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Dynamics of a fluid-filled curvilinear pipeline

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Abstract

A mathematical model is presented, and numerical experiments are performed to describe the mechanics of the slow movement of a pipeline. The problem reduction algorithm to one-dimensional formulation is offered. Results of numerical experiment for the model problem are adduced. The proposed mathematical model is found to adequately describe the dynamics of known phenomena of pipes. The cross-sections of the extended curvilinear thin-walled pipeline are numerically demonstrated to experience warping, which has experimental confirmation in the literature.

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Abbreviations

v 0 :

fluid velocity vector

L :

pipe length

ϑ s0 :

fluid velocity at the inlet

R 0 :

pipe radius

v s , v θ , v R :

components of the velocity of the fluid along the coordinates s, θ, and R, respectively

A,B :

coefficients of the first fundamental form of the middle surface of the pipe

p :

pressure in the fluid

h :

pipe wall thickness

s :

arc length along the pipe axis

u, v, w :

displacements of the median surface of a pipe along the coordinates s, θ, and R, respectively

x, y, z :

Cartesian coordinates

s, θ,R :

curvilinear coordinates

g ij :

components of a metric tensor

g :

acceleration due to gravity

H i :

Lamé coefficients

k i :

main curvatures of the median sur-face

Re :

Reynolds number

E :

Young’s modulus of the pipe mate-rial

p e :

ambient pressure

p a :

atmosphere pressure

ℓ:

characteristic scale of length

μ :

viscosity

ρ :

density

Φ(ϑ s0):

specific hydraulic resistance

κ 0(s), κ(s, t):

initial and current curvatures of the axial line of the pipe

θ :

angular curvilinear coordinate

ζ:

axial dimensionless coordinate

ω:

characteristic scale of time

ν :

Poisson’s ratio

Γ:

axial line of the pipe

ρ 0(s):

initial radius of curvature of the axis of the pipe

f:

fluid stream conditions

t:

pipe conditions

e:

external medium conditions.

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Authors and Affiliations

  1. Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Khabarovsk, 680000, Russia

    V. A. Rukavishnikov & O. P. Tkachenko

Authors
  1. V. A. Rukavishnikov
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  2. O. P. Tkachenko
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Correspondence to O. P. Tkachenko.

Additional information

Citation: Rukavishnikov, V. A. and Tkachenko, O. P. Dynamics of a fluid-filled curvilinear pipeline. Applied Mathematics and Mechanics (English Edition) (2018) https://doi.org/10.1007/s10483-018-2338-9

Project supported by the Russian Science Foundation (No. 18-11-00021)

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Rukavishnikov, V.A., Tkachenko, O.P. Dynamics of a fluid-filled curvilinear pipeline. Appl. Math. Mech.-Engl. Ed. 39, 905–922 (2018). https://doi.org/10.1007/s10483-018-2338-9

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  • DOI: https://doi.org/10.1007/s10483-018-2338-9

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