1 Introduction
###### Abstract

We consider incompressible flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier-Stokes equations in the limit when both the amplitude of vibrations and the thickness of the Stokes layer are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions, for the averaged flow. In the leading order, the averaged flow is described by the stationary Navier-Stokes equations with an additional term which contains the leading-order Stokes drift velocity. In a slightly different context (for a flow induced by an oscillating conservative body force), the same equations had been derived earlier by Riley [3]. The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and travelling plane waves. In particular, in the case of waves travelling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers.

Steady streaming between two vibrating planes

at high Reynolds numbers

Konstantin Ilin 111Department of Mathematics, University of York, Heslington, York, YO10 5DD, U.K.. Electronic mail: konstantin.ilin@york.ac.uk and Andrey Morgulis 222Department of Mathematics, Mechanics and Computer Science, The Southern Federal University, Rostov-on-Don, and South Mathematical Institute, Vladikavkaz Center of RAS, Vladikavkaz, Russian Federation. Electronic mail: amor@math.rsu.ru

## 1 Introduction

In this paper we study oscillating flows of a viscous incompressible fluid between two solid walls produced by the transverse vibrations of the walls. It is well-known that high-frequency oscillations of the boundary of a domain occupied by a viscous fluid generate not only an oscillating flow but also a (relatively) weak steady flow, which is usually called the steady streaming (see, e.g., the review papers [1] and [2, 3]). Recently such flows attracted considerable attention in the context of application of steady streaming to micro-mixing [4, 5, 6] and to drag reduction in channel flows [7].

The basic parameters of the problem are the inverse Strouhal number and the streaming Reynolds number , defined as

 ϵ=V∗0ωd,Rs=V∗20ων∗ (1)

where is the amplitude of the velocity of the oscillating walls, is the mean distance between the walls, is the angular frequency of the vibrations and is the kinematic viscosity of the fluid. Parameter measures the ratio of the amplitude of the displacement of the vibrating wall to the mean distance between the walls and is assumed to be small: . Note that the relation between the standard Reynolds number based on and and the streaming Reynolds number is given by , so that corresponds to when is small. The present paper deals with the flow regimes with . This implies that the amplitude of the wall displacement is of the same order of magnitude as the thickness of the Stokes layer.

Steady streaming at (and even for large ) induced by translational oscillations of a rigid body in a viscous fluid had been studied by many authors (see [8, 9, 10, 11, 12, 13, 14]). In the case of a body in an infinite fluid, the problem can be reduced to an equivalent problem about a fixed body placed in an oscillating flow, which considerably simplifies the analysis. A related problem of steady streaming and mass transport produced by water waves had been treated in [17, 18]) where flow regimes with large were considered. Previous studies of flows produced by transverse oscillations of solid walls had been mostly focused on the problem of peristaltic pumping in channels and pipes under the assumption of low Reynolds numbers () and small amplitude-to-wavelength ratio (see, e.g., [15, 16, 6]). Although there are a few paper where the case of large Reynolds numbers () had been considered [4, 5, 7], these studies still correspond to . As far as we know, the flow regimes with have not been treated before.

The aim of this paper is to construct an asymptotic expansion of the solution of the Navier-Stokes equation in the limit and . The procedure includes the derivation of the equations and boundary conditions for a steady component of the flow (steady streaming) that persists everywhere except thin layers near the vibrating walls (whose thickness is of the same order as the amplitude of vibrations). Since the thickness of the Stokes layer and the amplitude of the displacements of the walls are of the same order of magnitude, the boundary conditions on the moving walls cannot be transferred to the fixed mean positions of the walls. This is what makes the problem difficult.

To obtain an asymptotic expansion, we employ the Vishik-Lyusternik method (see, e.g., [19, 20]) rather than the standard method of matched asymptotic expansions. In comparison with the latter, the Vishik-Lyusternik method does not require the procedure of matching the inner and outer expansions and the boundary layer part of the expansion satisfies the condition of decay at infinity (in boundary layer variable) in all orders of the expansion (this is not so in the method of matched asymptotic expansions where the boundary layer part usually does not decay and may even grow at infinity). The Vishik-Lyusternik method had been used to study viscous boundary layers at a fixed impermeable boundary by Chudov [21]. Recently, it has been applied to viscous boundary layers in high Reynolds number flows through a fixed domain with an inlet and an outlet [22], to viscous flows in a half-plane produced by tangential vibrations on its boundary [23] and to the steady streaming between two cylinders [24]. A similar technique had been used in [25] to construct an asymptotic expansion in a problem of vibrational convection.

The outline of the paper is as follows. In Section 2, we formulate the mathematical problem. In Section 3, the asymptotic equations and boundary conditions are derived. Section 4 outlines the construction of the asymptotic solution. In Section 5, we consider examples of the steady streaming induced by vibrations of the walls in the form of standing or travelling waves. Finally, discussion of results and conclusions are presented in Section 6.

## 2 Formulation of the problem

We consider a three-dimensional viscous incompressible flow between two parallel walls produced by normal, periodic (in time) vibrations of the walls. Let be the Cartesian coordinates, the time, the velocity of the fluid, the pressure, the fluid density, the kinematic viscosity and the mean distance between the walls. We introduce the non-dimensional quantities

where and are the angular frequency and the amplitude of the vibrations. In these variables, the Navier-Stokes equations take the form

 vτ+ϵ(v⋅∇)v=−∇p+ϵ2ν∇v,∇⋅v=0. (2.1)

where

Parameter is the ratio of the amplitude of vibrations to the mean distance between the walls, where is the Reynolds number based on the amplitudes of the velocity and the normal displacement of the vibrating boundary (the streaming Reynolds number). The walls are described by the equations

 z=ϵf(x,y,τ)andz=1+ϵg(x,y,τ), (2.2)

where and are given functions that are -periodic in and have zero mean value, i.e.

 ¯f(x,y)≡12π2π∫0f(x,y,τ)dτ=0,¯g(x,y)≡12π2π∫0g(x,y,τ)dτ=0. (2.3)

We also assume that and are periodic both in and in with periods and , respectively. Boundary conditions for the velocity at the walls are the standard no-slip conditions:

 v∣∣z=ϵf=fτ(x,y,τ)ez,v∣∣z=1+ϵg=gτ(x,y,τ)ez. (2.4)

The incompressibility of the fluid implies that functions and must satisfy an additional condition which, in the periodic case, is given by

 Lx∫0Ly∫0f(x,y,τ)dxdy=Lx∫0Ly∫0g(x,y,τ)dxdy. (2.5)

The non-dimensional equations and boundary conditions depend only on two parameters and . In what follows we are interested in the asymptotic behaviour of periodic solutions of Eqs. (2.1), (2.4) in the limit when and . In other words, we consider the situation where the amplitude of the vibration is small and of the same order as the thickness of the Stokes layer.

It is convenient to separate vertical and horizontal components of the velocity as follows:

 v=u+wez,u=uex+vey,

i.e. is the projection of the velocity onto the horizontal plane (the plane). We will also use the notation: and .

We seek a solution of (2.1), (2.4) in the form

 u=ur(r,z,τ,ϵ)+ua(r,ξ,τ,ϵ)+ub(r,η,τ,ϵ), w=wr(r,z,τ,ϵ)+ϵwa(r,ξ,τ,ϵ)+ϵwb(r,η,τ,ϵ), p=pr(r,z,τ,ϵ)+pa(r,ξ,τ,ϵ)+pb(r,η,τ,ϵ). (2.6)

Here and are the boundary layer variables; all functions are assumed to be periodic in , and with periods , and , respectively. Functions , , represent a regular expansion of the solution in power series in (an outer solution), and and correspond to boundary layer corrections (inner solutions) to this regular expansion. We assume that the boundary layer parts of the expansion rapidly decay outside thin boundary layers, namely:

 ua,wa,pa=o(ξ−s)asξ→∞andub,wb,pb=o(η−s)asη→∞ (2.7)

for every . In other words, we require that at a distance of order unity from the wall, the boundary layer corrections are smaller than any power of . This assumption will be verified a posteriori. We begin with the regular part of the expansion.

## 3 Asymptotic expansion

### 3.1 Regular part of the expansion

Let

 vr=vr0+ϵvr1+ϵ2vr2+…,pr=pr0+ϵpr1+ϵ2pr2+…, (3.1)

where and (). The successive approximations , () satisfy the equations:

 ∂τvrk=−∇prk+Frk(x,τ),∇⋅vr0=0, (3.2)

where , and

 Frk(x,τ)=−k−1∑j=0(vrj⋅∇)vrk−j−1+ν∇2vrk−2(k=2,3…).

In what follows, we will use the following notation: for any -periodic function ,

 f(τ)=¯f+~f(τ),¯f=12π2π∫0f(τ)dτ (3.3)

where is the mean value of and, by definition, is the oscillating part of that has zero mean value.

#### 3.1.1 Leading-order equations

Consider Eqs. (3.2) for . We seek a solution which is periodic in . It can be written as

 vr0=¯vr0+~v0,~vr0=∇ϕ0 (3.4)

where has zero mean value and is the solution of the boundary value problem

 ∇2ϕ0=0,ϕ0(x+Lx,y,z)=ϕ0(x,y,z),ϕ0(x,y+Ly,z)=ϕ0(x,y,z), ϕ0z∣∣z=0=fτ(x,y,τ),ϕ0z∣∣z=1=gτ(x,y,τ). (3.5)

Boundary conditions for at and will be justified later. On averaging Eqs. (3.2) for , we obtain

 (¯vr0⋅∇)¯vr0+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(~vr0⋅∇)~vr0=−∇¯pr1,∇⋅¯vr0=0. (3.6)

Since, according to (3.4), is irrotational, we have: . Therefore, we can rewrite (3.6) as

 (¯vr0⋅∇)¯vr0=−∇Π0,∇⋅¯vr0=0, (3.7)

where . Equations (3.7) represent the time-independent Euler equations that describe steady flows of an inviscid incompressible fluid. It will be shown later that boundary conditions for are

 ¯vr0∣∣z=0=0,¯vr0∣∣z=1=0. (3.8)

Thus, is a steady solution of the Euler equations (3.7), which is periodic in and and subject to zero boundary condition at and . We choose zero solution:

 ¯vr0≡0 (3.9)

(evidently, it satisfies the Euler equations as well as the boundary conditions). This choice implies that there is no steady streaming in the leading order of the expansion.

#### 3.1.2 First-order equations

Separating the oscillatory part of Eqs. (3.2) for , we find that

 ∂τ~vr1=−∇~pr1−˜(vr0⋅∇)vr0,∇⋅~vr1=0, (3.10)

Using (3.9) and the fact that is irrotational, (3.10) can be written in the form

 ∂τ~vr1=−∇Q1,∇⋅~vr1=0, (3.11)

where . It follows from (3.11) that

 vr1=¯vr1+~v1,~vr1=∇ϕ1 (3.12)

where has zero mean value and is the solution of the problem

 ∇2ϕ1=0,ϕ1(x+Lx,y,z)=ϕ1(x,y,z),ϕ1(x,y+Ly,z)=ϕ1(x,y,z), (3.13)

where functions and will be defined later.

On averaging equation for and using (3.4) and (3.9), we find that

 (¯vr1⋅∇)¯vr1=−∇Π∗3+ν∇2¯vr1−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~ωr2×∇ϕ0,∇⋅¯vr1=0. (3.14)

where and . It is clear from Eq. (3.14) that in order to obtain a closed equation for , we need to find .

#### 3.1.3 Second-order equations

Consider Eqs. (3.2) for . Averaging yields

 0=−∇¯pr2−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(vr0⋅∇)vr1−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(vr0⋅∇)vr1,∇⋅¯vr2=0.

In view of (3.4), (3.9) and (3.12), the first of these reduces to the equation that can be integrated to obtain

 ¯pr2=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∇ϕ0⋅∇ϕ1+const. (3.15)

The oscillatory part of (3.2) for gives us the equations for :

 ~vr2τ=−∇Q2−(~vr0⋅∇)¯vr1−(¯vr1⋅∇)~vr0,∇⋅~vr2=0, (3.16)

where , and we have used (3.4), (3.9) and (3.12). Taking curl of the first equation and using (3.4), we obtain

 ~ωr2τ=[~vr0,¯ωr1] (3.17)

where and where for any divergence-free vector fields and . Now let be such that

 ζτ=~vr0,¯ζ=0. (3.18)

Then, it follows from (3.17) that

 ~ωr2=[ζ,¯ωr1]. (3.19)

#### 3.1.4 A closed system of equations for ¯vr1

In view of (3.18) and (3.19), we have

 −¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~ωr2×∇ϕ0=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ζτ×~ωr2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ζτ×[ζ,¯ωr1]=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ζ×[ζτ,¯ωr1]=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ζ×[¯ωr1,ζτ], (3.20)

where we have used the fact that for any periodic functions and . It follows from (3.20) that . With the help of the identity (which is valid for any divergence-free , and ), this can be simplified to

 −¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~ωr2×∇ϕ0=12¯¯¯¯¯¯¯¯¯¯¯¯¯[ζτ,ζ]×¯ωr1+∇(¯ωr1⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(ζτ×ζ)).

Finally, substituting the last formula into Eq. (3.14), we obtain

 (¯vr1⋅∇)¯vr1=−∇Π3+ν∇2¯vr1+V×¯ωr1. (3.21)

where and

 V=12¯¯¯¯¯¯¯¯¯¯¯¯¯[ζτ,ζ]. (3.22)

When is zero, (3.21) coincides with the stationary Navier-Stokes equations. In the two-dimensional case, (3.22) reduces to the equation derived earlier in [3]. It was also noticed in [3] that represents the Stokes drift velocity of fluid particles.

### 3.2 Boundary layer equations

In this subsection we will obtain asymptotic equations that describe boundary layers near the vibrating walls.

Boundary layer at the bottom wall. To derive boundary layer equations near the bottom wall, we ignore , and , because they are supposed to be small relative to any power of everywhere except a thin boundary layer near , and assume that

 u=ur0+ua0+ϵ(ur1+ua1)+…,w=wr0+ϵ(wr1+wa0)+…,p=pr0+pa0+ϵ(pr1+pa1)+… (3.23)

We insert (3.23) into Eq. (2.1) and take into account that , , () satisfy the equations (3.2). Then we make the change of variables , expand every function of in Taylor’s series at and collect terms of the equal powers in . This produces the following sequence of equations:

 ∂τuak+wr0∣∣z=0∂ξuak+∇∥pak−ν∂2ξuak=Fak,∂ξpak=Gak,∇∥⋅uak+∂ξwak=0 (3.24)

for Here . Functions and are defined in terms of lower order approximations. In particular, , , and

 (3.25)

Boundary layer at the upper wall. A similar procedure leads to the following equations of the boundary layer near the upper wall:

 ∂τubk−wr0∣∣z=1∂ηubk+∇∥pbk−ν∂2ηubk=Fbk,∂ηpbk=Gbk,∇∥⋅ubk−∂ηwbk=0 (3.26)

for , where , , and

 (3.27)

In accordance with (2.7), we require that

 uak,wak,pak=o(ξ−s)asξ→∞andubk,wbk,pbk=o(η−s)asη→∞ (3.28)

for every and for each .

### 3.3 Boundary conditions

Now we substitute (3.23) in (2.4), expand all functions corresponding to the outer flow in Taylor’s series at and collect terms of equal powers in . This leads to the following boundary conditions at the bottom wall:

 ur0∣∣z=0+ua0∣∣ξ=f=0,wr0∣∣z=0=fτ; (3.29) (3.30) wrk∣∣z=0=−wak−1∣∣ξ=f−k∑n=1fnn!∂nzwrk−n∣∣z=0(k=1,2,…). (3.31)

A similar procedure yields the following boundary conditions at the upper wall:

 ur0∣∣z=1+ub0∣∣η=−g=0,wr0∣∣z=1=gτ; (3.32) (3.33) wrk∣∣z=1=−wbk−1∣∣η=−g−k∑n=1gnn!∂nzwrk−n∣∣z=1(k=1,2,…). (3.34)

Note that the boundary conditions for at and in problem (3.5) follow directly from (3.29) and (3.32).

## 4 Construction of the asymptotic solution

### 4.1 Leading order equations

Oscillatory outer flow. In the leading order, the oscillatory outer flow is irrotational, and the velocity potential is a solution the boundary value problem (3.5) (which is unique up to a constant).

Boundary layer at the bottom wall. In the leading order, the boundary layer is described by Eqs. (3.24) with . The decay condition (3.28) for and the second equation (3.24) imply that . Hence, the equation for reduces to

 ua0τ+fτua0ξ=νua0ξξ. (4.1)

Here we have used boundary condition (3.29) for . Let

 s=ξ−f(r,τ)andua0(r,ξ,τ)=Ua0(r,s(r,ξ,τ),τ).

Then Eq. (4.1) simplifies to the standard heat equation for :

 Ua0τ=νUa0ss. (4.2)

Boundary conditions for are the decay condition at infinity and the condition

 Ua0∣∣s=0=−ur0∣∣z=0, (4.3)

which follows from (3.29). Averaging Eq. (4.2), we obtain the equation . The only solution of this equation that satisfies the decay condition at infinity is zero solution . [Note that this doesn’t mean that , because is averaged for fixed , and - for fixed . This fact, however, is inessential for what follows.]

A periodic (in ) solution of Eq. (4.2) satisfying the decay condition at infinity and boundary condition (4.3) and having zero mean can be found by standard methods (see, e.g., [26]). Once is found, the normal velocity is determined from the continuity equation (the third equation (3.24)) that can be written as

 ∇∥⋅Ua0−Ua0s⋅∇∥f+Wa0s=0 (4.4)

where function is defined by the relation . Integration of Eq. (4.4) in yields

 Wa0(r,s,τ)=Ua0⋅∇∥f+∇∥⋅∞∫sUa0(r,s′,τ)ds′, (4.5)

where the constant of integration was chosen so as to guarantee that as .

Boundary condition (3.31) for is

 (4.6)

This means that function in the boundary value problem (3.13) is given by

 a1(x,τ)=−˜fwr0z∣∣z=0−~Wa0∣∣s=0. (4.7)

Boundary layer at the upper wall. Exactly the same analysis as in the case of the bottom wall leads to the heat equation

 Ub0τ=νUb0qq, (4.8)

where and are defined as

 q=η+g(r,τ)andub0(r,η,τ)=Ub0(r,q(r,η,τ),τ).

Boundary conditions for are the decay condition at infinity and the condition

 Ub0∣∣q=0=−ur0∣∣z=1, (4.9)

that follows from (3.32). Equation (4.8) has a consequence that . This, together with the condition of decay at infinity (3.28), implies that .

Again, an oscillatory solution of Eq. (4.8) can be found by standard methods. The normal velocity is determined from the continuity equation (the third equation (3.26)):

 Wb0(r,q,τ)=Ub0⋅∇∥g−∇∥⋅∞∫qUb0(r,q′,τ)dq′, (4.10)

where function is defined by . and where the constant of integration was chosen so as to guarantee that as .

Boundary condition (3.34) for can be written as

 (4.11)

This means that function in the boundary value problem (3.13) is given by

 b1(x,τ)=−˜gwr0z∣∣z=1−~Wb0∣∣q=0. (4.12)

Averaged outer flow. Averaging boundary conditions (3.29) and (3.32), we find that

 ¯ur0∣∣z=0=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ua0∣∣ξ=f=−¯Ua0∣∣s=0=0,¯wr0∣∣z=0=¯¯¯¯¯fτ=0, ¯ur0∣∣z=1=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ub0∣∣η=−g=−¯Ub0∣∣q=0=0,¯wr0∣∣z=1=¯¯¯¯¯gτ=0.

This justifies our earlier assumption (3.8) and our conclusion (3.9) that there is no steady streaming in the leading order of the expansion.

### 4.2 First-order equations

Oscillatory outer flow. The oscillatory part of the (first-order) outer flow is determined by the boundary value problem (3.13) for the Laplace equation. Since functions and , which appear in the boundary conditions for , are now known, problem (3.13) can be solved, thus giving us .

Boundary layer at the bottom wall. Consider now the first-order boundary layer equations (Eqs. (3.24) for ). Again, the condition of decay at infinity for and the second equation (3.24) imply that , so that the first equation (3.24) simplifies to

 ua1τ+fτua1ξ−νua1ξξ=Fa1. (4.13)

We are looking for a solution of (4.13) which satisfies the decay condition at infinity (3.28) and boundary condition (3.30) for . To find it, we again employ variable and rewrite (4.13) and (3.30) in the form

 Ua1τ−νUa1ss=Ha1, (4.14) Ua1∣∣s=0=−ur1∣∣z=0−fur0z∣∣z=0, (4.15)

where is such that and where

 Ha1 = −(wr1+(s+f)wr0z)∣∣z=0Ua0s−(ur0⋅∇)∣∣z=0Ua0+(ur0⋅∇f)∣∣z=0Ua0s (4.16) −(Ua0⋅∇)ur0∣∣z=0−(Ua0⋅∇)Ua0+(Ua0⋅∇f)Ua0s−Wa0Ua0s.

Averaging (4.14), we find that

 ν¯Ua1ss=−¯Ha1(r,s). (4.17)

We integrate this equation two times, choosing constants of integration so as to satisfy the condition of decay at infinity in variable . As a result, we have

 ¯Ua1=−1ν∞∫s∞∫s′¯Ha1(r,s′′)ds′′ds′. (4.18)

Note that (4.18) is the unique solution of Eq. (4.17) that decays at infinity. The oscillatory part of can also be found from Eqs. (4.14) and (4.15), but we will not do it here as we are only interested in the averaged part of the flow.

Boundary layer at the upper wall. A similar analysis leads to the equations

 Ub1τ−νUb1qq=Hb1, (4.19) Ub1∣∣q=0=−ur1∣∣z=1−gur0z∣∣z=1, (4.20)

where is defined by the relation and where

 Hb1 = (wr1−(q−g)wr0z)∣∣z=1Ub0q−(ur0⋅∇)∣∣z=1Ub0−(ur0⋅∇g)∣∣z=1Ub0q (4.21) −(Ub0⋅∇)ur0∣∣z=1−(Ub0⋅∇)Ub0−(Ub0⋅∇g)Ub0q+Wb0Ub0q.

Averaging (4.19), we obtain

 ν¯Ub1qq=−¯Hb1(r,q). (4.22)

Integration of (4.22) yields

 ¯Ub1=−1ν∞∫q∞∫q′¯Hb1(r,q′′)dq′′dq′. (4.23)

Here again the constants of integration are chosen so as to satisfy the condition of decay at infinity.

Averaged outer flow. Averaging the boundary conditions (3.30) and (3.33) for , we obtain

 ¯ur1∣∣z=0=−¯Ua1∣∣s=