The dynamics of viscous incompressible flow inside elastic pipe

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The dynamics of viscous incompressible flow inside elastic pipe

Evdokimova E.V.1

Рассмотрены проблемы движения вязкой несжимаемой жидкости в упругой цилиндрической оболочке при воздействии ряда факторов. При рассмотрении данных проблем необходимо рассматривать систему «жидкость-упругое тело» при различных источниках движения. Для исследования динамических характеристик таких систем, необходимо решить связанную задачу динамики упругой оболочки и вязкой несжимаемой жидкости. Фактором, упрощающим данную задачу, как в теории смазки, является малость радиуса оболочки по сравнению с ее длиной. В рассматриваемой постановке задачи также водится предположение о малости упругих перемещений оболочки по сравнению с ее радиусом, но не с толщиной оболочки. В этих условиях уравнения гидродинамики становятся линейными. При этом уравнения динамики упругих элементов могут быть как линейными, так и нелинейными. Если нет возможности найти точное решение, применяется метод итерации для решения уравнений в рамках гидродинамической теории смазки.

Ключевые слова: вязкая жидкость, упругая оболочка, колебания, волны

The dynamics of viscous incompressible flow inside elastic pipe

Evdokimova E.V.1, Mogilevich L.I.2, Popov V.S.3

1Yuri Gagarin State Technical University of Saratov

2Yuri Gagarin State Technical University of Saratov

3Yuri Gagarin State Technical University of Saratov

The problems of viscous incompressible liquid movement in an elastic cylinder shell under the impact of some factors are considered. All these problems contain the systems «liquid - elastic body» movement sources. To reveal dynamic characteristic of these systems, it is necessary to solve the connected problems of an elastic shell and viscous incompressible liquid dynamics. The factor, making the problem simpler, is a small size of a shell radius in comparison with its length, as in lubrication theory. The problems under consideration contain this supposition about a small size of the shell deflection in comparison with a shell radius, but not with a shell thickness. Under these conditions hydrodynamics equations become linear. The elastic elements dynamics equations remain non-liner, if they were such initially. If it is not possible to find the exact solution, the iteration method for solving the lubrication theory hydrodynamics equations is applied.

Keywords: viscous liquid, elastic shell, oscillations, waves

fluid shell motion

Introduction

The problem solution can be defined in terms of all problem parameters time dependence in case of shells linear theory and time harmonic impacts on the system. It becomes possible due to the liquid viscosity, causing oscillations damping and solution quick time burning out thanks to initial conditions and going out to stationary oscillations. These conditions allow finding fluid mechanics linear equations exact solutions. They are written down in the form of lubrication theory equations, wherein a local inertia member for any lubrication Reynolds number meaning within laminar movement being taken into consideration. In the case mentioned above, a normal liquid stress (pressure) on elastic body surface is significantly larger then a shear one and the latter is usually neglected [1-7].

Let us consider the possibility of obtaining approximated solutions of fluid dynamics equations by means of iteration method, neglecting inertia local member at the first step and considering it at the second one. It is possible to prove that iteration method convergence under the condition that Reynolds oscillations number is less than 1 [1]. This approach makes it possible to refuse from time-harmonic law of all liquid and elastic elements parameters. It comes possible to solve hydroelastic problem both for shells dynamics non-linear equations and for non- time-harmonic laws of movement sources, as well.

Setting the Problem.Fig. 1 presents an elastic cylinder shell 2 with viscous uncompressible liquid 3 inside [5]. The inner radius of shell R1 is significantly less than its length l that is and small parameter is introduced. The elastic shell deflection is significantly less than thickness of liquid 3 inside that is , и , of order .

Let us consider laminar flow in the elastic cylinder shell. Liquid dynamics equations in axis-symmetric case take the form of [8]

(1)

Here are the liquid velocity vector projections on longitudinal axis and normal to this axis in cylinder coordinate system , x is the longitudinal coordinate along central shell axis, r is the distance from central axis of shell with inner radius , and length l.

Fig. 1

The boundary conditions are no-slip ones and the conditions of velocity limitation at the central axis of channel, as well as the pressure conditions at the channel edges are

, at ,2)

, at r = 0,

atand.

The first condition for pressure can be replaced with a pressure symmetric one

atx = 0.(3)

Here U is the shell longitudinal elastic displacement, W is the shell deflection, positive to curvature center, R1 is the inner radius of shell with thickness h0 and length l.

Surface loading is defined by liquid stress

, .(4)

Let us introduce dimensionless variables and small parameters

, , , , ,

, , ,

, , .(5)

By taking into account the presence of small parameters, we obtain liquid dynamics equations (within lubrication theory, but with considering inertia local member) and boundary conditions be come

, , ,

, at , at

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at.(6)

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Considering time-harmonic dependence on pressure and liquid velocity, as well as shell elastic displacements, we find

,

. (7)

In this case

(8)

,

, , ,

, ,

, .

Here ber,bei are Kelvin functions of zero order, a stroke is meaning derivative.

We may notice, that

, at

б = 1, , , at .

We apply iteration method to problem (6) by refusing time-harmonic dependence at flow parameters, which means non- time-harmonic law of pressure , and taking with as characteristic time.

We neglect inertia local member at the first step of iteration and then get

,

.

At the second step of iteration we obtain

,

.(9)

It is possible to introduce frequency dependent coefficients б(щ) and г(щ) and consider Re to be taken of any value. In this case we get

,

. (10)

These expressions formally coincide with (7) for medium parameters time-harmonic laws changes, but allow using these formula as for non- time-harmonic law of pressure , as for non- linear shell equations.

It is noteworthy, that reference [1] presents convergence of iteration method.

The study was supported by Russian Foundation for Basic Research (RFBR) according to the Research Project No16-01-00175-a

References

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