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Influence of wall properties on the peristaltic flow of a dusty Walter’s B fluid

  • A. A. Khan
  • H. Tariq
Technical Paper

Abstract

This paper explores the impact of wall properties on the Walter’s B fluid with fine solid particles in a uniform channel. Traveling sinusoidal wave is imposed on the channel walls which induces peristaltic motion in the fluid. A regular series is employed to obtain analytic solution by taking small wave number. The phenomenon is modeled in the form of stream function for both fluid and solid particles. Impact of pertinent parameters such as viscoelastic parameter \(\kappa\), wave number \(\delta\), the elastic tension, i.e., rigidity of the wall (E1), the mass characterizing parameter, i.e., stiffness of the wall (E2) and the damping nature of the wall (E3) are discussed through graphs for both fluid and dust particles. It is observed that the size of the trapped bolus increases on the right-hand side of the channel for both fluid and solid particles by increasing viscoelastic parameter. The flow rate of fluid particles increases by increasing viscoelastic parameter, while it decreases for solid particles.

Keywords

Walter’s B fluid Dusty fluid Wall properties Peristaltic motion 

List of symbols

\(\bar{P}\)

Isotropic pressure of the fluid

\(\rho\)

Fluid density

\(\bar{S}\)

Extra stress tensor

e

The rate of the stress tensor

\(\frac{\delta e}{\delta t}\)

The rate of stress tensor to the material in motion

\(\eta_{0}\)

Limiting viscosity at small shear rate

\(k_{0}\)

Short memory coefficient

k

Resistance coefficient of the solid particles

N

Number density of the solid particles

m

Mass of the solid particles

\(\delta\)

Wave number

\(\kappa\)

Viscoelastic parameter

\(E_{1}\)

Tension membrane parameter \(\left( {{\raise0.7ex\hbox{${Td^{4} }$} \!\mathord{\left/ {\vphantom {{Td^{4} } {\nu \rho }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\nu \rho }$}}} \right)\)

E2

Mass characterizing parameter \(\left( {{\raise0.7ex\hbox{${m^{\prime}d^{2} }$} \!\mathord{\left/ {\vphantom {{m^{\prime}d^{2} } {\lambda^{3} \rho }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda^{3} \rho }$}}} \right)\)

E3

Damping parameter \(\left( {{\raise0.7ex\hbox{${cd^{3} }$} \!\mathord{\left/ {\vphantom {{cd^{3} } {\lambda^{2} \nu \rho }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda^{2} \nu \rho }$}}} \right)\)

d

Half width of the passage

\(\lambda\)

Wavelength

a

Amplitude of the wave

t

Time

c

Velocity of the wave

T

Tension in the membrane

C

Coefficient of the viscous damping forces

\(m^{\prime}\)

Mass per unit area

L

Operator representing the motion of stretched membrane with damping forces

\(p_{0}\)

Pressure outside the wall

\(A_{1}\)

Non-dimensional parameter \(\left( {{\raise0.7ex\hbox{${kNd^{2} }$} \!\mathord{\left/ {\vphantom {{kNd^{2} } \mu }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\mu $}}} \right)\)

R

Non-dimensional parameter \(\left( {{\raise0.7ex\hbox{${kd^{2} }$} \!\mathord{\left/ {\vphantom {{kd^{2} } {m\nu }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${m\nu }$}}} \right)\)

\(\nu\)

Kinematic coefficient of viscosity of the fluid

\(\epsilon\)

Geometric parameter \(\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a d}}\right.\kern-0pt} \!\lower0.7ex\hbox{$d$}}} \right)\)

\(u,v\)

Velocity components of the fluid

\(\left( {u_{\text{s}} ,v_{\text{s}} } \right)\)

Velocity components of the dust particles

V

Velocity of the fluid

\(\psi ,\phi\)

Stream functions

\(Q\)

Flow rate of the fluid

\(Q_{\text{s}}\)

Flow rate of the solid dust particles

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsInternational Islamic UniversityIslamabadPakistan

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