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A Non-Fourier’s and Non-Fick’s Approach to Study MHD Mixed Convective Copper Water Nanofluid Flow over Flat Plate Subjected to Convective Heating and Zero Wall Mass Flux Condition

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Abstract

A theoretical model of MHD mixed convective Cu–water nanofluid boundary layer flow over flat vertical plate has been developed and investigated. As a novelty, firstly, modified Buongiorno’s model is utilized to include the effects of Brownian motion, thermophoresis and volume fraction for nanofluid. Secondly, thermal energy equation and concentration equation are modeled with the help of Cattaneo–Christov theory of heat and mass flux, respectively. Due to this non-Fourier’s and non-Fick’s approach, two parameters namely, thermal relaxation parameter and solutal relaxation parameter were introduced in thermal energy equation and concentration equation, respectively. In addition, the surface of flat plate is subjected to suction, convective heating and zero wall mass flux condition. Authors have used the similarity method and through analysis it is shown that transport equations can be converted to ODEs with the help of suitable similarity transformations. The analysis and computed results shows that various dimensional and non-dimensional parameters influence the velocity, temperature and concentration profiles. The pattern and behavior of boundary layer is depicted graphically. The results for skin friction coefficient and heat transfer coefficient are outlined in tabular form. The result of passive control of nanoparticles at the surface is that Brownian motion parameter does not influence the temperature profiles of nanofluid flow and heat transfer rate at the surface. Heat transfer coefficient is positively correlated to thermal relaxation parameter and Biot number, whereas thermophoresis parameter causes it to decrease. The flow of nanofluid is aided by buoyancy ratio parameter and thermophoresis parameter but contrary behavior is seen for magnetic parameter. The effect of volume fraction and suction parameter is to increase the value of skin friction coefficient.

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Data Availability Statement

All the data generated and materials during this study are included in this article.

Abbreviations

a :

Constant

\(A_{o} ,B_{o}\) :

Constant

\(B\) :

Magnetic field

Bi:

Biot number

C :

Concentration of the fluid

C f * :

Skin friction coefficient

\(C_{o} ,D_{o}\) :

Constant

\(D_{B}\) :

Brownian diffusion coefficient

\(D_{T}\) :

Thermophoretic diffusion coefficient

\(E_{o} ,F_{o}\) :

Constant

\(f^{{\prime}}\) :

Dimensionless velocity

\(\hbox{Gr}_{x}\) :

Local Grashof number

g :

Acceleration due to gravity

h :

Heat transfer coefficient of base fluid

j :

Mass flux

k :

Thermal conductivity

l :

Characteristic length

M :

Magnetic parameter

Nb :

Brownian motion parameter

Nr :

Buoyancy ratio parameter

Nt :

Thermophoretic parameter

\({\text{Nu}}_{x}^{*}\) :

Local Nusselt number

Pr:

Prandtl number

q :

Heat flux

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

Local Reynolds number

s :

Wall temperature related parameter

Sc:

Schmidt number

Su:

Suction parameter

T :

Temperature of the fluid

u :

Velocity component in x-direction

v :

Velocity component in y-direction

v :

Velocity

v w :

Velocity of mass transfer

x, y :

Cartesian coordinate system

\(\beta\) :

Thermal expansion coefficient

\(\gamma_{t}\) :

Thermal relaxation time parameter

\(\gamma_{c}\) :

Solutal relaxation time parameter

\(\eta\) :

Similarity variable

\(\Theta\) :

Dimensionless temperature

\(\lambda\) :

Mixed convection parameter

\(\mu\) :

Dynamic viscosity

\(\nu\) :

Kinematic viscosity

\(\rho\) :

Effective density

\(\rho c_{p}\) :

Heat capacity

\(\sigma\) :

Electrical conductivity

\(\tau\) :

Ratio of the effective heat capacity of the nanomaterial and base fluid

\(\tau_{t}\) :

Heat flux relaxation time

\(\tau_{c}\) :

Mass flux relaxation time

\(\varphi\) :

Dimensionless concentration

\(\phi\) :

Volume fraction of nanoparticle

\(^{\prime}\) :

Derivative w.r.t η

f :

Base fluid

nf :

Nanofluid

p :

Solid particle

w :

Wall

\(\infty\) :

Ambient condition

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Acknowledgements

This research was supported by UGC, Government of India (F. No. 16-6 (Dec. 2017)/2018 (NET/CSIR)) and thankfully recognized by the first author.

Funding

This research was supported by UGC, Government of India (F. No. 16-6 (Dec. 2017)/2018 (NET/CSIR)) and thankfully recognized by the first author.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India

    Sawan Kumar Rawat, Shubham Negi & Manoj Kumar

  2. Department of Allied Sciences, Graphic Era Hill University, Haldwani, Uttarakhand, India

    Himanshu Upreti

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  1. Sawan Kumar Rawat
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Rawat, S.K., Negi, S., Upreti, H. et al. A Non-Fourier’s and Non-Fick’s Approach to Study MHD Mixed Convective Copper Water Nanofluid Flow over Flat Plate Subjected to Convective Heating and Zero Wall Mass Flux Condition. Int. J. Appl. Comput. Math 7, 246 (2021). https://doi.org/10.1007/s40819-021-01190-4

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