# Nonlinear radiative peristaltic flow of Jeffrey nanofluid with activation energy and modified Darcy’s law

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## Abstract

The present communication addresses the magnetoperistalsis of Jeffrey nanomaterial in a vertical asymmetric compliant channel walls. Flow modeling is based upon mixed convection, non-Darcy’s resistance, thermal radiation, Brownian motion and thermophoresis, chemical reaction and activation energy. Nonlinear thermal radiation is taken instead of classical linear radiation consideration. Buongiorno model is used for nanofluid analysis. Channel boundaries are associated with no-slip, compliant characteristics and convective heat and mass transfer effects. Lubrication approach is followed and problems are numerically solved. Quantities of interest are analyzed physically. It is observed that velocity enhances for Hall and Darcy parameters. Temperature decreases with radiation parameter. It is found that concentration increases by increasing activation energy parameter.

## Keywords

Jeffrey nanoliquid Convective boundary condition Nonlinear radiative heat flux Energy activation Non-Darcy’s resistance## List of symbols

- \(\left( {\tilde{u},\tilde{v}} \right)\)
Velocity components in dimensional form

- \(\left( {\tilde{x},\tilde{y}} \right)\)
Coordinate axes in dimensional form

- \(H_{1} , H_{2}\)
Lower and upper wall shapes

- \(h_{1} , h_{2}\)
Wall shapes in wave frame

- \(\lambda\)
Wave length

- \(\hat{p}\)
Pressure in

- \(\phi_{1}\)
Phase difference

- \({\mathbf{J}}\)
Joule current

- \({\mathbf{B}}\)
Magnetic field

- \(B_{0}\)
Constant transverse magnetic field

- \(a_{1} , b_{1}\)
Amplitudes of upper and lower waves

- \(\tilde{t}\)
Time

- \(\varvec{\tau}\)
Cauchy stress tensor

- \(\varvec{A}\)
First Rivlin–Ericksen tensor

- \(S_{{\tilde{x}\tilde{x}}} ,\,S_{{\tilde{x}\tilde{y}}} ,\,S_{{\tilde{y}\tilde{y}}}\)
Components of extra stress tensor

- \(\tau^{\prime }\)
Tension in membrane

- \(d^{{^{\prime } }}\)
Viscous damping coefficient

- \(k_{1}\)
Permeability of porous media

- \(q_{r}\)
Radiative heat flux

- \(\beta_{1}^{{^{\prime } }}\), \(\beta_{2}^{{^{\prime } }}\)
Heat transfer coefficients

- \(T_{0}\), \(T_{1}\)
Constant temperatures of walls

- \(k_{r}^{2}\)
Chemical reaction

- \(D_{B}\)
Thermophoresis diffusion coefficients

- \(\psi\)
Stream function

*Re*Reynolds number

- M
Hartmann number

- \(E_{1}\), \(E_{2}\), \(E_{3}\)
Compliant wall parameters

*Pr*Prandt number

- \(Br\)
Brinkmann number

- \(N_{b}\)
Brownian motion parameter

- \(G_{c}\), \(G_{r}\)
Mass and thermal Grashof numbers

- \(\lambda_{1}\)
Ratio of relaxation to retardation number

- \(R_{n}\)
Radiation parameter

- \(N_{t}\)
Thermophoresis parameter

- \(\beta_{1}\), \(\beta_{2}\)
Convective heat Biot numbers

- \(\gamma_{1}\), \(\gamma_{2}\)
Convective mass Biot numbers

- \(E\)
Activation energy parameter

- \(\phi\)
Dimensionless concentration

- \(\left( {u, v} \right)\)
Velocity components in dimensionless form

- \(\left( {x,y} \right)\)
Coordinate axes in dimensionless form

- \(\alpha , \beta\)
Coefficients of thermal and concentration expansion

- \(\varvec{g}\)
Gravity vector

- \(d_{1}\), \(d_{2}\)
Channel widths

- \(\rho_{f}\), \(\rho_{p}\)
Densities of base and nanoparticle

- \(\kappa\)
Thermal conductivity

- \(\mu\)
Dynamic viscosity

- \(C_{f}\), \(C_{p}\)
Specific heats of base fluid and nanoparticle

- \(T\), \(C\)
Temperature and concentration of fluid

- \(\varvec{I}\)
Identity tensor

- \(\varvec{S}\)
Extra stress tensor

- \(\frac{d}{{{\text{d}}t}}\)
Material derivative

- \(\sigma\)
Electric conductivity

- \(m^{\prime }\)
Mass per unit area

- \(p_{0}\)
Pressure on the outside wall

- \(T_{m}\)
Fluid mean temperature

- \(\gamma_{1}^{{^{\prime } }}\), \(\gamma_{2}^{{^{\prime } }}\)
Mass transfer coefficients

- \(C_{0}\), \(C_{1}\)
Constant concentration on walls

- \(E_{a}\)
Activation energy

- \(D_{B}\)
Brownian motion coefficient

- \(\delta\)
Wave number

- \(D_{a}\)
Darcy resistance

- \(m\)
Hall parameter

- \(E_{c}\)
Eckert number

- \(N_{t}\)
Thermophoresis parameter

- \(\sigma^{*}\)
Stefan–Boltzmann constant

- \(n\)
Fitted rate constant

- \(a\), \(b\)
Amplitude in dimensionless form

- \(S_{c}\)
Schmidt number

- \(\xi\)
Dimensionless chemical reaction

- \(\theta\)
Dimensionless temperature

- \(p\)
Dimensionless pressure

- \(\lambda_{2}\)
Retardation time

- \(k^{*}\)
Mass spectral absorption coefficient

- \(k\)
Boltzmann constant

- \(d\)
Dimensionless width

## Notes

## References

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