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A couple stress of peristaltic motion of Sutterby micropolar nanofluid inside a symmetric channel with a strong magnetic field and Hall currents effect

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Abstract

The current study investigates the peristaltic transport of an incompressible micropolar non—Newtonian nanofluid following the Sutterby model. The heat and mass transfer inside the two-dimensional symmetric vertical channel is considered. The system is affected by a strong magnetic field together with thermal radiation, couple stress, chemical reaction, Joule heating, heat generation, Dufour, Soret and Hall current effects. The governing equations of motion are analytically solved by utilizing the long wavelength and low Reynolds number approximations. Furthermore, the resulted boundary—value problem is solved by means of the Homotopy perturbation method (HPM). An illustration of the influence of the various physical parameters in the foreign distributions; such as Hall currents, magnetic field, Sutterby, couple stress, Brownian motion, thermophoresis and slip parameters is obtained throughout a set of graphs and tables. It is observed that the axial velocity enhances with the increase in the Sutterby parameter. Furthermore, the temperature decreases with the larger values of a heat transfer Biot number. While, the concentration enlarges with the increase in the values of mass transfer Biot numbers. Moreover, the trapping phenomenon is discussed throughout a set of figures. This depicts the variation of the streamlines under the impact of couple stress, amplitude ratio, and magnetic field parameters. It is noticed that the size of the trapped bolus increases with the increase in the foregoing three parameters.

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Abbreviations

\(\underline{{A_{1} }}\) :

The first Rivilin Ericksen tensor

\(B\) :

Material constant

\(a\) :

Wave amplitude

\(B_{0}\) :

Magnetic field strength

\(B_{i1} ,B_{i2}\) :

Heat transfer Biot numbers

\(B_{r}\) :

Brinkman number

\(B_{t}\) :

Volumetric coefficient of expansion.

\(C\) :

Fluid concentration

\(C_{0}\) :

Mass concentration of the left wall

\(C_{1}\) :

Mass concentration of the right wall

\(c_{f}\) :

Specific heat parameter of fluid

\(c_{p}\) :

Specific heat parameter of nanoparticle

\(c_{s}\) :

Concentration susceptibility

\(c\) :

Speed of wave

\(D\) :

Coefficient of mass diffusivity

\(D_{a}\) :

Darcy number

\(D_{B}\) :

Brownian diffusion coefficient

\(D_{T}\) :

Thermophoretic diffusion coefficient

\(D_{u}\) :

Dufour number

\(d\) :

Half-channel width

\(E_{c}\) :

Eckert number

\(e\) :

Electric charge

\(f\) :

Dimensionless concentration

\(\underline{g} = \left( { - g,0,0} \right)\) :

Acceleration gravity

\(h_{1}\) :

Heat transfer coefficients on the left wall

\(h_{2}\) :

Heat transfer coefficients on the right wall

\(\underline{J}\) :

Current density

\(J_{1}\) :

Microinertia constant

\(k^{*}\) :

Thermophoretic coefficient

\(K_{1}\) :

Vortex viscosity coefficient

\(K_{c}\) :

Thermal conductivity

\(K_{m}\) :

Constant of chemical reaction

\(K_{p}\) :

Permeability of porous medium

\(K_{R}\) :

Mean absorption coefficient

\(K_{T}\) :

Thermal diffusion ratio

\(L_{e}\) :

Lewis number

\(L_{n}\) :

Nano Lewis number

\(L_{1}\) :

Mass transfer coefficients on the left wall

\(L_{2}\) :

Mass transfer coefficients on the right wall

\({\text{M}}\) :

Magnetic field parameter

\(M_{i1} ,M_{i2}\) :

Mass transfer Biot numbers

\(m\) :

The power index of the material

\(m_{1}\) :

Hall parameter

\(n\) :

Constant associated the couple stress

\(n_{e}\) :

Number density of electrons

\(N_{b}\) :

Brownian motion parameter

\(N_{t}\) :

Thermophoresis parameter

\(p\) :

Fluid pressure

\(P_{r}\) :

Prandtl number

\(\underline{q}\) :

Radiative heat flux

\(Q_{0}\) :

Constant of heat addition and absorption.

\(R_{N}\) :

Nanoparticle Rayleigh number

\(R_{T}\) :

Thermal Rayleigh number

\(R_{m}\) :

Basic density Rayleigh number

\(R_{c}\) :

Non-dimension chemical reaction parameter

\(R_{d}\) :

Radiation parameter

\(R_{e}\) :

Reynold’s number

\(\underline{S}\) :

Cauchy stress tensor

\(S_{r}\) :

Soret number

\(t\) :

Time

\(T\) :

Fluid temperature

\(T_{0}\) :

Temperature of the left wall

\(T_{1}\) :

Temperature of the right wall

\(T_{m}\) :

Mean fluid temperature

\(T_{r}\) :

Reference temperature

\(u\) :

Axial velocity component in the wave frame

\({\text{v}}\) :

Transverse velocity component in the wave frame

\(\underline{{\text{V}}}\) :

Velocity vector

\({\text{V}}_{{\text{T}}}\) :

Thermophertic velocity

\({\text{(x,y)}}\) :

Cartesian coordinate system in the wave frame

\({\text{x}}\) :

Axial coordinate

\({\text{y}}\) :

Transverse coordinate

\(\alpha_{1}\) :

Non-dimensional slip parameter for nanoparticles

\(\alpha_{f}\) :

Thermal diffusivity of the fluid

\(\alpha^{*} ,\beta^{*}\) :

Micropolar material

\(\beta_{1}\) :

Mean absorption coefficient,

\(\beta_{M}\) :

Micropolar dimensionless viscosity

\(\beta_{s}\) :

Sutterby parameter

\(\Gamma\) :

The material constants

\(\gamma\) :

Couple stress, non-dimensional parameter

\(\gamma_{m}\) :

Spin-gradient viscosity

\(\overline{\gamma }_{m}\) :

Microrotation parameter

\(\delta\) :

Wave number

\(\varepsilon\) :

Amplitude ratio parameter

\(\eta {\text{(x,t)}}\) :

Displacement of the right wall

\(- \eta {\text{(x,t)}}\) :

Displacement of the left wall

\(\theta\) :

Dimensionless temperature

\(\lambda\) :

Wave length

\(\mu\) :

Fluid viscosity

\(\nu\) :

Kinematic viscosity

\(\rho_{f}\) :

Fluid density

\(\rho_{p}\) :

Nanoparticle density

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

Heat capacity of the fluid

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

Effective heat capacity of the nanoparticlematerial

\(\sigma\) :

Electrical conductivity

\(\sigma^{ * }\) :

Stefan Boltzmann constant

\(\tau\) :

Ratio between the effective heat & fluid capacities

\(\tau^{ * }\) :

Thermophoretic parameter

\(\Phi\) :

Dimensionless nanoparticle phenomena

\(\phi\) :

Nanoparticle phenomena

\(\phi_{0}\) :

Nanoparticle at the left wall

\(\phi_{1}\) :

Nanoparticle at the right wall

\(\xi\) :

The second invariant strain tensor

\(\psi \left( {x,y} \right)\) :

Stream function

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Authors and Affiliations

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

    Nabil T. M. El-Dabe, Galal M. Moatimid, Mona A. A. Mohamed & Yasmeen M. Mohamed

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  1. Nabil T. M. El-Dabe
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El-Dabe, N.T.M., Moatimid, G.M., Mohamed, M.A.A. et al. A couple stress of peristaltic motion of Sutterby micropolar nanofluid inside a symmetric channel with a strong magnetic field and Hall currents effect. Arch Appl Mech 91, 3987–4010 (2021). https://doi.org/10.1007/s00419-021-01990-6

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