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Mixed Convection and Double Diffusion Impacts on Williamson–Sutterby Nanofluid with Activation Energy, Cattaneo–Christov Heat Flux, and a Magnetic Dipole

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Abstract

Current theoretical and computational investigation within the Darcy–Forchheimer medium through electromagnetic fields reveals the heat and mass transportation characteristics for the flow of Williamson–Sutterby nanofluid under the effects of Cattaneo–Christov double diffusion, radiation heat flux, magnetic dipole and convective boundary over a stretchy surface are taken into account. Cattaneo–Christov heat mass flux model is applied to shape the thermal and nanoparticle concentration equations. The prime purpose of the current study is to enhance the rate of heat transfer in industrial processes. Mechanism of thermal insulation, pharmacological processes, pace technology, crushing, nuclear reactor cooling, pharmaceutical procedures, geothermal reservoirs to enhance oil recovery with chemically reactive systems has great importance in mass transport. Moreover, the mass transportation is made by Arrhenius activation energy and binary chemical reaction. Additionally, the influences of bioconvection of self-propelled microorganisms are considered. The nanofluids with swimming microorganisms have great significance in microfluidics devices, medicine, cancer therapy, biotechnology applications such as biofuels and enzyme biosensor. The amalgamation of Sutterby–Williamson nanoparticles and microorganism with slight homogeneous diffusion exhibit the novelty of present work. Nonlinear PDEs are transformed into coupled ODEs by using similarity functions. The attained equations are then solved numerically with the help of software. Hartmann, Sutterby Reynolds, Sutterby Deborah, bioconvection Rayleigh numbers, mixed convection, porosity, ferrohydrodynamic interaction, buoyancy ratio and inertial resistance parameters slow down the flow of fluids. But reverse effects were observed for temperature, concentration and motile density profiles for Hartmann, porosity and ferrohydrodynamic interaction parameters. These theoretical outcomes play a vital role in industrial applications to enhance the heat/mass process, biomedical flows, heating and cooling systems, chemical processes and mining industries.

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Abbreviations

\(\alpha ^{'}\) :

Dimensionless distance

\(\alpha ^{*}\) :

Mixed convection parameter

\(\beta _\text {d}\) :

Ferrohydrodynamic interaction variable

\(\delta _1\) :

Temperature difference parameter

\(\epsilon _1\) :

Temperature difference variable

\(\eta \) :

Similarity variable

\(\Gamma \) :

Fluid relaxation time

\(\lambda \) :

Williamson parameter

\(\lambda ^{'}\) :

Viscous dissipation factor

\(\lambda _\text {o}\) :

Magnetic permeability

\(\mu \) :

Dynamic viscosity

\(\Omega \) :

concentration difference of motile microorganism

\(\phi \) :

Dimensionless nanoparticle concentration

\(\Phi ^*\) :

Magnetic scalar potential

\(\Psi \) :

Dimensionless density of microorganism

\(\psi \) :

Stream function

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

Base liquid density

\(\rho _\text {m}\) :

Density of microorganisms

\(\rho _\text {p}\) :

Nanoparticle mass density

\(\sigma \) :

Chemical reaction rate constant

\(\sigma ^*\) :

Stefan–Boltzmann constant

\(\sigma _\text {m}\) :

Magnetic permeability

\(\tau \) :

Parameter defined by ratio between \((\rho C)_\text {p}\) and \((\rho C)_\text {f}\)

\(\tau _\text {w}\) :

Wall shear stress

\(\theta \) :

Dimensionless temperature of the fluid

\(\infty \) :

Ambient

p:

Nanoparticles

w:

On the wall

a :

Positive constant

\(B_0\) :

Uniform transverse magnetic field

\(b_1\) :

Chemotaxis constant

c :

Distance

\(C_\infty \) :

Ambient concentration

\(C_\text {F}\) :

Forchheimer quantity

\(C_\text {p}\) :

Specific heat

\(C_\text {w}\) :

Wall concentration

\(C_\text {{fx}}\) :

Velocity gradient

\(D_\text {B}\) :

Brownian coefficient

\(D_\text {N}\) :

Microorganism diffusion coefficient

\(D_\text {T}\) :

Thermophoretic diffusion coefficient

De :

Deborah number

\(E_0\) :

Electric field

\(E_1\) :

Electric parameter

Ea:

Activation energy

Ec:

Eckert number

\(F^*\) :

Inertia parameter

H :

Magnetic field

Ha:

Hartmann number

k :

Thermal conductivity

\(k^{'}\) :

Absorbent medium of permeability

\(K_1\) :

Pyromagnetic coefficient

\(k_2\) :

Concentration jump length

\(k_3\) :

Motile density jump length

\(k_\text {o}\) :

Thermal jump length

Lb:

Bioconvection Lewis number

M :

Magnetization

m :

Fitted rate constant

N :

Density of microorganism

\(N_\infty \) :

Ambient density

\(N_\text {b}\) :

Brownian movement variable

\(N_\text {o}\) :

Slip length

\(N_\text {r}\) :

Buoyancy ratio parameter

\(N_\text {t}\) :

Thermophoresis variable

\(N_\text {w}\) :

Wall density

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

Density gradient

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

Thermal gradient

Pe:

Peclet number

Pr:

Prandtl number

\(q_\text {m}\) :

Local mass flux

\(q_\text {n}\) :

Motile microorganism flux

\(q_\text {w}\) :

Local heat flux

Rb:

Rayleigh number

Rd:

Radiation parameter

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

Local Reynolds number

S :

Flow deportment index

Sc:

Schmidt number

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

Solutal gradient

\(T_\infty \) :

Ambient temperature

\(T_\text {w}\) :

Surface Temperature

u, v :

Velocity components

\(U_\text {w}\) :

Wall velocity

\(W_\text {c}\) :

Maximum cell swimming speed

x, y :

Space coordinates

ODEs:

Ordinary differential equations

PDEs:

Partial differential equations

BVP:

Boundary value problem

CCDD:

Cattaneo–Christov double diffusion

CCHF:

Cattaneo–Christov heat flux

MHD:

Magnetohydrodynamics

RKF:

Runge–Kutta–Fehlberg

BCs:

Boundary conditions

BL:

Boundary layer

MF:

Magnetic field

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  1. Department of Mathematics, University of Engineering and Technology, Lahore, 54890, Pakistan

    Samina Saeed Khan, Muhammad Mushtaq & Kanwal Jabeen

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Khan, S.S., Mushtaq, M. & Jabeen, K. Mixed Convection and Double Diffusion Impacts on Williamson–Sutterby Nanofluid with Activation Energy, Cattaneo–Christov Heat Flux, and a Magnetic Dipole. Arab J Sci Eng (2023). https://doi.org/10.1007/s13369-023-08367-7

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