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

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## 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

- 2
*d* mean width 2

*d*- (
*u*_{f},*v*_{f}) 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

- (
*u*_{p} *v*_{p}), 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

*a*_{0}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

- 〈V
_{fx}〉 net time axial velocity of the fluid phase

*f*_{w}frequency of wave

*V*_{fx}dimensional net axial velocity of the fluid phase

- Δ
*p* excess pressure

*c*_{o}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

*ρ*_{f}actual density of the materials constituting liquid

*ρ*_{p}actual density of the particulate phase

- λ
wavelength

- (1 −
*C*)*ρ*_{f} liquid phase density

*C*ρ_{p}particulate phase density

*μ*_{s}(*C*)particle liquid mixture viscosity, i.e., effective viscosity of the suspension

*ρ*constant density at the reference pressure

*μ*_{0}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## Preview

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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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