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Applied Mathematics and Mechanics

, Volume 38, Issue 1, pp 137–154 | Cite as

Interaction between compressibility and particulate suspension on peristaltically driven flow in planar channel

  • I. M. Eldesoky
  • S. I. AbdelsalamEmail author
  • R. M. Abumandour
  • M. H. Kamel
  • K. Vafai
Article

Abstract

The peristaltic pumping of a viscous compressible liquid mixed with rigid spherical particles of the same size in a channel is theoretically investigated. The momentum equations for the compressible flow are solved with a perturbation analysis. The analysis is carried out by duly accounting for the nonlinear convective acceleration terms for the fluid part on the wavy wall. The zeroth-order terms yield the Poiseuille flow, and the first-order terms give the Orr-Sommerfeld equation. The explicit expression for the net axial velocity is derived. The effects of the embedded parameters on the axial fluid velocity are studied through different engineering applications. The features of the flow characteristics are analyzed and discussed in detail. The obtained results are evaluated for various parameters associated with the blood flow in the blood vessels with diameters less than 5 500 mm, whereas the particle diameter has been taken to be 8 mm. This study provides a scope to evaluate the effect of the theory of two-phase flow characteristics with compressible fluid problems, and is helpful for understanding the role of engineering applications of pumping solid-fluid mixture by peristaltically driven motion.

Keywords

two-phase flow peristaltic transport compressible liquid perturbation method 

Nomenclature

2d

mean width 2d

(uf, vf)

liquid phase velocities

x

Cartesian coordinate measuring the direction of the wave propagation

y

Cartesian coordinate measuring the direction normal to the mean position of the channel walls

C

volume fraction density of the particles

(up

vp), particulate phase velocities

p

pressure

S

drag coefficient of interaction for the force exerted by one phase on the other

k*

compressibility of the liquid

a0

radius of the particle

T

absolute temperature (K)

a

amplitude

c

wave speed

Re

suspension Reynolds number

M, N

suspension parameters

Z(y)

mean-velocity perturbation function

〈Vfx

net time axial velocity of the fluid phase

fw

frequency of wave

Vfx

dimensional net axial velocity of the fluid phase

Δp

excess pressure

co

speed of sound in the oil

P

average power output generated at the acoustic source

G

shear modulus of the porous medium

D

distensibility of the channel

f

frequency of the acoustic source

Greek symbols

ρf

actual density of the materials constituting liquid

ρp

actual density of the particulate phase

λ

wavelength

(1 − C)ρf

liquid phase density

Cρp

particulate phase density

μs(C)

particle liquid mixture viscosity, i.e., effective viscosity of the suspension

ρ

constant density at the reference pressure

μ0

fluid viscosity

μ

viscosity of the liquid

η

transverse displacement of the wall

χ

compressibility number

α

wave number

ε

amplitude ratio

Chinese Library Classification

O354 

2010 Mathematics Subject Classification

76Nxx 35Q80 74F10 76T20 94A40 

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Notes

Acknowledgements

S. I. ABDELSALAM thanks the Binational Fulbright Commission in Egypt and the Council for International Exchange of Scholars in U. S.A. for the honor of the Fulbright Egyptian Scholar Award for the year (2015–2016). In addition, the first two authors declare that they contributed to this work equally.

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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • I. M. Eldesoky
    • 1
  • S. I. Abdelsalam
    • 2
    • 3
    Email author
  • R. M. Abumandour
    • 1
  • M. H. Kamel
    • 4
  • K. Vafai
    • 2
  1. 1.Department of Basic Engineering Sciences, Faculty of EngineeringMenoufia UniversityMenoufiaEgypt
  2. 2.Department of Mechanical EngineeringUniversity of CaliforniaRiversideUSA
  3. 3.Basic Science Department, Faculty of EngineeringThe British University in EgyptCairoEgypt
  4. 4.Department of Engineering Mathematics and Physics, Faculty of EngineeringCairo UniversityGizaEgypt

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