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The combined effects of induced magnetic field, thermophoresis and Brownian motion on double stratified nonlinear convective-radiative Jeffrey nanofluid flow with heat source/sink

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Abstract

This communication explores the theoretical investigation of induced magnetic field, thermophoresis and Brownian motion on an unsteady, laminar and incompressible nonlinear convective flow of Jeffrey nanofluid through a rectangular channel in presence of thermal radiation and heat source/sink. Further, we also considered double stratification and heat and mass flux conditions at the boundaries. The governing flow field equations are shortened into coupled non-dimensional system ordinary differential equation using an appropriate transformation, then carried out a numerical solution using fifth-order Runge–Kutta–Fehlberg scheme along with shooting technique. The effect of various non dimensional form of the characteristic function of flow, heat and mass transfers, induced magnetic field and skin friction are plotted with respect to the prominent physical parameters and calculated heat and mass transfer rates at the plates. The results so obtained are compared with the existing published work for viscous case which leads a considerable agreement.

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Abbreviations

\( a \) :

Suction/injection ratio (mixed suction \( \left| {V_{2} } \right| \ge \left| {V_{1} } \right| \)) \( 1 - \frac{{V_{1} }}{{V_{2} }} \)

t:

Time

\( U_{0} \) :

Arbitrary velocity at \( x = 0 \)

\( \overline{B} \) :

Total magnetic field vector

h:

Width of the channel

\( B_{0} \) :

Applied magnetic field strength

\( B_{y} \) :

Radial induced magnetic field component

\( B_{x} \) :

Axial induced magnetic field component

\( V_{1} e^{i\omega t} \) :

Injection velocity

\( V_{2} e^{i\omega t} \) :

Suction velocity

C:

Concentration

P:

Pressure of the fluid

\( \overline{q} \) :

Velocity of the fluid

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

Intensities of thermal and solutal stratification

c:

Specific heat at constant temperature

\( c_{s} \) :

Concentration susceptibility

\( C^{*} \) :

Dimensionless concentration, \( \frac{{C - C_{1} e^{i\omega t} }}{{\left( {C_{2} - C_{1} } \right)e^{i\omega t} }} \)

\( C_{1} e^{i\omega t} \) :

Concentrations at the lower plate

\( C_{2} e^{i\omega t} \) :

Concentrations at the upper plate

\( k \) :

Thermal conductivity of the fluid

\( S_{t} \) :

Thermal stratification \( \frac{{d_{1} x}}{{T_{2} - T_{1} }} \)

\( S_{s} \) :

Solutal stratification \( \frac{{d_{2} x}}{{C_{2} - C_{1} }} \)

u:

Axial velocity component

v:

Radial velocity component

\( D_{B} \) :

Mass diffusion coefficient

\( T^{*} \) :

Non-dimensional temperature, \( \frac{{T - T_{1} e^{i\omega t} }}{{\left( {T_{2} - T_{1} } \right)e^{i\omega t} }} \)

\( T_{m} \) :

Reference temperature

T:

Temperature

\( T_{1} e^{i\omega t} ,T_{2} e^{i\omega t} \) :

Temperatures at the bottom and top plates, respectively

Pr:

Prandtl number, \( \frac{\mu c}{k} \)

Re:

Suction/injection Reynold’s number, \( \frac{{\rho V_{2} h}}{\mu } \)

Sc :

Schmidt number, \( \frac{\mu }{\rho D} \)

Ec :

Eckert number, \( \frac{{\mu V_{2} }}{{\rho hc\left( {T_{2} - T_{1} } \right)}} \)

Sh :

Sherwood number, \( \frac{{\dot{n}_{A} }}{{h\upsilon \left( {C_{2} - C_{1} } \right)}} \)

Gr :

Thermal Grashof number, \( \frac{{\rho g\beta_{0} \left( {T_{2} - T_{1} } \right)h^{2} }}{{\mu V_{2} }} \)

Gm :

Solutal Grashof number, \( \frac{{\rho g\beta_{2} \left( {C_{2} - C_{1} } \right)h^{2} }}{{\mu V_{2} }} \)

Rm :

Magnetic Reynolds number, \( \sigma \mu_{e} hV_{2} \)

St :

Strommer’s number (magnetic force number), \( \frac{{B_{0} }}{{V_{2} }}\sqrt {\frac{{\mu_{e} }}{\rho }} \)

Nb:

Brownian motion parameter \( \frac{{\tau_{1} D_{B} \left( {C_{2} - C_{1} } \right)}}{\alpha } \)

Nt:

Thermophoresis parameter \( \frac{{\tau_{1} D_{T} \left( {T_{2} - T_{1} } \right)}}{{T_{2} \alpha }} \)

\( 1/\sigma \mu_{e} \) :

Magnetic diffusivity

\( \bar{J} \) :

Current density

\( q_{w} ,q_{m} \) :

Heat and mass fluxes per unit area at the plate

\( \dot{n}_{A} \) :

Mass transfer rate

hf :

Thermal stratification parameters \( \frac{{hq_{w} }}{{\left( {T_{2} - T_{1} } \right)k}} \)

mf :

Solutal stratification parameters \( \frac{{hq_{m} }}{{\left( {C_{2} - C_{1} } \right)D_{B} }} \)

Rd:

Thermal radiation \( \frac{{16\sigma_{1} T_{1}^{3} }}{{3k_{3} k}} \)

Hs:

Heat source/sink parameter \( \frac{{Q_{h} h^{2} }}{k} \)

\( \bar{E} \) :

Electric field

\( k_{3} \) :

Mean absorption coefficient

\( \rho \) :

Density of the fluid

\( \alpha \) :

Thermal diffusivity \( {k \mathord{\left/ {\vphantom {k {\rho c}}} \right. \kern-0pt} {\rho c}} \)

\( \upsilon \) :

Kinematic viscosity

\( \mu_{e} \) :

Magnetic permeability

\( \mu \) :

Dynamic viscosity

\( \sigma \) :

Electric conductivity of the fluid

\( \sigma_{1} \) :

Stefan–Boltzmann constant

\( \lambda_{1} \) :

Ratio of relaxation to retardation times

\( \lambda_{2} \) :

Relaxation time

\( \lambda \) :

Non-dimensional variable, \( \frac{y}{h} \)

\( \theta \) :

Frequency, \( \omega t \)

\( \tau \) :

Shear stress

\( \tau_{1} \) :

The ratio of nanoparticle heat capacity and the base fluid heat capacity

\( \beta_{0} ,\beta_{1} \) :

Coefficients of thermal expansions

\( \beta_{2} ,\beta_{3} \) :

Coefficients of solutal expansions

\( \eta_{1} \) :

Nonlinear thermal convection parameter \( \frac{{\beta_{1} }}{{\beta_{0} }}\left( {T_{2} - T_{1} } \right) \)

\( \eta_{2} \) :

Nonlinear solutal convection parameter \( \frac{{\beta_{3} }}{{\beta_{2} }}\left( {C_{2} - C_{1} } \right) \)

\( \beta \) :

Deborah number, \( \frac{{\lambda_{2} V_{2} }}{h} \)

\( \xi \) :

Dimensionless axial variable, \( \left( {\frac{{U_{0} }}{{aV_{2} }} - \frac{x}{h}} \right) \)

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Acknowledgements

One of the authors (A.R) is thankful to the DRDO, Government of India for providing financial support as SRF (No. DIAT/F/Acad (PhD)/1613/F-15-52-09).

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Authors and Affiliations

  1. Department of Applied Mathematics, Defence Institute of Advanced Technology (Deemed University), Pune, 411025, India

    Adigoppula Raju, Odelu Ojjela & Pravin Kashyap Kambhatla

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  1. Adigoppula Raju
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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Correspondence to Odelu Ojjela.

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Raju, A., Ojjela, O. & Kambhatla, P.K. The combined effects of induced magnetic field, thermophoresis and Brownian motion on double stratified nonlinear convective-radiative Jeffrey nanofluid flow with heat source/sink. J Anal 28, 503–532 (2020). https://doi.org/10.1007/s41478-019-00187-z

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