w is a sufficiently smooth function from [0,T] such that w(t) ∈ Kct for a.a. t ∈ [0,T]. Using the form a) of the diver- gence of the dynamic stress tensor in (7), multiplying equation (1) by the difference w−v, integrating in Ω×(0,T), applying the integration by parts and using the equality w −v = 0 on Γ0 ∪ Γ1 × (0,T) and the boundary condition (5), we get∫ T 0 ∫ Ω [∂tv + v ·∇v] · (w−v) dx dt + ∫ T 0 ∫ Ω ν∇v : ∇(w−v) dx dt = ∫ T 0 ∫ Ω f · (w−v) dx dt + ∫ T 0 ∫ Γ2 g · (w−v) dS dt. (14) The term, which contains the derivative ∂tv, satisfies∫ T 0 ∫ Ω ∂tv · (w−v) dx dt = ∫ T 0 ∫ Ω ∂t(v−w) · (w−v) dx dt + ∫ T 0 ∫ Ω ∂tw · (w−v) dx dt = 1 2 ‖w(0) −v(0)‖22 − 1 2 ‖w(T) −v(T)‖22 + ∫ T 0 ∫ Ω ∂tw · (w−v) dx dt ≤ 1 2 ‖w(0) −v0‖22 + ∫ T 0 ∫ Ω ∂tw · (w−v) dx dt. (15) Let w ∈ K c(0,T) further on. Since∫ Ω ∂tw · (w−v) dx = 〈∂tw,w−v〉,∫ Ω f · (w−v) dx = 〈f,w−v〉, (15) and (14) yield∫ T 0 〈 ∂tw,w−v 〉 dt + ∫ T 0 ∫ Ω v ·∇v · (w−v) dx dt + ∫ T 0 ∫ Ω ν∇v : ∇(w−v) dx dt ≥ ∫ T 0 〈f,w−v〉 dt + ∫ T 0 ∫ Γ2 g · (w−v) dS dt − 1 2 ‖w(0) −v0‖22. (16) Since (16) is an inequality, we have the possibility to choose another condition and to impose it on the solution v: we require that the inclusion v(t) ∈ Kct holds for a.a. t ∈ (0,T). 3.3. Definition of the initial–boundary value problem (P) Let v∗ext be the extension of function v∗ with the properties (i) and (ii) from paragraph 3.2. Let v0 ∈ L2(Ω), div v0 = 0 in Ω in the sense of dis- tributions, v0 ·n = 0 on Γ0 and v0 ·n = v∗(0) ·n on Γ1 in the sense of traces. Let f ∈ L2(0,T ; W−1,2(Ω)) and g ∈ L2(0,T; L4/3(Γ2)). One looks for v ∈ L∞(0,T ; L2(Ω))∩L2(0,T ; W 1,2(Ω)) such that v(t) ∈ Kct for a.a. t ∈ (0,T) and v satisfies inequality (16) for all w ∈ K c(0,T). It is well known that for v0 ∈ L2(Ω), such that div v0 ∈ L2(Ω) (which it definitely satisfies if v0 is 92 vol. 61 Special Issue/2021 Modeling of flows through a channel divergence–free in the sense of distributions), the scalar product v · n makes sense on ∂Ω, as an ele- ment of W−1/2,2(∂Ω). (This follows e.g. from [21, Theorem III.2.2].) Thus, the conditions v0 ·n = 0 on Γ0 and v0 ·n = v∗(0) ·n are assumed to hold on Γ0 and Γ1 as equalities in W−1/2,2(Γ0) and W−1/2,2(Γ1), respectively. The next theorem provides the information on the existence of a solution of a problem (P) on an arbi- trarily long time interval (0,T). Theorem 1. Let v0, v∗ext, f and g be the functions with the aforementioned properties. Then problem (P) is solvable. The solution can be expressed in the form v = v∗ext + u, where u ∈ L∞(0,T; L 2(Ω)) ∩ L2(0,T; V ) satisfies the inequality ‖u(t)‖22 + ν ∫ t 0 ‖∇u(s)‖22 ds ≤ ‖u(0)‖22 − ∫ t∗ 0 ∫ Γ1 (v∗ ·n) |v∗|2 dS dt + c1 ∫ t 0 ‖u(s)‖22 ds + ∫ t 0 [ c2 ‖f(s)‖2−1,2 + c3 ‖v∗ext(s)‖ 2 1,2 + c4 ‖∂tv ∗ ext(s)‖ 2 −1,2 + c5 ‖g(s)‖24/3; Γ2 ] ds (17) for all t in(0,T), where all the constants c1–c5 are independent of v0, v∗, v∗ext, f, g and u. Note that there is a minus sign in front of the first inte- gral on the right hand side, because n is the outer nor- mal vector and therefore −12 ∫ t∗ 0 ∫ Γ1 (v∗ ·n) |v∗|2 dS dt represents the inflow of the kinetic energy to Ω through Γ1 in the time interval (0, t∗). An analogous theorem, with a different convex set Kct, has been proven in [15]. 3.4. The principle of the proof and a priori estimates The complete way Theorem 1 can be proven consists of these main steps: 1) construction of appropriate approximations vn (for n ∈ N) of solution v, 2) deriva- tion of a series of estimates of the approximations, 3) derivation of various types of convergence of a subse- quence of {vn} in various spaces, 4) verification that the limit is the solution v. Among others, one also needs the strong convergence in L2(0,T; W 1,2(Ω)), which follows from an estimate of a fractional deriva- tive with respect to t of vn and from the Lions–Aubin lemma, see e.g. [22]. Since the complete estimates of the approximations are laborious, technically com- plicated and necessarily influenced by the technique, used just for the construction of the approximations, we show below on an a priori level how one can directly obtain from the variational inequality the estimates of u in L∞(0,T ; L2(Ω)) and in L2(0,T ; W 1,2(Ω)). The advantage of a priori estimates is that they enable one to abstract from the whole machinery, which is necessary in the proof of existence of the approxima- tions. On the other hand, we assume, just inside the procedure, that u is smooth. (This formal assumption is naturally satisfied on the level of approximations.) Thus, let t∗ ∈ (0,T), α ∈ (0, 1) and δ > 0 be so small that t∗+δ < T . Define function η of one variable t by the formulas η(t) =   α for 0 < t ≤ t∗, α + (1 −α) δ (t− t∗) for t∗ < t < t∗ + δ, 1 for t∗ + δ ≤ t < T. (Function η is continuous on (0,T), constant on (0, t∗] and on [t∗ + δ,T) and linear on [t∗, t∗ + δ].) Solution v can be expressed in the form v = v∗ext + u, where u ∈ V . Put w := v∗ext + ηu = (1 −η)v∗ext + ηv. (As set K c(0,T) is convex and 0 < η ≤ 1, w belongs to K c(0,T).) Then w−v = (η − 1)u (which equals 0 on the interval [t∗ + δ,T)). Substituting this to the first term in (16), we obtain ∫ T 0 〈 ∂tw,w−v 〉 dt = ∫ t∗ 0 〈 ∂t(v∗ext + αu), (α− 1)u 〉 dt + ∫ t∗+δ t∗ 〈 ∂t(v∗ext + ηu), (η − 1)u 〉 dt = (α− 1) ∫ t∗ 0 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) ∫ t∗ 0 〈 ∂tu, u 〉 dt + ∫ t∗+δ t∗ 〈 ∂tv ∗ ext, (η − 1)u 〉 dt + ∫ t∗+δ t∗ 〈 ∂t(ηu), ηu 〉 dt− ∫ t∗+δ t∗ 〈 η̇u, u 〉 dt − ∫ t∗+δ t∗ 〈 η ∂tu, u 〉 dt = ∫ t∗+δ 0 (η − 1) 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) 2 ( ‖u(t∗)‖22 −‖u(0)‖ 2 2 ) + 1 2 ( ‖η(t∗ + δ)u(t∗ + δ)‖22 −‖η(t ∗)u(t∗)‖22 ) − 1 −α δ ∫ t∗+δ t∗ ‖u‖22 dt − 1 2 ∫ t∗+δ t∗ η d dt ‖u‖22 dt = ∫ t∗+δ 0 (η − 1) 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) 2 ( ‖u(t∗)‖22 −‖u(0)‖ 2 2 ) + 1 2 ( ‖η(t∗ + δ)u(t∗ + δ)‖22 −‖η(t ∗)u(t∗)‖22 ) 93 Stanislav Kračmar, Jiří Neustupa Acta Polytechnica − 1 −α δ ∫ t∗+δ t∗ ‖u‖22 dt − 1 2 ( η(t∗ + δ)‖u(t∗ + δ)‖22 −η(t ∗)‖u(t∗)‖22 ) + 1 2 ∫ t∗+δ t∗ η̇‖u‖22 dt = ∫ t∗+δ 0 (η − 1) 〈 ∂tv ∗ ext, u 〉 dt + α(α− 1) 2 ( ‖u(t∗)‖22 −‖u(0)‖ 2 2 ) + 1 2 ( ‖η(t∗ + δ)u(t∗ + δ)‖22 −‖η(t ∗)u(t∗)‖22 ) − 1 −α δ ∫ t∗+δ t∗ ‖u‖22 dt − 1 2 ( η(t∗ + δ)‖u(t∗ + δ)‖22 −η(t ∗)‖u(t∗)‖22 ) + 1 −α 2δ ∫ t∗+δ t∗ ‖u‖22 dt. Considering δ → 0+, we get∫ T 0 〈 ∂tw,w−v 〉 dt = (α− 1) ∫ t∗ 0 〈∂tv∗ext, u〉 dt + α2 − 1 2 ‖u(t∗)‖22 − α(α− 1) 2 ‖u(0‖22. Substituting this to (16), using v = v∗ext + u and w = v∗ext + ηu in all other terms in (16), considering δ → 0+, dividing the whole inequality by α−1 (which is negative), and considering finally α → 0+, we obtain∫ t∗ 0 〈∂tv∗ext,u〉 dt + 1 2 ‖u(t∗)‖22 + ∫ t∗ 0 ∫ Ω (v∗ext + u) ·∇(v ∗ ext + u) ·u dx dt + ∫ t∗ 0 ∫ Ω ν∇(v∗ext + u) : ∇u dx ≤ ∫ t∗ 0 〈f,u〉 dt + ∫ t∗ 0 ∫ Γ2 g ·u dS dt + 1 2 ‖u(0)‖22, 1 2 ‖u(t∗)‖22 + ∫ t∗ 0 ν‖∇u‖22 dt + ∫ t∗ 0 ∫ Ω (v∗ext + u) ·∇(v ∗ ext + u) · (v ∗ ext + u) dx dt + ∫ t∗ 0 〈∂tv∗ext,u〉 dt ≤ ∫ t∗ 0 ∫ Ω (v∗ext + u) ·∇(v ∗ ext + u) ·v ∗ ext dx dt + ∫ t∗ 0 ∫ Ω ν∇v∗ext : ∇u dx dt + ∫ t∗ 0 〈f,u〉 dt + ∫ t∗ 0 ∫ Γ2 g ·u dS dt + 1 2 ‖u(0)‖22, 1 2 ‖u(t∗)‖22 + ∫ t∗ 0 ν‖∇u‖22 dt + 1 2 ∫ t∗ 0 ∫ Γ1 (v∗ ·n) |v∗|2 dS dt + 1 2 ∫ t∗ 0 ∫ Γ2 [(v∗ext + u) ·n] |v ∗ ext + u| 2 dS dt + ∫ t∗ 0 〈∂tv∗ext,u〉 dt ≤ ∫ t∗ 0 ∫ Ω ( v∗ext ·∇v ∗ ext ·v ∗ ext + v ∗ ext ·∇u ·v ∗ ext + u ·∇v∗ext ·v ∗ ext + u ·∇u ·v ∗ ext ) dx dt + ∫ t∗ 0 ∫ Ω ν∇v∗ext : ∇u dx dt + ∫ t∗ 0 〈f,u〉 dt + ∫ t∗ 0 ∫ Γ2 g ·u dS dt + 1 2 ‖u(0)‖22. Since ∣∣∣∣ ∫ t∗ 0 ∫ Ω u ·∇u ·v∗ext dx dt ∣∣∣∣ ≤ ∫ t∗ 0 ‖u‖3 ‖∇u‖2 ‖v∗ext‖6 dt ≤ c ∫ t∗ 0 ‖u‖3 ‖∇u‖2 ‖v∗ext‖1,2 dt ≤ c ∫ t∗ 0 ‖u‖3 ‖∇u‖2 dt ≤ c ∫ t∗ 0 ‖u‖1/22 ‖u‖ 1/2 6 ‖∇u‖2 dt ≤ c ∫ t∗ 0 ‖u‖1/22 ‖∇u‖ 3/2 2 dt ≤ ∫ t∗ 0 ( ξ‖∇u‖22 + c(ξ)‖u‖ 2) dt, where c is a generic constant, we get 1 2 ‖u(t∗)‖22 + (ν − ξ) ∫ t∗ 0 ‖∇u‖22 dt + 1 2 ∫ t∗ 0 ∫ Γ1 (v∗ ·n) |v∗|2 dS dt + ∫ T 0 〈∂tv∗ext,u〉 dt ≤ 1 2 ∫ t∗ 0 ∫ Γ2 [(v∗ext + u) ·n]− |v ∗ ext + u| 2 dS dt + ∫ t∗ 0 ∫ Ω ( v∗ext ·∇v ∗ ext ·v ∗ ext + v ∗ ext ·∇u ·v ∗ ext + u ·∇v∗ext ·v ∗ ext ) dx dt + c(ξ) ∫ t∗ 0 ‖u‖2 dt + ∫ t∗ 0 ∫ Ω ν∇v∗ext : ∇u dx dt + ∫ t∗ 0 〈f,u〉 dt + ∫ T 0 ∫ Γ2 g ·u dS dt + 1 2 ‖u(0)‖22. (18) (Note that ξ > 0 can be chosen arbitrarily small.) The first integral on the right hand side satisfies the 94 vol. 61 Special Issue/2021 Modeling of flows through a channel inequality∫ Γ2 [(v∗ext + u) ·n]− |v ∗ ext + u| 2 dS ≤ (∫ Γ2 [(v∗ext + u) ·n] 3 − dS )1 3 · (∫ Γ2 |v∗ext + u| 3 dS )2 3 . (19) Since v∗ext + u ∈ K c t, there exists a sequence {uk} in Kct, such that uk → u (for k → ∞) in the norm of W 1,2(Ω). Then we also have(∫ Γ2 [(v∗ext + u) ·n] 3 − dS )1 3 = lim k→∞ (∫ Γ2 [(v∗ext + uk) ·n] 3 − dS )1 3 . To each function uk, there exist finite families {θki}Nki=1 and {uki}Nki=1 in [0, 1] and Kt, respectively, such that Nk∑ i=1 θki = 1 and uk = Nk∑ i=1 θkiuki. Then, applying Minkowski’s inequality, we get(∫ Γ2 [(v∗ext + uk) ·n] 3 − dS )1 3 = (∫ Γ2 [Nk∑ i=1 θki(v∗ext + uki) ·n ]3 − dS )1 3 ≤ (∫ Γ2 Nk∑ i=1 [θki(v∗ext + uki) ·n] 3 − dS )1 3 ≤ Nk∑ i=1 (∫ Γ2 [θki(v∗ext + uki) ·n] 3 − dS )1 3 = Nk∑ i=1 θki (∫ Γ2 [(v∗ext + uki) ·n] 3 − dS )1 3 ≤ Nk∑ i=1 θkiζ = ζ. Hence (∫ Γ2 [(v∗ext + u) ·n] 3 − dS )1 3 ≤ ζ, (20) too. Note that this is a crucial part, where we use the fact that v∗ext +u lies in K c t. The estimates, following from this information, are not available if one deals with the Navier–Stokes equation instead of the Navier– Stokes variational inequality (16). As there exists a continuous operator of traces from the Sobolev– Slobodetski space W 5/6,2(Ω) to L3(Γ2), which can be deduced e.g. by means of [23], we have(∫ Γ2 [(v∗ext + u) ·n] 3 − dS )1 3 (∫ Γ2 |v∗ext + u| 3 dS )2 3 ≤ ζ‖v∗ext + u‖ 2 3; Γ2 ≤ c‖v∗ext + u‖ 2 5/6,2 ≤ cζ‖v∗ext + u‖ 1 3 2 ‖v ∗ ext + u‖ 5 3 1,2 ≤ cζ‖v∗ext + u‖ 1 3 2 ( ‖v∗ext‖ 5 3 1,2 + ‖∇u‖ 5 3 2 ) ≤ ξ‖∇u‖22 + c(ξ) ζ 6 ‖v∗ext + u‖ 2 2 + ‖v ∗ ext‖ 2 1,2. Recall that ζ ∈ L∞(0,T). Substituting to (18), and using also the estimates∫ t∗ 0 ∫ Ω ν∇v∗ext : ∇u dx dt ≤ ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ) ν 2 ∫ t∗ 0 ‖∇v∗ext‖ 2 2 dt = ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ,v ∗ ext),∣∣∣∣ ∫ t∗ 0 〈∂tv∗ext,u〉 dt ∣∣∣∣ ≤ ∫ t∗ 0 ‖∂tv∗ext‖−1,2 ‖u‖1,2 dt ≤ c ∫ t∗ 0 ‖∂tv∗ext‖−1,2 ‖∇u‖2 dt ≤ ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ) ∫ t∗ 0 ‖∂tv∗ext‖ 2 −1,2 dt = ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ,v ∗ ext),∫ t∗ 0 ∫ Ω v∗ext ·∇u ·v ∗ ext dx dt ≤ ∫ t∗ 0 ‖∇u‖2 ‖v∗ext‖ 2 4 dt ≤ ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ) ∫ t∗ 0 ‖v∗ext‖ 4 4 dt = ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ,v ∗ ext),∫ t∗ 0 ∫ Ω ( v∗ext ·∇v ∗ ext ·v ∗ ext + u ·∇v ∗ ext ·v ∗ ext ) dx dt ≤ c(v∗ext) + ∫ t∗ 0 ∫ Ω ‖u‖4 ‖∇v∗ext‖2 ‖v ∗ ext‖4 dt ≤ ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ,v ∗ ext),∫ t∗ 0 〈f,u〉 dt ≤ ∫ t∗ 0 ‖f‖−1,2 ‖u‖1,2 dt ≤ ∫ t∗ 0 ξ‖∇u‖22 dt + c(ξ,f),∫ t∗ 0 ∫ Γ2 g ·u dS dt ≤ ∫ t∗ 0 ‖g‖4/3; Γ2 ‖u‖4; Γ2 dt ≤ ∫ t∗ 0 ‖g‖4/3; Γ2 ‖u‖1,2 dt ≤ c ∫ t∗ 0 ‖g‖4/3; Γ2 ‖∇u‖2 dt ≤ ∫ t∗ 0 ξ‖∇u‖22 + c(ξ,g), 95 Stanislav Kračmar, Jiří Neustupa Acta Polytechnica where c is independent of t∗, we obtain 1 2 ‖u(t∗)‖22 + (ν − 9ξ) ∫ t∗ 0 ‖∇u‖22 dt + 1 2 ∫ t∗ 0 ∫ Γ1 (v∗ ·n) |v∗|2 dS dt ≤ c(ξ) ∫ t∗ 0 ‖u‖2 dt + c(ξ,v∗ext,f,g)+ + c(ξ) ∫ t∗ 0 ζ6(t) ( ‖v∗ext + u‖ 2 2 + ‖v ∗ ext‖ 2 1,2 ) dt + 1 2 ‖u(0)‖22. (21) Choose ξ so small that ξ < 118ν. Evaluating pre- cisely the right hand side (which concerns especially c(ξ,v∗ext,f,g)), we can rewrite the inequality in the form (17). Omitting at first the second term on the left hand side (i.e. the integral of ‖∇u‖22) and applying the generalized Gronwall inequality, we obtain the es- timate of u in L∞(0,T ; L2(Ω)) in terms of the norms of ζ(t), v∗ext, f and g in appropriate spaces, which are all finite. Then, omitting the first term on the left hand side in (21) and considering t∗ → T−, we obtain the estimate of the norm of u in L2(0,T; W 1,2(Ω)). 3.5. Remark By analogy with [15], one can show that if v is a solution of a problem (P) then there exists an as- sociated pressure p as a distribution in Ω × (0,T). The pair (v,p) satisfies the equations (1), (2) in the sense of distributions in Ω × (0,T). If, more- over, ∂tv ∈ L1(0,T; W−1,2(Ω)) and one prescribes p ∈ L1(0,T) then the pressure p can be chosen so that∫ Ω p(t) dx = p(t) for a.a. t ∈ (0,T). Suppose now that the solution v has these a pos- teriori properties: v ∈ L2(0,T; W 2,2(Ω)), ∂tv, v · ∇v, f ∈ L2(0,T; L2(Ω)) and there exists �3 > 0 such that all φ ∈ v(t) + V , whose distance from v(t) in the W 1,2–norm is less than �3, belong to Kct for a.a. t ∈ (0,T). (The last condition means that v(t) lies “uniformly” in the interior of Kct.) Then one can prove that the distribution p is regular and can be represented by a function from L2(0,T ; W 1,2(Ω)). Moreover, one can also find a function ϑ ∈ L2(0,T) so that ν ∂v ∂n − (p + ϑ)n = g (22) holds a.e. in Γ2 ×(0,T ). This shows that the concrete pressure, obtained from the variational inequality and satisfying the outflow boundary condition on Γ2 × (0,T), is unique in the sense that it cannot be changed by adding an arbitrary constant (or a function of t). 4. The Navier–Stokes inequality – the steady case In both the non-steady and steady cases, the solu- tion’s proof of existence relies on the construction of appropriate approximations, the estimations of the approximations that in some sense copy a priori esti- mates, the deduction of various types of convergence of a sequence (or a subsequence) of approximations to some limit function, and the demonstration that the limit is a solution whose existence one wants to prove. As we have already mentioned in subsections 1.2 and 3.4, the crucial part is the derivation of a priori estimates. In order to obtain appropriate estimates, in the non–steady case, one can apply Gronwall–type inequalities in order to obtain a uniform (in time) estimate of the L2–norm of the solution and the es- timate of ∫T 0 ‖∇u‖ 2 2 dt (see subsection 3.4). In the steady case, the estimates substantially depend on the properties of the extended function v∗ext, intro- duced in subsection 3.1. Moreover, as follows from estimate (25), we are able to prove the existence of the steady solution only if ζ (which is now just a positive number) is “sufficiently small” in comparison to ν. (Recall the ζ estimates possible reverse flows on the outflow part Γ2 of the boundary, see (13).) The extended function v∗ext should now be naturally time–independent, and should be constructed so that the integral ∫ Ω u ·∇u ·v ∗ ext dx is “sufficiently small” in comparison with ‖∇u‖22 for all u ∈ V . The reasons are the same as in the case of the steady Navier–Stokes problem with inhomogeneous Dirichlet–type boundary condition on the whole boundary of Ω, see e.g. [21, Chapter IX] for the detailed explanation. It follows from the paper [14] that this condition of “sufficient smallness” of the aforementioned integral is in fact not an obstacle. Concretely, it is shown in [14] that if v∗ satisfies the condition (?) v∗ can be extended from Γ1 onto the whole bound- ary ∂Ω so that the extended function belongs to W 1/2,2(∂Ω) is equal to zero on Γ0 and its flux through ∂Ω is zero, then the extension v∗ext can be constructed so that given δ > 0, v∗ext ∈ W 1,2(Ω), v∗ext is divergence–free and∫ Ω u1 ·∇u2 ·v∗ext dx ≤ δ‖∇u1‖2 ‖∇u2‖2 (23) for all u1, u2 ∈ V . This is the analogue of the so called Leray–Hopf inequality, see [21]. Let us show how the a priori estimate looks. Ob- viously, in the steady case, ζ is just a number and set Kct is independent of t. Hence we further on use the notation Kc instead of Kct. The “steady state version” of inequality (16) is∫ Ω v ·∇v · (w−v) dx + ∫ Ω ν∇v ·∇(w−v) dx ≥ 〈f,w−v〉 + ∫ Γ2 g · (w−v) dS. (24) The solution v lies in Kc and the inequality is required to be satisfied for all w ∈ Kc. Writing v in the form v∗ext + u, where u ∈ V , using inequality (24) with 96 vol. 61 Special Issue/2021 Modeling of flows through a channel w = v∗ext, and applying (19), (20) and (23), we obtain ∫ Ω v ·∇v · (w−v) dx + ∫ Ω ν∇v ·∇(w−v) dx ≥ 〈f,w−v〉 + ∫ Γ2 g · (w−v) dS, ν‖∇u‖22 + ∫ Ω (v∗ext + u) ·∇(v ∗ ext + u) · (v ∗ ext + u) dx ≤ ∫ Ω [ ν∇v∗ext : ∇u + (v∗ext + u) ·∇(v ∗ ext + u) ·v ∗ ext ] dx + 〈f,u〉 + ∫ Γ2 g ·u dS, ν‖∇u‖22 + 1 2 ∫ Γ1 (v∗ ·n) |v∗|2 dS + 1 2 ∫ Γ2 [ (v∗ + u) ·n ] |v∗ + u|2 dS ≤ ∫ Ω [ ν∇v∗ext : ∇u + v ∗ ext ·∇v ∗ ext ·v ∗ ext + v∗ext ·∇u ·v ∗ ext + u ·∇v ∗ ext ·v ∗ ext + u ·∇u ·v∗ext ] dx + 〈f,u〉 + ∫ Γ2 g ·u dS, ν‖∇u‖22 ≤ − 1 2 ∫ Γ1 (v∗ ·n) |v∗|2 dS + 1 2 ∫ Γ2 [ (v∗ + u) ·n ] − |v ∗ + u|2 dS + ∫ Ω [ ν∇v∗ext : ∇u + v ∗ ext ·∇v ∗ ext ·v ∗ ext + v∗ext ·∇u ·v ∗ ext + u ·∇v ∗ ext ·v ∗ ext ] dx + δ‖∇u‖22 + 〈f,u〉 + ∫ Γ2 g ·u dS, ≤ − 1 2 ∫ Γ1 (v∗ ·n) |v∗|2 dS + 1 2 ζ‖v∗ + u‖23; Γ2 + ∫ Ω [ ν∇v∗ext : ∇u + v ∗ ext ·∇v ∗ ext ·v ∗ ext + v∗ext ·∇u ·v ∗ ext + u ·∇v ∗ ext ·v ∗ ext ] dx + δ‖∇u‖22 + 〈f,u〉 + ∫ Γ2 g ·u dS, ≤ − 1 2 ∫ Γ1 (v∗ ·n) |v∗|2 dS + ζ 2 c6 ‖v∗ext + u‖ 2 1,2 + ∫ Ω [ ν∇v∗ext : ∇u + v ∗ ext ·∇v ∗ ext ·v ∗ ext + v∗ext ·∇u ·v ∗ ext + u ·∇v ∗ ext ·v ∗ ext ] dx + δ‖∇u‖22 + 〈f,u〉 + ∫ Γ2 g ·u dS, where c6 = c6(Ω). Writing only the terms with second powers of u, which are decisive for the estimates, and involving all other terms to a generic constant c, we obtain ν‖∇u‖22 ≤ δ‖∇u‖ 2 2 + c7 ζ‖∇u‖ 2 2 + c, (25) where c7 = c7(Ω). As δ > 0 can be chosen to be arbi- trarily small, we observe that these inequalities yield an a priori estimate of ‖∇u‖2 in terms of v∗, f and g, provided that ζ > 0 is so small that c7ζ < ν. Obvi- ously, in this case one also obtains an a priori estimate of ‖v‖1,2 ≡‖v∗ext + u‖1,2. Under the aforementioned condition on ζ, one can prove the existence of a weak solution v ∈ Kc of the variational inequality (24), ap- plying the procedure sketched at the beginning of this section. (See also [14] for the construction of appropri- ate approximations and the detailed derivation of the estimates on the level of approximations. However, the convex set, used in paper [14], differs from Kc used here.) Thus, we can formulate the theorem: Theorem 2. Let functions v∗ ∈ W 1/2,2(Γ1) (satisfy- ing condition (?)), f ∈ W−1,2(Ω) and g ∈ L4/3(Γ2) be given. Let number ζ be so small that c7ζ < ν. Then there exists v ∈ Kc, such that the variational inequality (24) is satisfied for all w ∈ Kc. Recall that ζ is used in the definition of the convex set Kc, see (12) and (13). The smaller is ζ, the smaller is Kc and the narrower space is left for possible reverse flows on Γ2. 5. Conclusion The paper provides a mathematical model of flows through a channel with an artificial boundary condi- tion (5) on the outflow. Both unsteady and steady cases are considered. The core of the model is the variational inequalities (16) (in the unsteady case) and (24) (in the steady case). Solutions are sought in an appropriate closed convex subsets of relevant func- tion spaces, defined by means of restrictions, imposed on possible reverse flows on the outflow. The restrict- ing conditions bound the kinetic energy, brought back to Ω through Γ2 by the reverse flows. Consequently, they enable one to derive energy–type a priori esti- mates. Then, applying a relatively standard technique (based e.g. on construction of appropriate approxima- tions or some of the fixed point theorems), one can come to the conclusion on the existence of solutions. This confirms the sense of the used model and asso- ciated variational inequalities, in contrast to models based just on equations, where the existence of weak or strong solutions is generally an open problem. Except for the discussion on various boundary con- ditions of the “do nothing” type (see paragraphs 2.2 and 2.3) and some a posteriori properties of solutions (paragraph 3.5), we present a detailed description of a priori estimates of solutions. These estimates clarify, on the formal level, how the information that the solu- tions belong to L∞(0,T ; L2(Ω)) ∩L2(0,T ; W 1,2(Ω)) (in the unsteady case) or W 1,2(Ω) (in the steady case) directly follows from the used variational inequalities, regardless of other technicalities, connected e.g. with possible approximations. Analogous estimates have been obtained in a completely different and much 97 Stanislav Kračmar, Jiří Neustupa Acta Polytechnica more technical way (i.e. at first on the level of approx- imations and then considering an appropriate limit transition) in papers [14] and [15]. However, it must be noted that while the convex set, corresponding to our Kct, is defined in a rather artificial way in [14] and [15], our Kct has a good physical sense. Naturally, the change of set Kct requires a new technique in the derivation of approximations. We do not present any numerical justification of our model. 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Manuscipta Mathematica 58:47–65, 1987. doi:10.1007/BF01169082. 98 http://dx.doi.org/10.1007/s00013-012-0387-x http://dx.doi.org/10.1051/m2an/1996300708151 http://dx.doi.org/10.1002/mma.755 http://dx.doi.org/10.1007/s00021-013-0135-4 http://dx.doi.org/10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y http://dx.doi.org/10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y http://dx.doi.org/10.1016/j.amc.2015.05.066 http://dx.doi.org/10.1108/HFF-03-2016-0104 http://dx.doi.org/10.1023/A:1006185601807 http://dx.doi.org/10.4208/jcm.1405-m4347 http://dx.doi.org/10.1007/s10492-011-0016-1 http://dx.doi.org/10.1016/S0362-546X(01)00534-X http://dx.doi.org/10.1002/mana.201700228 http://dx.doi.org/10.1002/mana.200310167 http://dx.doi.org/10.1007/978-3-642-65161-8 http://dx.doi.org/10.1137/1.9781611971088.BM http://dx.doi.org/10.1007/BF01169082 Acta Polytechnica 61(SI):89–98, 2021 1 Introduction 1.1 The considered initial–boundary value problem 1.2 On some previous related existential results 2 Several boundary conditions of the ``do nothing'' type 2.1 Three equivalent forms of the dynamic stress tensor in equation (1) 2.2 Variational formulations of the initial–boundary value problem 2.3 The momentum equation with the Bernoulli pressure 2.4 Which artificial boundary condition is the best? 3 The Navier–Stokes inequality – the non-steady case 3.1 Notation 3.2 A formal derivation of the variational inequality 3.3 Definition of the initial–boundary value problem (P) 3.4 The principle of the proof and a priori estimates 3.5 Remark 4 The Navier–Stokes inequality – the steady case 5 Conclusion Acknowledgements References