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Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls

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Abstract

Effects of compliant wall properties on the peristaltic flow of a non-Newtonian fluid in an asymmetric channel are investigated. The rheological characteristics are characterized by the constitutive equations of a power-law fluid. Long wavelength and low Reynolds number approximations are adopted in the presentation of mathematical developments. Exact solutions are established for the stream function and velocity. The streamlines pattern and trapping are given due attention. Salient features of the key parameters entering into the present flow are displayed and important conclusions are pointed out.

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Abbreviations

V :

velocity

u, v :

velocity components in x, and y-directions, respectively

T :

Cauchy stress tensor

S :

extra stress tensor

D :

symmetric part of velocity gradient

A 1 :

first Rivlin-Ericksen tensor

L :

gradient of velocity

L T :

transpose of the gradient of velocity

C :

coefficient of viscous damping

B :

flexural rigidity of the plate

a 1 :

amplitude of the wave along the upper channel wall

a 2 :

amplitude of the wave along the lower channel wall

t :

time

x :

spatial coordinate along the channel walls

y :

spatial coordinate normal the channel walls

m :

power-law exponent

−pI :

indeterminate part of the stress

c :

wave speed

k :

spring stiffness coefficient

p :

pressure

d 1 :

upper half channel width

d 2 :

lower half channel width

a :

dimensionless amplitude ratio of the wave along upper wall

b :

dimensionless amplitude ratio of the wave along lower wall

E 1 :

dimensionless number corresponding to the plate mass per unit area

E 2 :

dimensionless number corresponding to the coefficient of viscous damping

E 3 :

dimensionless number corresponding to the flexural rigidity of the plate

E 4 :

dimensionless number corresponding to the elastic tension in the membrane

E 5 :

dimensionless number corresponding to the spring stiffness coefficient

h :

channel width ratio

\( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{m} \) :

plate mass per unit area

Re :

Reynolds number

η 1 :

vertical displacement corresponding to the upper wall

η 2 :

vertical displacement corresponding to the lower wall

μ :

dynamic viscosity

μ app :

apparent viscosity

ν :

kinematic viscosity

ψ:

stream function

ρ :

fluid density

δ :

dimensionless wave number

θ :

phase difference

τ :

elastic tension in the membrane

λ :

wave length

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan

    T. Hayat & M. Javed

  2. Department of Mathematics, College of Sciences, King Saud University, Riyadh, 11451, Saudi Arabia

    T. Hayat

Authors
  1. T. Hayat
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  2. M. Javed
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Correspondence to M. Javed.

Additional information

Communicated by Chuan-jing LU

Project supported by the Higher Education Commission (HEC) of Pakistan

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Hayat, T., Javed, M. Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls. Appl. Math. Mech.-Engl. Ed. 31, 1231–1240 (2010). https://doi.org/10.1007/s10483-010-1356-7

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  • DOI: https://doi.org/10.1007/s10483-010-1356-7

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