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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- 2d :
-
mean width 2d
- (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 :
-
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
- 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
- 〈Vfx〉:
-
net time axial velocity of the fluid phase
- f w :
-
frequency of wave
- V fx :
-
dimensional net axial velocity of the fluid phase
- Δp :
-
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
- 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
- Cρp :
-
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
References
Latham, T. W. Fluid Motion in a Peristaltic Pump, M. Sc. dissertation, Massachusetts Institute of Technology, Cambridge (1966)
Rath, H. J. Peristaltische Stromungen, Springer, Berlin (1980)
Srivastava, L. M. and Srivastava, V. P, Peristaltic transport of blood: Casson model-II. Journal of Biomechanics 17, 821–830 (1984)
Srivastava, L. M. and Srivastava, V. P, Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube. Journal of Biomechanical Engineering 18, 247–253 (1985)
Srivastava, L. M. and Srivastava, V. P, Peristaltic transport of a particle-fluid suspension. Journal of Biomechanical Engineering 111, 157–65 (1989)
Srivastava, L. M. and Srivastava, V. P, Effects of poiseuille flow on peristaltic transport of a particulate suspension. Zeitschrift für Angowandte Mathematik und Physik 46, 655–679 (1995)
Srivastava, V. E. and Saxena, M. A, Two-fluid model of non-Newtonian blood flow induced by peristaltic waves. Rheologica Acta 34, 406–414 (1995)
Hung, T. K. and Brown, T. D, Solid-particle motion in two-dimensional peristaltic flows. Journal of Fluid Mechanics 73, 77–96 (1976)
Mekheimer, K. S., El-Shehawey, E. F., and Elaiw, A. M, Peristaltic motion of a particle-fluid suspension in a planar channel. International Journal of Theoretical Physics 37, 2895–2920 (1998)
El-Misery, A. M., El-Shehawey, E. F., and Hakeem, A. A, Peristaltic motion of an incompressible generalized Newtonian fluid in a planar channel. Journal of Physical Society 65, 3524–3529 (1996)
Aarts, A. C. T. and Ooms, G, Net Flow of compressible viscous liquids induced by traveling waves in porous media. Journal of Engineering Mathematics 34, 435–450 (1998)
Antanovskii, L. K. and Ramkissoon, H, Long-wave peristaltic transport of a compressible viscous fluid in a finite pipe subject to a time dependent pressure drop. Fluid Dynamics Research 19, 115–123 (1997)
Tsiklauri, D. and Beresenev, I, Non-Newtonian effects in the peristaltic flow of a Maxwell fluid. Physics Review E 64, 0363031–0363035 (2001)
Castulik, P, The compressible boundary layer and the Boltzmann equation. The compressible boundary layer and the Boltzmann equation 15, 17–23 (2002)
Kawashita, M, On global solutions of Cauchy problems for compressible Navier-Stokes equations. On global solutions of Cauchy problems for compressible Navier-Stokes equations 48, 1087–1105 (2002)
Eldesoky, I. M. and Mousa, A. A. Peristaltic pumping of fluid in cylindrical tube and its applications in the field of aerospace. 13th International Conference on Aerospace Sciences and Aviation Technology, Military Technical College, Cairo, 1–14 (2009)
Eldesoky, I. M, Influence of slip condition on pristaltic transport of a compressible Maxwell fluid through porous medium in a tube. Influence of slip condition on pristaltic transport of a compressible Maxwell fluid through porous medium in a tube 8, 99–117 (2012)
Mekheimer, K. S. and Abdel-Wahab, A. N, Net annulus flow of a compressible viscous liquid with peristalsis. Net annulus flow of a compressible viscous liquid with peristalsis 25, 660–669 (2012)
Mekheimer, K. S., El-Komy, S. R., and Abdelsalam, S. I, Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a micro-channel. Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a micro-channel 22, 1247021–12470210 (2013)
Felderhof, B. U, Dissipation in peristaltic pumping of a compressible viscous fluid through a planar duct or a circular tube. Dissipation in peristaltic pumping of a compressible viscous fluid through a planar duct or a circular tube 83, 0463101–0463106 (2011)
Elshehawey, E. F., El-Saman, E. R., El-Shahed, M., and Dagher, M, Peristaltic transport of a compressible viscous liquid through a tapered pore. Peristaltic transport of a compressible viscous liquid through a tapered pore 169, 526–543 (2005)
Hayat, T., Ali, N., and Asghar, S, An analysis of peristaltic transport for flow of a Jeffrey fluid. An analysis of peristaltic transport for flow of a Jeffrey fluid 193, 101–112 (2007)
El-Shehawy, E. F., El-Dabe, N. T., and Eldesoky, I. M, Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid. Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid 186, 141–159 (2006)
Fung, Y. C. and Yih, C. S, Peristaltic transport. Peristaltic transport 35, 669–675 (1968)
Drew, D. A, Stability of a Stokes layer of a dusty gas. Stability of a Stokes layer of a dusty gas 22, 2081–2086 (1979)
Srivastava, L. M. and Srivastava, V. P, On two-phase model of pulsatile blood flow with entrance effects. On two-phase model of pulsatile blood flow with entrance effects 20, 761–777 (1983)
Anderson, J. D. Modern Compressible Flow: With Historical Perspective, McGraw Hill Higher Company, New York (1990)
Batchelor, G. K, Transport properties of two-phase materials with random structure. Transport properties of two-phase materials with random structure 6, 277–255 (1974)
Batchelor, G. K, Diffusion of particles with hydrodynamic interaction. Diffusion of particles with hydrodynamic interaction 74, 1–29 (1976)
Tam, C. K. W, The drag on a cloud of spherical particles in low Reynolds number flow. The drag on a cloud of spherical particles in low Reynolds number flow 38, 537–546 (1969)
Charm, S. E. and Kurland, G. S. Blood Flow and Micro-Circulation, John Wiley, New York (1974)
Waller, B. F., Orr, C. M., Slack, J. D., Pinkerton, C. A., van Tassel, J., and Peters, T, Anatomy, histology, and pathology of coronary arteries: a review relevant to new interventional and imaging techniques-part I. Anatomy, histology, and pathology of coronary arteries: a review relevant to new interventional and imaging techniques-part I 15, 451–457 (1992)
Lighthill, J. Waves in Fluids, Cambridge University Press, Cambridge (1978)
Chung, S. and Vafai, K, Mechanobiology of low-density lipoprotein transport within an arterial wall-impact of hyperthermia and coupling effects. Mechanobiology of low-density lipoprotein transport within an arterial wall-impact of hyperthermia and coupling effects 47, 137–147 (2014)
Keangin, P., Vafai, K., and Rattanadecho, P, Electromagnetic field effects on biological materials. Electromagnetic field effects on biological materials 65, 389–399 (2013)
Goodarzi, M., Safaei, M. R., Vafai, K., Ahmadi, G., Dahari, M., Kazi, S. N., and Jomhari, N, Investigation of nanofluid mixed convection in a shallow cavity using a two-phase mixture model. Investigation of nanofluid mixed convection in a shallow cavity using a two-phase mixture model 75, 204–220 (2014)
Hayat, T., Saleem, N., Abd Elmaboud, Y., Asghar, S, Peristaltic flow of a second-order fluid in the presence of an induced magnetic field. Peristaltic flow of a second-order fluid in the presence of an induced magnetic field 67, 537–558 (2011)
Mekheimer, K. S. and Abd Elmaboud, Y, Peristaltic transport of a particle-fluid suspension through a uniform and non-uniform annulus. Peristaltic transport of a particle-fluid suspension through a uniform and non-uniform annulus 5, 47–57 (2008)
Kamel, M. H., Eldesoky, I. M., Maher, B. M., and Abumandour, R. M, Slip effects on peristaltic transport of a particle-fluid suspension in a planar channel. Slip effects on peristaltic transport of a particle-fluid suspension in a planar channel 2015, 1–14 (2015)
Chinjung, Y. and Eugene, C. E, Transient lateral transport of platelet-sized particles in flowing blood suspensions. Biophysical Journal 66, 1706–1716 (1994)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eldesoky, I.M., Abdelsalam, S.I., Abumandour, R.M. et al. Interaction between compressibility and particulate suspension on peristaltically driven flow in planar channel. Appl. Math. Mech.-Engl. Ed. 38, 137–154 (2017). https://doi.org/10.1007/s10483-017-2156-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-017-2156-6