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Effects of velocity second slip model and induced magnetic field on peristaltic transport of non-Newtonian fluid in the presence of double-diffusivity convection in nanofluids

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Abstract

The significance of velocity second slip model of non-Newtonian fluid on peristaltic pumping in existence of double-diffusivity convection in nanofluids and induced magnetic field is deliberated. Mathematical modelling of current problem is defined in fixed frame of reference and then abridges under well- known conjecture of long wavelength and low but finite Reynolds number approximation. Precise results of coupled nonlinear partial differential equations are presented. Graphical results exhibit the performance of various supportive parameters. The phenomenon of stream functions with different wave forms is also discussed in detail. The effects of thermal energy, solute concentration, and nanoparticle fraction are also described using graphical representation.

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Abbreviations

U, V :

Velocities in x and y directions in fixed frame

p :

Pressure

g :

Acceleration due to gravity

\(\varTheta \) :

Nanoparticle volume fraction

\(a_{1} \) and \(b_{1} \) :

Amplitudes of waves

\(\sigma \) :

Electrical conductivity

\(\mu _\mathrm{e}\) :

Magnetic permeability

\(\lambda _{{1}} \) :

Ratio of relaxation to retardation times

\(\rho _{\mathrm{f}_{0} } \) :

Fluid density at \(T_{0} \)

Re:

Reynolds number

\(D_\mathrm{B} \) :

Brownian diffusion coefficient

\(D_\mathrm{s} \) :

Solutal diffusivity

\(D_\mathrm{CT} \) :

Soret diffusivity

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

Heat capacity of fluid

Q :

Volume flow rate

\(\gamma \) :

Dimensionless solutal (species) concentration

\(\theta \) :

Temperature of fluid in dimensionless form

\(G_\mathrm{rF} \) :

Nanoparticle Grashof number

C :

Solutal concentration

\(N_\mathrm{t}\) :

Thermophoretic parameter

\(N_\mathrm{CT}\) :

Soret parameter

Ln:

Nanofluid Lewis number

\(\mu \) :

Fluid viscosity

\(\lambda \) :

Wave length

T :

Denotes temperature

k :

Thermal conductivity

\(d_{{1} } +d_{{2} }\) :

Width of channel

\(\dot{{\gamma }}\) :

Shear rate

\(\in \) :

Magnetic diffusivity

\(\lambda _{{2} }\) :

Retardation time

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

Fluid density

\(\rho _{p} \) :

Nanoparticle mass density

\(D_\mathrm{T} \) :

Thermophoretic diffusion coefficient

\(D_\mathrm{TC} \) :

Dufour diffusivity

\(\delta \) :

Dimensionless wave number

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

Effective heat capacity of nanoparticle,

\(\varPsi \) :

Stream function

\(\varOmega \) :

Nanoparticle volume fraction

\(G_\mathrm{rT} \) :

Thermal Grashof number

\(G_\mathrm{rc} \) :

Solutal Grashof number

Pr:

Prandtl number

\(R_\mathrm{m}\) :

Magnetic Reynolds number

\(N_\mathrm{b}\) :

Brownian motion parameter

\(N_\mathrm{TC}\) :

Dufour parameter

Le:

Regular Lewis number

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Authors and Affiliations

  1. MCS, National University of Sciences and Technology, Islamabad, Pakistan

    Safia Akram, Alia Razia & Farkhanda Afzal

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  1. Safia Akram
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Appendix

Appendix

$$\begin{aligned} \xi _{0}= & {} G_{\mathrm{rF}} \left( {-\frac{e^{-h_{1} m_{2} }N_\mathrm{t} }{N_\mathrm{b} \left( {e^{-h_{2} m_{2} }-e^{-h_{1} m_{2} }} \right) }+\frac{h_{1} N_\mathrm{t} }{\left( {h_{2} -h_{1} } \right) \,N_\mathrm{b} }+\frac{h_{1} }{h_{2} -h_{1} }} \right) \\&+\,G_{\mathrm{rc}} \left( {\frac{N_{\mathrm{CT}} e^{-h_{1} m_{2} }}{e^{-h_{2} m_{2} }-e^{-h_{1} m_{2} }}-\frac{h_{1} N_{\mathrm{CT}} }{h_{2} -h_{1} }-\frac{h_{1} }{h_{2} -h_{1} }} \right) -\frac{G_{\mathrm{rt}} e^{-h_{1} m_{2} }}{e^{-h_{2} m_{2} }-e^{-h_{1} m_{2} }}, \\ \xi _{1}= & {} \frac{e^{\left( {h_{1} +h_{2} } \right) \,m_{2} }\left( {N_\mathrm{b} \left( {G_{\mathrm{rt}} -N_{\mathrm{CT}} G_{\mathrm{rc}} } \right) +G_{\mathrm{rF}} N_\mathrm{t} } \right) }{N_\mathrm{b} \left( {e^{h_{1} m_{2} }-e^{h_{2} m_{2} }} \right) },\\ \xi _{2}= & {} \frac{G_{\mathrm{rF}} \left( {N_\mathrm{b} +N_\mathrm{t} } \right) -N_\mathrm{b} \left( {N_{\mathrm{CT}} +1} \right) \,G_{\mathrm{rc}} }{\left( {h_{1} -h_{2} } \right) \,N_\mathrm{b} }, \\ \xi _{3}= & {} \frac{m_{2}^{2} }{m_{1}^{2} \left( {m_{1}^{2} -m_{2}^{2} } \right) }-\frac{1}{m_{1}^{2} -m_{2}^{2} },_{{}} \xi _{4} =E+\frac{\xi _{0} }{m_{1}^{2} },\\ K_{0}= & {} \frac{e^{h_{1} m_{1} }\left( {\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) +1} \right) }{\lambda _{1} +1}, \\ K_{1}= & {} \frac{e^{h_{2} m_{1} }\left( {\lambda _{1} -m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) +1} \right) }{\lambda _{1} +1}, \\ K_{2}= & {} \frac{e^{-h_{1} m_{1} }\left( {\lambda _{1} +\eta _{2} m_{1}^{2} -\eta _{1} m_{1} +1} \right) }{\lambda _{1} +1}, \\ K_{3}= & {} \frac{e^{-h_{2} m_{1} }\left( {\lambda _{1} +m_{1} \left( {\eta _{1} -\eta _{2} m_{1} } \right) +1} \right) }{\lambda _{1} +1}, \\ K_{4}= & {} \frac{\xi _{1} e^{-h_{2} m_{2} }\left( {-\lambda _{1} +\eta _{2} m_{2}^{2} -\eta _{1} m_{2} -1} \right) }{\left( {\lambda _{1} +1} \right) \,\left( {m_{2}^{2} -m_{1}^{2} } \right) }, \\ K_{5}= & {} -\frac{\xi _{1} e^{-h_{1} m_{2} }\left( {\lambda _{1} +\eta _{2} m_{2}^{2} -\eta _{1} m_{2} +1} \right) }{\left( {\lambda _{1} +1} \right) \,\left( {m_{2}^{2} -m_{1}^{2} } \right) }, \\ K_{6}= & {} \xi _{3} \left( {K_{1} -K_{0} } \right) \,\left( {\xi _{3} \left( {\frac{e^{-h_{1} m_{1} }}{m_{1} }-\frac{e^{-h_{2} m_{1} }}{m_{1} }} \right) -K_{2} \left( {h_{2} \xi _{3} -h_{1} \xi _{3} } \right) } \right) , \\ K_{7}= & {} \xi _{3} \left( {K_{3} -K_{2} } \right) \,\left( {\xi _{3} \left( {\frac{e^{h_{2} m_{1} }}{m_{1} }-\frac{e^{h_{1} m_{1} }}{m_{1} }} \right) -K_{0} \left( {h_{2} \xi _{3} -h_{1} \xi _{3} } \right) } \right) , \\ K_{8}= & {} \frac{\xi _{1} \left( {e^{-h_{1} m_{2} }-e^{-h_{2} m_{2} }} \right) }{m_{2} \left( {m_{1}^{2} -m_{2}^{2} } \right) }+\frac{\xi _{2} \left( {h_{2}^{2} -h_{1}^{2} } \right) }{2m_{1}^{2} }+(h_{2} -h_{1} )\xi _{4}, \\ K_{9}= & {} \frac{\xi _{3} \left( {e^{h_{2} m_{1} }-e^{h_{1} m_{1} }} \right) }{m_{1} }-K_{0} \xi _{3} \left( {h_{2} -h_{1} } \right) , \\ K_{10}= & {} -\frac{\eta _{1} \xi _{2} }{m_{1}^{2} \left( {\lambda _{1} +1} \right) }+\frac{h_{2} \xi _{2} }{m_{1}^{2} }+K_{4} +\xi _{4} +1, \\ K_{11}= & {} \frac{\xi _{2} \eta _{1} }{m_{1}^{2} \left( {\lambda _{1} +1} \right) }+\frac{h_{1} \xi _{2} }{m_{1}^{2} }+K_{5} +\xi _{4} +1, \\ K_{12}= & {} \xi _{3} \left( {K_{1} -K_{0} } \right) \,\left( {\xi _{3} \left( {F+K_{8} } \right) -K_{11} \left( {h_{2} -h_{1} } \right) \,\xi _{3} } \right) ,\\ K_{13}= & {} e^{h_{1} m_{1} }\xi _{3} \left( {K_{10} -K_{11} } \right) \,\left( {h_{1} m_{1} \left( {\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) +1} \right) -\lambda _{1} -1} \right) ,\\ K_{14}= & {} e^{-h_{1} m_{1} }\left( {h_{1} m_{1} (1+\lambda _{1} -\eta _{1} m_{1} +\eta _{2} m_{1}^{2} )+\lambda _{1} +1} \right) ,\\ K_{15}= & {} \frac{K_{13} \left( {K_{3} -K_{2} } \right) \,\xi _{3} }{\left( {\lambda _{1} +1} \right) \,m_{1} \left( {K_{1} -K_{0} } \right) \,\xi _{3} },\\ K_{16}= & {} e^{2h_{2} m_{1} }\left( {-\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) -1} \right) \,\left( {\lambda _{1} +\eta _{2} m_{1}^{2} -\eta _{1} m_{1} +1} \right) \\&-\,e^{2h_{1} m_{1} }\left( {-\lambda _{1} +\eta _{2} m_{1}^{2} -\eta _{1} m_{1} -1} \right) \,\left( {\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) +1} \right) , \\ K_{17}= & {} e^{h_{2} m_{1} }\left( {-\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) -1} \right) +e^{h_{1} m_{1} }\left( {\lambda _{1} +m_{1} \left( {\eta _{1} +\eta _{2} m_{1} } \right) +1} \right) ,\\ K_{18}= & {} \xi _{3} \left( {K_{1} -K_{0} } \right) \,\left( {\xi _{3} \left( {F+K_{8} } \right) -K_{11} \left( {h_{2} -h_{1} } \right) \,\xi _{3} } \right) -K_{9} \xi _{3} \left( {K_{10} -K_{11} } \right) , \end{aligned}$$

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Akram, S., Razia, A. & Afzal, F. Effects of velocity second slip model and induced magnetic field on peristaltic transport of non-Newtonian fluid in the presence of double-diffusivity convection in nanofluids. Arch Appl Mech 90, 1583–1603 (2020). https://doi.org/10.1007/s00419-020-01685-4

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