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A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects

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Abstract

In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms’ density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.

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Abbreviations

\(\bar{u},\bar{v},\bar{w}\) :

Velocity components

\(\bar{r},\bar{z}\) :

Cylindrical coordinate

Re :

Reynolds number

\(\tilde{t}\) :

Time

\(\bar{C}\) :

Nanoparticle volume fraction

\(\bar{P}\) :

Pressure

F w :

Suction/injection parameter

\(\bar{T}_{\infty }\) :

Atmosphere temperature

\(\bar{C}_{\infty }\) :

Atmosphere concentration

\(\bar{T}_{w}\) :

Surface temperature

N t :

Thermophoresis parameter

N b :

Brownian motion parameter

N 1 :

Velocity slip parameter

\(\bar{n}_{w}\) :

Surface density of motile-organism

\(\bar{k}_{\text{p}}\) :

Permeability of porous medium

\(\bar{t}\) :

Time

J :

Heat flux of microorganisms

a 0 (>0):

Constant

c F :

Forchheimer coefficient

Q 0 :

Heat source/sink

\(\bar{D}_{1}\) :

Thermal slip factor

\(D_{{\bar{T}}}\) :

Thermophoretic diffusion coefficient

\(\bar{n}\) :

Density of motile microorganisms

\(\bar{b}\) :

Chemotaxis constant

W c :

Maximum cell swimming speed

S :

Unsteady parameter

D B :

Brownian diffusion coefficient

\(D_{{\bar{n}}}\) :

A micro-organism diffusivity

Pr :

Prandtl number

Sc :

Schmidt number

Sb :

Bio-convection Schmidt number

Pe :

Peclet number

k f :

Forchheimer number

M :

Magnetic parameter

R d :

Thermal radiation parameter

H S :

Heat source/sink parameter

q w :

Surface heat flux

q M :

Surface mass flux

q N :

Motile surface microorganism flux

k′:

Mean absorption coefficient

\(C_{{F\bar{x}}}\) :

Skin friction coefficient

\(Nu_{{\bar{x}}}\) :

Nusselt number

\(Sh_{{\bar{x}}}\) :

Sherwood number

\(N_{{n\bar{x}}}\) :

Density number of motile microorganisms

\(\bar{\beta }\) :

Contraction expansion strength

\(\bar{\alpha }_{m}\) :

Thermal conductivity

(ρc)f :

Heat capacity of the fluid

(ρc)p :

Heat capacity of nanoparticle

σ :

Electrical conductivity

μ :

Viscosity of nanofluid

θ :

Temperature profile

ϕ :

Nanoparticle concentration profile

Φ:

Motile microorganism density profile

\(\bar{\sigma }\) :

Stefan–Boltzmann constant

β :

Velocity slip

τ w :

Shear stress

ρ p :

Density of nanoparticles

σ :

Electrical conductivity

β T :

Thermal slip

ρ :

Density

ν :

Kinematic viscosity

γ :

Chemical reaction parameter

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    M. M. Bhatti

  2. Department of Mathematics, Siksha ‘O’ Anusandhan University, Khandagiri, Bhubaneswar, Odisha, 751030, India

    S. R. Mishra

  3. Department of Mathematics, Quaid-I-Azam University, Islamabad, 44000, Pakistan

    T. Abbas

  4. Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai, 201804, China

    M. M. Rashidi

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  1. M. M. Bhatti
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  2. S. R. Mishra
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  3. T. Abbas
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  4. M. M. Rashidi
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Correspondence to M. M. Bhatti.

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Bhatti, M.M., Mishra, S.R., Abbas, T. et al. A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects. Neural Comput & Applic 30, 1237–1249 (2018). https://doi.org/10.1007/s00521-016-2768-8

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  • DOI: https://doi.org/10.1007/s00521-016-2768-8

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