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Homotopy perturbation method for couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid

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Abstract

An analytical study is presented for couple stresses effects on MHD peristaltic transport of a non-Newtonian Jeffery nanofluid. The fluid flows through a porous media between co-axial tubes under long-wavelength assumption. The inner tube is uniform, while the outer flexible tube has a sinusoidal wave traveling down its wall. Homotopy perturbation method is used to obtain analytical solutions which satisfies the governing equations. Numerical results for the behaviors of the axial velocity, temperature and nanoparticles with other physical parameters are obtained. Several graphs for these results of physical interest are displayed and discussed in detail.

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Abbreviations

B :

The wave amplitude

\( \underline{B} \) :

The magnetic field \( = (B_{0} ,0,0) \)

c :

The propagation velocity

D :

The radius of outer tube

Da :

Darcy number, defined by Eq. (20)

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

\( \underline{E} \) :

The electric field

Ec :

Eckert number, defined by Eq. (20)

F :

The nanoparticle phenomena

f 0 :

Nanoparticles at \( {\text{r}} = {\text{r}}_{1} \)

f 1 :

Nanoparticles at \( {\text{r}} = {\text{r}}_{2} \)

\( \underline{I} \) :

The identity tensor

\( \underline{J} \) :

The current density

k R :

The mean absorption coefficient

K :

Thermal conductivity

\( \kappa^{*} \) :

The permeability parameter

M :

The magnetic field parameter, defined by Eq. (20)

N b :

Brownian motion parameter, defined by Eq. (20)

N t :

The thermophoresis parameter, defined by Eq. (20)

P :

The fluid pressure

Pr:

Prandtl number, defined by Eq. (20)

Q :

The radiative heat flux

R :

Radiation parameter, defined by Eq. (20)

Re:

Reynolds number, defined by Eq. (20)

\( \underline{S} \) :

The stress tensor defined by Eqs. (2) and (3)

t :

The time

T :

The fluid temperature

T 0 :

Temperature at \( {\text{r}} = {\text{r}}_{1} \)

T 1 :

Temperature at \( {\text{r}} = {\text{r}}_{2} \)

\( \underline{V} \) :

The velocity vector

\( \alpha \) :

The couple stress fluid parameter, defined by Eq. (20)

σ :

The electrical conductivity

σ*:

Stefan Boltzmann constant

ε :

The amplitude ratio

\( \eta ,\,\overline{\eta } \) :

Couple stress constants

\( \dot{\gamma } \) :

The shear rate

λ :

The wavelength

\( \lambda_{1} \) :

The ratio of relaxation to retardation times

\( \lambda_{2} \) :

The retardation time

\( \mu_{f} \) :

The dynamic viscosity of fluid

µ e :

The magnetic permeability

\( \rho_{f} \) :

The fluid density

\( \rho_{p} \) :

The particle density

\( (\rho c)_{f} \) :

Heat capacity of the fluid

\( (\rho c)_{p} \) :

Effective heat capacity of the nanoparticle material

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo, 11757, Egypt

    Mohamed Y. Abou-zeid

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  1. Mohamed Y. Abou-zeid
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Correspondence to Mohamed Y. Abou-zeid.

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Appendix

Appendix

$$ a_{1} = g\alpha^{2} a_{49} /(5 + \eta^{\prime } )^{2} ;\,a_{2} = \alpha^{2} (ga_{50} (5 + \eta^{\prime } )^{2} - ga_{49} e)/(7 + \eta^{\prime } )^{2} (5 + \eta^{\prime } )^{2} ; $$
$$ a_{3} = - \alpha^{2} ea_{50} /(7 + \eta^{\prime } )^{2} (9 + \eta^{\prime } )^{2} ;\,a_{4} = a_{50} b(5 + \eta^{\prime } )^{2} /e;\,a_{5} = - a_{50} ;\,a_{6} = a_{46} ; $$
$$ a_{7} = c_{1} + c_{5} ;\,a_{8} = (c_{7} + c_{3} )/2;\,a_{9} = \alpha^{2} (4ga_{48} - a_{47} e)/64; $$
$$ a_{10} = - \frac{{\alpha^{2} ea_{48} ( - 11025 + 8785 \eta^{\prime } + 2766 \eta^{\prime 2} + 430 \eta^{\prime 3} + 33 \eta^{\prime 4} + \eta^{\prime 5} )}}{{(5 + \eta^{\prime } )(7 + \eta^{\prime } )(9 + \eta^{\prime } )}}; $$
$$ a_{11} = \alpha^{2} a_{46} e/2304;\,a_{12} = \alpha^{2} (96a_{46} g - 6a_{45} e + 5a_{48} e)/3456; $$
$$ a_{13} = \alpha^{2} (8a_{45} g - 4a_{48} g + 3a_{47} e - 2a_{51} e)/128;\,a_{14} = (2c_{2} - c_{3} + 2c_{6} - c_{7} )/4; $$
$$ a_{15} = c_{4} + c_{8} ; $$
$$ a_{16} = ( - ga_{57} (Nt + Nb)/2(2 + \eta^{\prime } )(\ln r_{1} - \ln r_{2} )) - EcPra_{49} (3 + 4\eta^{\prime } + \eta^{\prime 2} )/2(2 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right) + gEcM/4(2 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)^{2} ; $$
$$ a_{17} = gEc(Ma_{49} - (5 + 6\eta^{\prime } + \eta^{\prime 2} )Pra_{50} )/2(3 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{18} = gEcM/2(4 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{19} = PrEcMr_{1}^{{2 + 2\eta^{\prime } }} /4\left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)^{2} ; $$
$$ a_{20} = ( - gEcPr/4(r_{1}^{{1 + \eta^{'} }} - r_{2}^{{1 + \eta^{'} }} )^{2} ) - a_{56} g(Nb + Nt)/2(1 + \eta^{'} )(\ln r_{1} - \ln r_{2} ); $$
$$ a_{21} = - 2gEcMr_{1}^{{1 + \eta^{\prime } }} /(3 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)^{2} ; $$
$$ a_{22} = 2a_{47} gEcPr/(1 + \eta^{\prime } )\left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{23} = - 2gEcMr_{1}^{{1 + \eta^{\prime } }} /(3 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)^{2} ; $$
$$ a_{24} = - 2gEc( - (5 + \eta^{\prime } )a_{45} M + 2a_{48} M + 20Pra_{46} + 24a_{46} Pr\alpha + 4a_{46} Pr\alpha^{2} + a_{49} Mr_{1}^{{1 + \eta^{\prime } }} )/(5 + \eta^{\prime } )^{3} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{25} = 2gEcM\left( {a_{46} - a_{50} r_{1}^{{1 + \eta^{\prime } }} } \right)/(7 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{26} = 2gEc(a_{47} M - 2(1 + \eta^{\prime } )a_{48} )/(3 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{27} = - 24a_{48} gEcM lnr_{1} lnr_{2} /(\ln r_{1} - \ln r_{2} )(5 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{28} = g( - a_{55} (Nt + Nb)/(\ln r_{1} - \ln r_{2} )(3 + \eta^{\prime } ); $$
$$ a_{29} = - 2Ec(a_{47} M - (3 + \eta^{\prime } )a_{51} M + (1 + \eta^{\prime } )(2(3 + \eta^{\prime } )a_{46} + ( - 1 + \eta^{\prime } )a_{48} )Pr)lnr_{1} /(\ln r_{1} - \ln r_{2} )(3 + \eta^{\prime } )^{3} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{30} = 2Ec(2a_{47} M - (3 + \eta^{\prime } )a_{51} M + (1 + \eta^{\prime } )(2(3 + \eta^{\prime } )a_{46} + ( - 1 + \eta^{\prime } )a_{48} )Pr)lnr_{2} /(\ln r_{1} - \ln r_{2} )(3 + \eta^{\prime } )^{3} (r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} ); $$
$$ a_{31} = - ga_{53} (Nb + Nt)/3(\ln r_{1} - \ln r_{2} ); $$
$$ a_{32} = g(Nb + Nt + (c_{13} Nb + a_{52} \left( {Nb + Nt} \right))(\ln r_{1} - \ln r_{2} )/2(\ln r_{1} - \ln r_{2} )^{2} ; $$
$$ a_{33} = c_{9} + c_{11} ; $$
$$ a_{34} = - gEc Mr_{1}^{{1 + \eta^{\prime } }} a_{47} /2(r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} ); $$
$$ a_{35} = - gEc Mr_{1}^{{1 + \eta^{\prime } }} a_{48} /8(r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} ); $$
$$ a_{36} = - gEc Mr_{1}^{{1 + \eta^{\prime } }} a_{46} /18(r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} ); $$
$$ a_{37} = PrEcMr_{1}^{{1 + \eta^{\prime } }} (2a_{45} - a_{48} )/16(r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} ); $$
$$ a_{38} = PrEcMr_{1}^{{2 + 2\eta^{\prime } }} /4\left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)^{2} ;\,a_{39} = c_{10} + c_{12} ; $$
$$ a_{40} = a_{57} ;\,a_{41} = a_{53} ;\,a_{42} = a_{52} ;\,a_{43} = a_{54} ;\,a_{44} = Nt(1 - \ln r_{2} )/Nb(\ln r_{1} - \ln r_{2} ); $$
$$ g = \left( {1 + \frac{1}{\lambda }} \right);\,e = \left( {M + \frac{1}{Da}} \right);\,a_{45} = (c_{2} + c_{3} )/2; $$
$$ a_{46} = (\alpha^{2} /64)\left( { - P_{z} + e r_{1}^{{1 + \eta^{\prime } }} /\left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right)} \right);\,a_{47} = c_{1} ;\,a_{48 = } c_{3/2} ; $$
$$ a_{49} = b\alpha^{2} /(3 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right);\,a_{50} = e\alpha^{2} /(3 + \eta^{\prime } )^{2} (5 + \eta^{\prime } )^{2} \left( {r_{1}^{{1 + \eta^{\prime } }} - r_{2}^{{1 + \eta^{\prime } }} } \right); $$
$$ a_{51} = c_{4} ;\,a_{52} = 4c_{9} ;\,a_{53} = - 2g(Nt + Nb)/(\ln r_{1} - \ln r_{2} )^{2} ; $$
$$ a_{54} = gEcM(6 + 5\eta^{\prime } + \eta^{\prime 2} )r_{1}^{{2 + 2\eta^{\prime } }} ;\,a_{55} = - 8gEcM(2 + \eta^{\prime } )^{2} r_{1}^{{1 + \eta^{\prime } }} ; $$
$$ a_{56} = - gEcPr(6 + 5\eta^{\prime } + \eta^{\prime 2} );\,a_{57} = gEcM(3 + \eta^{\prime } )^{2} ;\,a_{58} = c_{10} ; $$

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Abou-zeid, M.Y. Homotopy perturbation method for couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid. Microsyst Technol 24, 4839–4846 (2018). https://doi.org/10.1007/s00542-018-3895-1

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