Graph model for the Stokes flow 1. Introduction ...

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Graph model for the Stokes flow M. O. Kovaleva Department of Higher Mathematics, ITMO University, Kronverkskiy pr. 49, St. Petersburg, 197101, Russian Federation E-mail: [email protected]

I. Yu. Popov Department of Higher Mathematics, ITMO University, Kronverkskiy pr. 49, St. Petersburg, 197101, Russian Federation E-mail: [email protected]

Quasi-one-dimensional model for the Stokes flow concentrated near a network is suggested. It is analogous to the quantum graph model in quantum mechanics. A version of the Harnack’s inequality for the flow is obtained. Keywords: Metric graph, Stokes flow, spectrum.

1. Introduction Quantum graph is a graph considered as a one-dimensional singular variety and equipped with a self-adjoint differential operator. Reasons for studying quantum graphs arise as simplified (due to the reduced dimensions) models in physics, chemistry and nanotechnology, when one considers propagation of waves of different nature through a quasi-one-dimensional system that looks like a thin neighborhood of a graph. The intensive investigation of quantum graphs started in 1980th ( [1, 2]). As for the state of the art in the field, see, e.g., [3]. The model has attractive features. From the one side, the method allows one to obtain explicit solutions of the model problem, from the other side, it gives one good approximation for ”realistic” physical solutions. It is a reason for seeking of new applications for the model. In this paper we suggest a version of the metric graph model for the Stokes flow. The Stokes approximation is often used in the cases when one can omit the inertial terms in the Navier-Stokes equation, i.e. when the Reynolds number is small. More precisely, our model is related with the Stokes flow through narrow channels. A channel is considered as a ”narrow” one if the ratio of its width and length is essentially less

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than one. We can note that real sizes of channels for different problems differ considerably. Particularly, the Stokes approximation is used for the description of flows through nanotubes (see, e.g., [4–6], blood flow [7–9] and geophysical flows [10, 11]. The problem becomes more complicated due to the viscosity variation and to the fact that we have to deal with a network of channels. In the present paper we try to construct a simplified quasi-one-dimensional model for the flow. There are several ways to construct quasi-one-dimensional model for the Stokes flow through a network. One way is based on the investigation of the Stokes equations (2D or 3D) in a system of narrow tubes and the consideration of the limiting procedure for the tubes widths tending to zero. This approach is similar to so-called model of thick quantum graphs (see, e.g., [12, 13]). We use another way initially suggested in the diffraction theory to describe waves concentrated near lines (see, e.g., [14, 15]). As for the procedure, it is as follows. We look for a solution in the form of formal asymptotic expansion in small parameter (channel width). By substitution of this series in the equation and collecting terms of identical orders, we get a chain of 1D problems for the coefficients of this expansion, particularly, a metric graph problem for the main term of the asymptotic expansion. Keeping in mind the background problem, one can see that the term ”Stokes graph” is appropriate for this metric graph model. 2. Model construction Consider the plane flow with varying viscosity and density, i.e. the system of coupled 2D Stokes and continuity equations. In the Cartesian coordinates ( x, y) it has the form ∂ 2 vx ∂ η ∂vx ∂ η ∂vy ∂P ∂ η ∂vx ∂ 2 vy ∂ 2 vx + η 2 + η + + − = − ρGx , (1) +2 2 η 2 ∂ x ∂ x ∂ x ∂ y ∂ y∂ x ∂ y ∂ y ∂ y ∂ x ∂ x η

∂ η ∂vx ∂ η ∂vy ∂ η ∂vy ∂ 2 vy ∂ 2 vy ∂ 2 vx ∂P + + +2 + 2 η 2 − = − ρGy , + η 2 ∂ x ∂ y ∂ x ∂ x ∂ y ∂ x ∂ x ∂ y ∂ y ∂ y ∂ y

(2)

ρvy ) ∂( ρvx ) ∂( + = 0. (3) ∂ x ∂ y Here (vx , vy ) is the flow velocity, η = η( x, y) is the viscosity, P is the pressure, ρ = ρ( x, y) is the density, (Gx , Gy ) is the gravitational force (we use here geophysical terminology, it can be another driving (external) force). Note that (3) is the continuity equation. At first, we consider the flow concentrated near the axis. More precisely, we deal with the case of smooth functions η and ρ having essentially greater values outside the strip {( x, y) : x ∈ (−∞, ∞), y ∈ (−ε, ε)} than inside it: y | ≤ ε, ηc , | y ), η2 ( y) = , ηw ηc , η( x, y) = η( x) η2 ( y ), | y | > 2ε, ηw (

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Graph model for the Stokes flow

ρ( x, y) = ρ( x) ρ2 ( y ), ρ2 ( y) =

277

y | ≤ ε, ρc , | , ρ w ρc . y ), | y| > 2ε, ρw (

y ) and ρw ( y) Here ε is a small parameter. We will assume (for simplicity) that ηw ( grows exponentially when | y| → ∞. It means that, really, the flow is only inside the strip. We introduce new coordinates x = x , y = y/ε. and seek the solutions in the form of formal asymptotic series in ε : vx = vx0 + vx1 ε + ..., vy = vy0 + vy1 ε + ..., P = P 0 + P 1 ε + .... Let us insert the series into equations (1), (2), (3) and collect terms of the same powers of ε. We get the chain of equations for the series coefficients. Terms of order ε−2 are in the Stokes equations (in new variables) only: η

∂ η ∂vx0 ∂ 2 vx0 = 0, + ∂y 2 ∂y ∂y

These simple equations give us vx0

fx (x) = η(x)

dy = u(x) η2 (y)

η

∂ 2 vy0 ∂ η ∂vy0 = 0. + ∂y 2 ∂y ∂y

dy , η2 (y)

vy0

fy (x) = η(x)

dy . η2 (y)

(4)

Here u(x), fy (x) are some functions of one variable. They should be determined later. Terms of order ε−1 are in three equations. The continuity equation gives us: ρ. To have no contradiction with (4) we conclude that vy0 = 0, i.e. the vy0 = gy (x)/ series for vy starts from another power than for vx . It is a conventional situation (see, e.g., [16]). The Stokes equations give us the following relations for the terms of this order: η

2 η

∂ 2 vy0 ∂ η ∂vy0 ∂ η ∂vx1 ∂ 2 vx1 + η + = 0, + ∂y 2 ∂y ∂y ∂y∂x ∂y ∂x

∂ 2 vy1 ∂ 2 vx0 ∂ η ∂vx0 ∂P 0 ∂ η ∂vy1 + η + − = 0. + 2 2 ∂y ∂y ∂y ∂y∂x ∂x ∂y ∂y

These equations are solvable in respect to vx1 , vy1 . The pressure term P 0 should be determined later. Consider the terms of zero order. We have the following three equations: ∂vy1 ∂ ρ 1 ∂ ρ 0 ∂v 0 vx + ρ x + vy + ρ = 0, ∂x ∂x ∂y ∂y 2 η

∂ 2 vy1 ∂ 2 vx2 ∂ η ∂vx2 ∂ η ∂vy1 ∂P 0 ∂ 2 vx0 ∂ η ∂vx0 + η + + − = − ρGx , + 2 + η ∂x2 ∂x ∂x ∂y 2 ∂y∂x ∂y ∂y ∂y ∂x ∂x η

∂ 2 vy2 ∂ η ∂vx1 ∂ η ∂vy2 ∂P 1 ∂ 2 vx1 + +2 + 2 η 2 − = − ρGy . ∂y∂x ∂x ∂y ∂y ∂y ∂y ∂y

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At present, we confine our attention by the first term of the asymptotic series for the velocity. The pressure is determined at the next stage (by taking into account 2 0 P ∂P 0 = ∂x∂y )). Here we insert vy0 = 0. The last equation the solvability condition ( ∂∂y∂x does not contain an information about v 0 . But the first two equations give us an interesting correlation. Taking into account the expressions for η, ρ, vx0 , one can see that it leads to the following equation for the function u(x): u −

ρ η ρ u = − Gx . η ρ η

(5)

Particularly, in the homogeneous case (Gx = 0) u −

η ρ u = 0. η ρ

(6)

we have the 1D Schr¨ odinger equation with a specific potential for the function u(x) corresponding to zero energy (the analogous problem for the quantum graph is known as the threshold resonance, see, e.g., [19]). The obtained correlation opens an intriguing perspective related with graphs. Let us consider a metric graph Γ with the Schr¨ odinger operator H on each edge: H =−

η ρ d2 . + 2 dx η ρ

(7)

The domain of the operator consists of functions from the Sobolev space H 2 at each edge satisfying boundary conditions at vertices (see below). Keeping in mind the above treatment of the Stokes flow, we will refer to the graph as to the Stokes graph. It is necessary to determine boundary conditions at vertices. Consider a vertex (say zero point) with n output edges. From physical conditions (see, e.g., [10]) one has ρ1 (0) = ρ2 (0) = ...ρn (0) = ρ(0) and u1 (+0) = u2 (+0) = ...un (+0) = u (0). The last condition is related with the most natural physical condition of the pressure continuity [17, 18]. Here uj (+0) is the derivative in the outgoing direction at the vertex 0. The continuity equation gives us (for this vertex under the assumption n that ρj (+0) = 0): j=1



n j=1

 −1  n   uj (+0) = − ρ(0)  ρj (+0)  u (0).

(8)

j=1

It is well-known δ -coupling condition for a quantum graph (see, e.g., [20–22]).

Discussion To show usefulness of the model we present two physically sound theorems. First, introduce two conventional definitions [17]. We consider a graph Γ and assume the Dirichlet or Neumann conditions at its boundary ( u|∂Γ = 0. or u |∂Γ = 0.) As for

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the relation with the flow properties, the Dirichlet condition corresponds to the absence of the flow through the boundary vertices and the Neumann condition, in contrast, corresponds to free flow through these vertices. Definition. We will say that the operator H preserve the sign of a solution on Γ if for any subgraph Γ0 ⊆ Γ (Γ0 = ∅) there is no solution u of the equation Hu = 0 with condition (8) at the vertices such that u(x) changes sign inside Γ0 . Note that this property is analogous to non-oscillation property from [17]. The difference is that the authors of [17] deal with continuous functions on Γ. Correspondingly, changing of sign means that there is a point where u(x) = 0. In our case the continuity is broken at vertices. Definition. Function u is a solution of the differential inequality Hu ≥ 0 if u is a solution of the equation Hu = f for some non-negative f ∈ C(Γ). Inequality analogous to the Harnack’s inequality for elliptic problem on manifold was obtained in [17] for continuous on the graph solutions of the differential inequality. We proved similar inequality for the Stokes graph, i.e. for the case of another condition at vertices corresponding to the absence of the continuity at these points (8). To prove the inequality in this case we assume additionally that the following condition takes place at any vertex: n ρj (+0) > 0. (9) j=1

Theorem. Let the operator H preserve the sign of a solution on Γ. If a solution u of the differential inequality Hu ≥ 0 is non-negative on Γ then for any locally compact (in respect to Γ ) subgraph Γ0 , Γ0 ⊂ Γ, the following inequality takes place: maxu(x) ≤ γ min u(x),

x∈Γ0

x∈Γ0

where the constant γ depends on Γ0 only. The structure of the proof is analogous to that in [17]. It will be published in another paper. We can mention that constant γ is estimated with using of the Green function of the operator H. This theorem gives us an interesting property of the Stokes flow velocity in this case. Namely, Let us consider the inhomogeneous equation (5). The solution (velocity), naturally, depends on the right hand side of the equation (i.e. on the driving forces). The theorem shows an interesting fact that the ratio of the maximal and the minimal values of the velocity (over any subgraph Γ0 , Γ0 ⊆ Γ , possibly, over the whole graph Γ if it is finite) is bounded by a constant which depends only on Γ0 and does not depend on the inhomogeneous term. The following theorem describes a critical case when 0 is an eigenvalue of the operator H.

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Theorem. Let the operator H preserve the sign of a solution on any subgraph Γ0 , Γ0 ⊂ Γ which does not coincide with Γ but do not preserve the sign of a solution on the whole Γ. Then each solution u of the differential inequality Hu ≥ 0, u|∂Γ ≥ 0, is a solution of the problem Hu = 0, u|∂Γ = 0. Spectral problem is one more source of interesting results for fluid mechanics. We will describe it in another publication. Acknowledgments This work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by Ministry of Science and Education of the Russian Federation (GOSZADANIE 2014/190, Project 14.Z50.31.0031), by grant of Russian Foundation for Basic Researches and grants of the President of Russia (state contracts 14.124.13.2045-MK and 14.124.13.1493-MK). Bibliography [1] N. Gerasimenko, B. Pavlov, Theor. Math. Phys. 74 230 (1988). [2] P. Exner, P. Seba, Rep. Math. Phys. 28 7 (1989). [3] P. Exner, J.P. Keating, P. Kuchment, T. Sunada, A. Teplyaev (eds.), Analysis on Graphs and Its Applications. Proc. Symp. Pure Math. 77 (Amer. Math. Soc., Providence, RI, 2008). [4] M. B. Belonenko, S. A. Chivilikhin, V. V. Gusarov, I. Yu. Popov, O. A. Rodygina, Europhys. Lett. 101 (6) 66001 (2013). [5] I. Yu. Popov, Phys. Scripta 83 045601/1-3 (2011). [6] T.-D. Li, J. Gao, R. Szoszkeiwicz, U. Landman, E. Riedo, Phys. Rev. B. 75 115415/1-6 (2007). [7] M. Sharan, A. S. Popel, Biorheology 38 415 (2001). [8] T. W. Secomb, A. V. El-Kareh, Biorheology 32 169 (1995). [9] V. P. Srivastava, Int. J. Applications and Appl. Math. 2 (1) 51 (2007). [10] T. Gerya, Introduction to Numerical Geodynamic Modelling (Cambridge University Press, Cambridge, 2010). [11] A. Ismail-Zadeh, P. Tackley, Computational methods in Geodynamics (Cambridge University Press, Cambridge, 2010). [12] O. Post, J. Phys. A: Math. Gen. 38 4917 (2005). [13] S. Molchanov, B. Vainberg, Comm. Math.Phys. 273 533 (2007). [14] V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short-wavelength Diffraction Theory (Alpha Science Series on Wave Phenomena) (Alpha Science Intl Ltd., Oxford, 2009). [15] A. I. Popov, V. M. Babich, J. Math. Sci. 173 (3) 243 (2011). [16] M. Van Dyke, Perturbation Methods in Fluid Mechanics (Parabolic Press, Stanford, 1975).

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[17] Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadev, A. V. Borovskikh, K. P. Lazarev, S. A. Shabrov, Differential Equations on Geometrical Graphs (Moscow, Fizmatlit, 2004). [18] A. V. Gudzovskii, Doklady RAS 358 (6) 765 (1998). [19] C. Cacciapuoti, P. Exner, J. Phys. A. 40 F511 (2007). [20] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, with an appendix by P. Exner, Solvable Models in Quantum Mechanics:Second Edition (AMS Chelsea Publishing. Providence, R.I. 2005). [21] P. Kuchment, Waves Random Media 14 S107 (2004). [22] A. Mikhailova, B. Pavlov, I. Popov, T. Rudakova, A. Yafyasov, Math. Nachr. 235 101 (2002).

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