Oscillations of laminar ow velocity in a channel induced by harmonic perturbation of mass injection velocity through the permeable wall Andrei Kulikovsky1a

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Oscillations of laminar ﬂow velocity in a channel induced by harmonic

perturbation of mass injection velocity through the permeable wall

Andrei Kulikovsky1, a)

Forschungszentrum J¨ulich GmbH

Theory and Computation of Energy Materials (IEK–13)

Institute of Energy and Climate Research,

D–52425 J¨ulich, Germany b)

(Dated: 14 February 2023)

Transient 2D Navier–Stokes equations for the laminar ﬂow of incompressible ﬂuid in a channel with permeable wall

are reduced to a single transient one–dimensional equation for the transversal proﬁle of longitudinal ﬂow velocity.

Small–amplitude harmonic perturbation of injection velocity induces oscillations of longitudinal velocity with the

peak at the walls. The peak amplitude increases with the distance along the channel. The eﬀect of increasing

amplitude is explained by formation of linearly growing along the channel perturbation of pressure gradient. With

the frequency growth, the peaks move toward to the walls.

Keywords: Laminar oscillating ﬂow in the channel, permeable wall, Berman’s model

I. INTRODUCTION

Laminar ﬂow of incompressible ﬂuid in channels with per-

meable wall(s) is of interest for ultraﬁltration applications1

and fuel cells2. In 1953, Berman 3reduced 2D problem for

the ﬂow between parallel permeable walls with constant ve-

locity of injection to a single ODE for the transversal shape

of longitudinal velocity and provided an elegant asymp-

totic solution. Later, Berman’s approach has been used

to solve the problem of a ﬂow in pipes and ducts with per-

meable walls and variable along the pipe/duct velocity of

suction/injection (Terrill and Thomas 4, Terrill 5, Kosinski

et al. 6, Granger et al. 7, Karode 8). So far, however, the

transient eﬀects due to time–dependent injection velocity

have not been considered.

In 1929, Richardson and Tyler 9reported measurements

of oscillating ﬂow in a pipe with impermeable wall induced

by harmonic longitudinal pressure gradient. They demon-

strated formation of a peak of velocity oscillations ampli-

tude close to the pipe wall. A year later, Sexl 10 developed

a model for oscillating ﬂow in a circular pipe and derived a

simple solution for the radial shape of velocity amplitude.

He showed that the oscillations amplitude is distributed

along the radius according to the Bessel function with a

peak (shoulder) positioned at the distance

l∗=rν

ω(1)

from the wall. Here νis the air kinematic viscosity and

ωis the angular frequency of perturbation. Harris et al. 11

provided accurate measurements conﬁrming the result of

Sexl. In all these works the velocity oscillations were induced

by harmonic variation of longitudinal pressure gradient.

a)Electronic mail: A.Kulikovsky@fz-juelich.de

b)Also at: Lomonosov Moscow State University, Research Com-

puting Center, 119991 Moscow, Russia

FIG. 1. Schematic of the channel. Hatched area indicates the

permeable wall through which the mass enters the channel.

Below, unsteady laminar ﬂow of incompressible ﬂuid be-

tween parallel walls with a variable and time–dependent rate

of mass injection through one of the walls is considered.

Following the Berman’s approach, the system of transient

Navier–Stokes equations is reduced to a single equation for

the transversal shape of longitudinal ﬂow velocity. The

equation is used to study the ﬂow response to a small–

amplitude harmonic perturbation of the injection velocity.

It is shown that such a perturbation induces perturbations

of the longitudinal ﬂow velocity. At high frequencies, close

to the walls a peak of oscillations amplitude forms which

increases dramatically with the distance along the channel.

The eﬀect is explained by linearly growing along the channel

perturbation amplitude of pressure gradient. The peaks are

located at the distance l∗from the nearest wall.

II. MODEL

A. Basic equations

Consider laminar ﬂow of incompressible ﬂuid between par-

allel walls separated by the distance 2hand let the upper

wall be permeable to mass injection (Figure 1). Navier–

Stokes equations for longitudinal uand transversal vﬂow

arXiv:2210.04731v3 [physics.flu-dyn] 13 Feb 2023

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OscillationsoflaminarowvelocityinachannelinducedbyharmonicperturbationofmassinjectionvelocitythroughthepermeablewallAndreiKulikovsky1,a)ForschungszentrumJulichGmbHTheoryandComputationofEnergyMaterials(IEK{13)InstituteofEnergyandClimateResearch,D{52425Julich,Germanyb)(Dated:14February2023)Transient...

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Oscillations of laminar ow velocity in a channel induced by harmonic perturbation of mass injection velocity through the permeable wall Andrei Kulikovsky1a

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