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Mathematical analysis of hybridized ferromagnetic nanofluid with induction of copper oxide nanoparticles in permeable channel by incorporating Darcy–Forchheimer relation

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Abstract

Current pagination is devoted to explicate flow attributes of magnetized ferronanoliquid in permeable media in an absorbent channel. Darcy–Forchheimer's relation is utilized to interpret the consideration of porosity. Two types of metallic oxide nanoparticles are used to study hybrid nano-ferrofluids influenced by convective conduction. Mathematical formulation of problem is attained in partial differential structuring and transmuted into ODE’s by incorporating transformation. Solution of manipulated formulation is attained by implementing shooting method. Influence of involved flow variables is elaborated in graphical manner. Variation in quantities like \(f''\)\((\eta )\) (wall friction coefficient) and \(\theta ^{\prime}(\eta )\) (heat flux) is disclosed in tables. It is observed from analysis that by incrementing Forchheimer number (F) and porosity parameter (r) momentum distribution declines. Thermal augmentation performance is explored with the variation of critical values of the hybrid-nanoparticles volume fractions along with expanding/contracting parameter. It is deduced from examination that addition of hybridized particle composed of \({\text{Fe}}_{{\text{3}}} {\text{O}}_{{\text{4}}}\)+CuO raises thermophysical features of base fluid rather than consideration of CuO particles separately. Novel findings associated with wall drag coefficient at lower boundary of channel in view of decrement in its aptitude versus Darcy–Forchheimer variable in case of contraction is depicted, whereas reverse behavior is observed for the situation of expansion in channel is attained. Decrease in convective thermal flux and wall drag coefficients are exhibited against Reynolds number. Optimum positive change in Nusselt number against Eckert number (Ec) is manifested at nanoparticle volume fraction magnitude equals to 0.05. With expansion in channel Nusselt and skin friction coefficients magnitude drops suddenly.

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Abbreviations

F :

Darcy porous medium parameter

r :

Porous medium parameter

F fd :

Ferrofluid

HFfd :

Hybrid ferrofluid

NP's :

Nanoparticles

FP's :

Ferroparticles

Pr:

Prandtl number

Re:

Reynolds number

Ec :

Eckert number

Nu:

Nusselt number

Pe:

Peclet number

k bf :

Shape factor for thermal conductivity base fluid

µ bf :

Dynamic viscosity of the base fluid

\(\upsilon\) hffd :

Kinematic viscosity of HFfd

α :

Thermal diffusivity

µ hffd :

Dynamic viscosity of HFfd

α hffd :

Thermal diffusivity of HFfd

ρ bf :

Water density (base fluid)

µ bf :

Water dynamic viscosity

x, y :

Coordinate axis

ρ hffd :

Density of HFfd

\(\upsilon\) :

Kinematic viscosity

µ :

Dynamic viscosity

ρ :

Density

ρ c P :

Specific heat capacity

T :

Temperature

(C p)hffd :

Specific heat of HFfd

k hffd :

Thermal conductivity of HFfd

N:

Size of particles

P :

Pressure

u, v :

Velocity components

ρ s 1, ρ s 2 :

Densities of FP's

k s 1, k s 2 :

Thermal conductivities for FP'sFe3o4, Cuo

η :

Independent similarity variable

φ 1, φ 2 :

Ferroparticle’s volume fraction

F η :

Dimensionless radial velocity profile

θ η :

Dimensionless temperature profile

K :

Thermal conductivity

dy :

Heat flux

A VP :

Axial velocity profile

R VP :

Radial velocity profile

T P :

Temperature profile

ς y :

Shear stresses

Rer :

Local Reynolds number

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Authors and Affiliations

  1. Department of Mathematics, Air University, P.A.F Complex, Sector E-9, Islambad, 44000, Pakistan

    S. Bilal

  2. Department of Mathematics, Air University, Multan Campus, Islamabad, 60000, Pakistan

    M. Zubair Akbar Qureshi

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Bilal, S., Qureshi, M.Z.A. Mathematical analysis of hybridized ferromagnetic nanofluid with induction of copper oxide nanoparticles in permeable channel by incorporating Darcy–Forchheimer relation. Math Sci (2021). https://doi.org/10.1007/s40096-021-00421-5

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