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Mixed bio-convection analysis on MHD Casson hybrid nanofluid flow over a spinning cone/plate embedded in a variable porosity medium: a comparative study

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Abstract

The study of heat and mass transport in MHD flows through various geometries is vital for the design of heat exchangers and other devices. With this initiative, a mathematical framework has been proposed to explore the two-dimensional mixed convective, heat, and mass transport dynamics of the magneto-hydrodynamic Casson hybrid nanofluid flow comprised of \(\text{Fe}_{3} \text{O}_{4}\)/TiO across two distinct geometries, the cone and plate revolving in an upright position within a variable porosity medium. The influence of a heat source/sink, Brownian motion, thermophoresis, bio-convection of gyrotactic microorganisms, chemical reaction, and nanoparticle shape effects are substantial physical features of the investigation. The governing equations are partial differential equations that are subsequently transferred into an appropriate structure of coupled nonlinear ordinary differential equations using the requisite similarity variables. In order to compute the transformed non-dimensional governing equation and their relevant boundary conditions, the Range–Kutta fourth-order methodology is used in conjunction with the shooting procedure. With the aid of graphs and tables, the influence of non-dimensional quantities on momentum, energy, solutal, and motile microorganism profiles, along with friction drag, heat transfer rate, Sherwood, and motile density numbers, is discussed. The results showed that flow over a plate containing nanoparticles in the shape of a blade improves heat transfer rate. Also, it has been revealed that the heat and mass transmission efficiency of a revolving cone is substantially higher than that of a rotating plate. Furthermore, the present findings are compared to prior studies, which show significant agreement.

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Data Availability Statement

No data associated in the manuscript.

Abbreviations

\(\Omega\) :

Angular velocity

Lb:

Bio-convection Lewis number

Pe:

Bio-convection Peclet number

\(D_{\rm{B}}\) :

Brownian motion constant

Nb:

Brownian motion parameter

\(\beta\) :

Casson parameter

\(b\) :

Chemotaxis constant

\(C\) :

Concentration

\(\alpha\) :

Cone half angle

\(\rho\) :

Density (kg m−3)

\(\varphi\) :

Dimensionless concentration

\(D_{\rm{m}}\) :

Diffusivity of microorganism

\(\chi\) :

Dimensionless motile density

\(\theta\) :

Dimensionless temperature

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\sigma\) :

Electrical conductivity (S m−1)

\(\text{Ha}^{2}\) :

Hartmann number

\(Q\) :

Heat source/sink

D − 1 :

, Inverse Darcy number

\(\nu\) :

Kinematic viscosity (m2 s−1)

Le:

Lewis number

B 0 :

Magnetic field (Wb m−1)

\(\gamma\) :

Microorganism buoyancy parameters

\(\omega\) :

Microorganism concentration difference parameter

\(\text{Gr}_{M}\) :

Microorganism Grashof numbers

\(M\) :

Motile density

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

Motile density number

\(g\) :

Non-dimensional circumferential velocity

\(M^{2}\) :

Non-dimensional magnetic parameter

\(h\) :

Non-dimensional normal velocity

\(\eta\) :

Non-dimensional similarity variable

\(f\) :

Non-dimensional tangential velocity

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

Nusselt number

\(K\) :

Permeability of porous medium

\(\varepsilon\) :

Porosity parameter

\(\Pr\) :

Prandtl number

\(r\) :

Radius of the cone

\(\tau\) :

Ratio between heat capacities of fluid and nanoparticle

\(K_{\rm{r}}\) :

Reaction rate constant

\(S\) :

Shape factor

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

Sherwood number

\(C_{fy}\) :

Skin friction along circumferential velocity

\(C_{fx}\) :

Skin friction along normal velocity

\(\delta\) :

Solutal buoyancy parameters

\(\text{Gr}_{\rm{C}}\) :

Solutal Grashof numbers

\(C_{\rm{p}}\) :

Specific heat capacity (J kg−1K −1)

\(W_{\rm{c}}\) :

Swimming cell speed

\(T\) :

Temperature

\(\lambda\) :

Thermal buoyancy parameter

\(\kappa\) :

Thermal conductivity (W m−1K−1)

\(\text{Gr}_{T}\) :

Thermal Grashof numbers

Nt:

Thermophoresis parameter

\(D_{T}\) :

Thermophoretic constant

\(u,v,w\) :

, Velocity components

\(\beta_{M}\) :

Volumetric coefficients of motile density expansion

\(\beta_{\rm{C}}\) :

Volumetric coefficients of solutal expansion

\(\beta_{T}\) :

Volumetric coefficients of thermal expansion (K−1)

\(\infty\) :

Ambient condition

f :

Base fluid

hnf :

Hybrid nanofluid

nf :

Nanofluid

np :

Nanoparticle

w :

Wall

BVP:

Boundary value problem

DEs:

Differential equations

HNF:

Hybrid nanofluid

MHD:

Magneto-hydrodynamics

ODEs:

Ordinary differential equations

PDEs:

Partial differential equations

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    N. M. Lisha & A. G. Vijaya Kumar

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Correspondence to A. G. Vijaya Kumar.

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Lisha, N.M., Vijaya Kumar, A.G. Mixed bio-convection analysis on MHD Casson hybrid nanofluid flow over a spinning cone/plate embedded in a variable porosity medium: a comparative study. Eur. Phys. J. Plus 138, 1042 (2023). https://doi.org/10.1140/epjp/s13360-023-04650-6

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