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Mixed Bioconvective Flow of Williamson Nanofluid Over a Rough Vertical Cone

  • Research Article-Mechanical Engineering
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Abstract

This paper investigates the mixed bio-convection in a flow of Williamson nanofluid with liquid oxygen diffusion. The main innovation is to study the bio-convection and surface roughness in a flow of non-Newtonian Williamson nanofluid via a nonsimilar approach. The applications of bio-convection in boundary layer analysis include biological sciences, drugs and biotechnology, wastewater treatment, biofuels processing, and food production. The nonlinear coupled partial differential equations are the governing equations of the given problem. Mangler’s nonsimilar transformations are utilized to get a non-dimensional form of governing equations. For mathematical simplifications, implicit finite difference scheme and Quasilinearization technique are used. Further, an analysis is carried out for various controlled parameters, and the results are depicted through graphs. The heat transfer rate for nanofluid is approximately about 4% more than that of Williamson nanofluid. Moreover, the heat transfer rate for a permeable surface (suction case) is about 44% more than for an impermeable surface. The wall suction augments the mass transfer rate of liquid oxygen, and it is about 140% for rough surface compared to the smooth surface. Microorganism’s density in the fluid is more for lower values of bio-convection Lewis and Peclet numbers. The microorganism’s density number enhances by about 78% when the value of the Peclet number increases from 1 to 2. The results are compared to earlier published papers and have an excellent agreement with each other.

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Abbreviations

\(u,\,\,v\) :

Velocity components

\(x,\,y\) :

Cartesian coordinates

\(U_{0}\) :

Reference velocity

\(U_{e}\) :

Velocity of the mainstream flow

\(Pr\) :

Prandtl number

\(Re\) :

Reynolds number

\(D_{T}\) :

Thermophoresis coefficient

\(Nt\) :

Thermophoresis parameter

\(D_{B}\) :

Brownian diffusion coefficient

\(Nb\) :

Brownian motion parameter

\(Nr\) :

Nanoparticles Buoyancy ratio parameter

\(J\) :

Ratio of heat capacity of nanoparticles to the fluid phase

\(C_{nf}\) :

Specific heat capacity of the nanoliquid

\(Sc\) :

Schmidt number

\(Le\) :

Lewis number

\(W\) :

Williamson parameter

\(Pe\) :

Bioconvection Peclet number

\(D_{N}\) :

Diffusivity of microorganisms

\(Lb\) :

Bioconvection Lewis number

\(Rb\) :

Bioconvection Rayleigh number

\(Wc\) :

Maximum cell speed of microorganisms

\(b\) :

Chemotaxis constant

\(m\) :

Exponent in the power law variation in the velocity of mainstream

\(A\) :

Suction parameter

\(F\) :

Dimensionless velocity

\(G\) :

Non-dimensional temperature

\(H\) :

Non-dimensional concentration

\(R\) :

Dimensionless nanoparticle’s volume fraction

\(S\) :

Dimensionless microorganisms’ density

\(C_{f}\) :

Skin friction coefficient

\(Re_{x}\) :

Local Reynolds number

\(Nu\) :

Nusselt number

\(Sh_{1}\) :

Mass transfer rate of liquid hydrogen

\(Sh_{2}\) :

Nanoparticle’s mass transfer rate

\(Nn\) :

Density number of motile microorganisms

\(\phi\) :

Half angle of the vertical cone

\(\Gamma\) :

Time constant

\(\nu\) :

Viscosity

\(\rho_{f}\) :

Density of nanofluid

\(\rho_{p}\) :

Density of the solid nanoparticles

\(\rho_{N}\) :

Density of microorganisms

\(\theta\) :

Nanoparticle’s volume fraction

\(\alpha\) :

Roughness parameter

\(\omega\) :

Frequency

\(\varepsilon\) :

Velocity ratio parameter

\(\gamma\) :

Average volume of a microorganisms

\(\sigma\) :

Microorganisms’ concentration difference

\(\lambda\) :

Mixed convection parameter

\(\xi ,\,\,\eta\) :

Transformed variables

\(p\) :

Nanoparticles

\(f\) :

Nanofluid

\(w\) :

Conditions at the cone’s surface

\(\infty\) :

Conditions at the boundary layer edge

\(\xi ,\,\,\eta\) :

Derivatives w.r.t these variables

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Acknowledgements

This work was partially supported by the Karnataka Science and Technology Promotion Society, Bengaluru, India, through a fellowship for Sunil Benawadi No. DST/KSTePS/Ph.D. Fellowship/OTH-03:2020-21.

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Authors and Affiliations

  1. Department of Mathematics, Karnatak University, Pavate Nagar, Dharwad, 580003, India

    P. M. Patil & Sunil Benawadi

  2. Wrangler Dr. D.C. Pavate Institute of Mathematical Sciences (PIMSci.), Karnatak University, Pavate Nagar, Dharwad, 580003, India

    P. M. Patil

  3. Department of Physics, JSS College, Vidyagiri, Dharwad, 580004, India

    V. T. Muttannavar

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  1. P. M. Patil
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Correspondence to P. M. Patil.

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Patil, P.M., Benawadi, S. & Muttannavar, V.T. Mixed Bioconvective Flow of Williamson Nanofluid Over a Rough Vertical Cone. Arab J Sci Eng 48, 2917–2928 (2023). https://doi.org/10.1007/s13369-022-07048-1

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