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Micro rheology of Jeffrey nanofluid through cilia beating subject to the surrounding temperature

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Abstract

The objective of this study is to discuss the micro rheology of mucus (Jeffrey nanofluid) which is a complex biological fluid that protects lungs from pollutants, bacteria, and allergens that can be inhaled during the breathing process. To see the insight of pollutants and effect of surrounding temperature in the mucus, momentum, energy, and concentration equations are modeled with the help of metachronal wave formed by cilia beating. The governing system of equations are modeled in the wave and fixed frame and simplified by the lubrication approach. The velocity profile for recovery and effective stroke is compared and it is analyzed that effective stroke possess high magnitude of velocity when compared with the recovery stroke. The flow of mucus with pollutants and surrounding temperature is calculated with homotopy perturbation method and software “MATHEMATICA.” The results within the given domain are convergent under the different parameters appearing into the system of equations. The physical interpretation of involved parameters is explained through graphs.

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Abbreviations

V :

Velocity field vector

U, W :

Velocity components in fixed frame

u, w :

Velocity components in wave frame

R, Z :

Cylindrical coordinates of ciliated tube in fixed frame

r, z :

Cylindrical coordinates of ciliated tube in wave frame

μ :

Dynamic viscosity of fluid

ν :

Kinematic viscosity of fluid

P :

Pressure in fixed frame

p :

Pressure in wave frame

τ :

Cauchy stress tensor

S :

Shear rate

c :

Wave speed

c p :

Specific heat capacity

ε :

Cilia length

ρ f :

Density of fluid

α :

Eccentricity of elliptical pair

β :

Wave number

λ1, λ2 :

Jeffrey viscoelastic parameters

A 1 :

Rivilian-Erickson tensor

T :

Temperature profile

T 0 :

Temperature at the center of the tube

T 1 :

Temperature at the ciliated wall

C :

Concentration profile

C 0 :

Concentration profile at the center of the tube

C 1 :

Concentration at ciliated wall

β 1 :

Coefficient of thermal expansion

\( {\beta}_1^{\ast } \) :

Coefficient of concentration expansion

g :

Gravitational acceleration

D B :

Coefficient of mass diffusivity

D T :

Constant ratio due to thermal diffusion

Re :

Reynolds’ number

Br :

Brinkman number

Nt :

Thermophoresis parameter

Nb :

Brownian motion parameter

Gr C :

Concentration Grashof number

Gr T :

Thermal Grashof number

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    Sidra Shaheen & Khadija Maqbool

  2. Department of Mathematics, York Campus, Pennsylvania State University, 1031 Edgecomb Avenue York, State College, PA, 17403, USA

    Abdul Majeed Siddiqui

Authors
  1. Sidra Shaheen
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  2. Khadija Maqbool
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  3. Abdul Majeed Siddiqui
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Correspondence to Khadija Maqbool.

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Appendix

Appendix

The constants calculated by software “MATHEMATICA” are given as

$$ {A}_1=-1-2\pi \varepsilon \alpha \beta \cos \left(2\pi z\right) $$
$$ {A}_2=\frac{Gr_T{h}^4{Nb}^2\left(1+{\lambda}_1\right)+16{Gr}_C{h}^2 Nt\left(1+{\lambda}_1\right)}{256 Nb}+\frac{64\left(\frac{dp}{dz}+4\right)\left(1+{\lambda}_1\right)+16{Gr}_C{h}^2\left(2+{\lambda}_1\right)}{256} $$
$$ \frac{Gr_T\left(-16{h}^2{\lambda}_1+{h}^4\right)\left(1+{\lambda}_1\right)\left( Nt- Br{\left(\frac{dp}{dz}\right)}^2\right)\left(1+{\lambda}_1\right)}{256} $$
$$ {A}_3=\frac{\left(2{Gr}_T Nb-{Gr}_C Nt\right)\left(1+{\lambda}_1\right)}{64 Nb} $$
$$ {A}_4=\frac{Gr_T\left(1+{\lambda}_1\right)\left( Nb+ Nt- Br{\left(\frac{dp}{dz}\right)}^2\left(1+{\lambda}_1\right)\right)}{2304},{A}_5=\frac{1}{4} $$
$$ {A}_6=\frac{1}{16}\left(\frac{Nb}{4}-\frac{Nb+ Nt}{4}\right)+\frac{Nt}{4}+\frac{1}{16} Br\left({Gr}_T+{Gr}_C\right){h}^2\frac{dp}{dz}\left(1+{\lambda}_1\right) $$
$$ -\frac{1}{64}\left(- Nb- Nt+ Br{\left(\frac{dp}{dz}\right)}^2\left(1+{\lambda}_1\right)\right) $$
$$ {A}_7=\frac{1}{36}\left(\frac{1}{32} Br\left({Gr}_T+{Gr}_C\right)\frac{dp}{dz}\left(1+{\lambda}_1\right)+\frac{1}{32}\left( Nb+ Nt\right)\left( Nb+ Nt- Br{\left(\frac{dp}{dz}\right)}^2\left(1+{\lambda}_1\right)\right)\right) $$
$$ {A}_8=\frac{1}{4}+\frac{Nt}{4 Nb}+\frac{\left( Nb+ Nt\right)}{4 Nb} $$
$$ {A}_9=\frac{Nt\left( Nb- Nt+ Br{\left(\frac{dp}{dz}\right)}^2\left(1+{\lambda}_1\right)\right)}{64 Nb},{A}_{10}=\frac{-1}{8}{h}^4\left(1+{\lambda}_1\right) $$
$$ {A}_{11}=\frac{5 Br{Gr}_T{h}^8{\left(\left(1+{\lambda}_1\right)\right)}^2}{3072} $$
$$ {A}_{12}={h}^2\frac{-3072 Nb w(h)+5{Gr}_T{h}^6{Nb}^2\left(1+{\lambda}_1\right)+5{Gr}_T{h}^6 NbNt\left(1+{\lambda}_1\right)}{3072 Nb} $$
$$ {h}^2\frac{64{Gr}_C{h}^4\left(2 Nb+ Nt\right)\left(1+{\lambda}_1\right)}{3072 Nb} $$

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Shaheen, S., Maqbool, K. & Siddiqui, A.M. Micro rheology of Jeffrey nanofluid through cilia beating subject to the surrounding temperature. Rheol Acta 59, 565–573 (2020). https://doi.org/10.1007/s00397-020-01222-8

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  • DOI: https://doi.org/10.1007/s00397-020-01222-8

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