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Nonlinear Radiation in Bioconvective Casson Nanofluid Flow

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Abstract

We investigate the impact of nonlinear thermal radiation and variable transport properties on the two-dimensional flow of an electrically conducting Casson nanofluid containing gyrotactic microorganisms along a moving wedge. In some previous studies, it has been assumed that the fluid viscosity and thermal conductivity are temperature dependent. However, this study assumes that the fluid viscosity, thermal conductivity, and the nanofluid properties, are dependent on the solute concentration. Some experimental studies have shown that the viscosity and thermal conductivity of nanofluids are strongly dependent on the volume fraction of nanoparticles rather than just the temperature. The spectral local linearization method is used to solve the conservation equations. We compare our results with those in the literature, and we discuss the convergence and accuracy of the spectral local linearization method. The impact of some parameters on the skin friction, heat and microorganisms mass transport is discussed.

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Abbreviations

\(\alpha \) :

\(\theta _w - 1\)

\(\beta \) :

Casson fluid parameter

\(\eta \) :

Independent similarity variable

\(\gamma \) :

Velocity ratio parameter

\(\lambda \) :

Wedge angle parameter

\(\mu (c)\) :

Variable dynamic viscosity of the flow (\(\frac{\mathrm{kg}}{\mathrm{ms}}\))

\(\mu _{\infty }\) :

Constant dynamic viscosity of the fluid (\(\frac{\mathrm{kg}}{\mathrm{ms}}\))

\(\mu _{B}\) :

Plastic viscosity \(\left( \frac{\mathrm{kg}}{\mathrm{ms}}\right) \)

\(\nu _{\infty }\) :

Constant kinematic viscosity of the fluid (\(\frac{\mathrm{m}^2}{\mathrm{s}}\))

\(\Omega \) :

Total angle of the wedge

\(\phi (\eta )\) :

Dimensionless nanoparticle volume fraction

\(\pi \) :

Component of product of rate of strain

\(\pi _{c}\) :

Critical value of product of rate of strain

\(\psi \) :

Stream function

\(\rho _{\infty }\) :

Constant density (\(\frac{\mathrm{kg}}{\mathrm{m}^3}\))

\(\sigma \) :

Electrical conductivity \(\left( \frac{\mathrm{S}}{\mathrm{m}}\right) \)

\(\sigma ^*\) :

Stefan–Boltzmann constant \(\left( \frac{\mathrm{W}}{\mathrm{m}^2 \mathrm{K}^4}\right) \)

\(\sigma _1\) :

Microorganisms concentration difference

\(\tau \) :

\(\frac{(\rho cp)_p}{(\rho cp)_f}\) Ratio of the effective heat capacity of the nanoparticle material to the fluid heat capacity

\(\tau _w\) :

Local wall shear stress \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(\tau _{ij}\) :

Component of the stress tensor \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(\theta (\eta )\) :

Dimensionless temperature

\(\theta _w\) :

Surface temperature parameter

\(\chi (\eta )\) :

Dimensionless number of motile microorganism

\((\rho cp)_f\) :

Heat capacity of the fluid (\(\frac{\mathrm{J}}{\mathrm{m}^3 \mathrm{K}}\))

\((\rho cp)_p\) :

Effective heat capacity of the nanoparticle medium \(\left( \frac{\mathrm{J}}{\mathrm{m}^3 \mathrm{K}}\right) \)

\(B_{0}\) :

Magnetic induction parameter (T)

C :

Nanoparticles volume fraction (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(C_{\infty }\) :

Concentration far away from the surface (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(C_w\) :

Concentration of the fluid (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(Cf_x\) :

Skin friction coefficient

\(D_{B,\infty }\) :

Constant mass diffusivity \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_{N,\infty }\) :

Constant microorganism diffusivity \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_B(c)\) :

Variable Brownian diffusion coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_N(c)\) :

Variable microorganism diffusivity coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_T\) :

Thermophoretic diffusion coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

M :

Magnetic field

N :

Number of motile microorganism (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(N_{\infty }\) :

Concentration density of motile microorganisms far away from the surface (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(N_w\) :

Concentration density of the motile microorganism (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

Nb :

Brownian motion

\(Nn_x\) :

Local density of motile microorganisms

Nt :

Thermophoresis diffusion

\(Nu_x\) :

Local Nusselt number

Pe :

Bioconvection Péclet Number

Pr :

Prandtl number

Rd :

Nonlinear thermal radiation

\(Re_x\) :

Reynold’s Number

S :

Suction/injection

Sb :

Bioconvection Lewis number

Sc :

Schmidt number

\(T_{\infty }\) :

Temperature far away from the surface (K)

\(T_w\) :

Fluid temperature (K)

\(U_\infty \) :

Constant velocity at the free stream \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(U_w\) :

Constant velocity at the surface \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(W_c\) :

Maximum cell swimming speed (\(\frac{\mathrm{m}}{\mathrm{s}}\))

b :

Chemotaxis constant (m)

\(c_{5}\) :

Dimensionless variable viscosity parameter \((c_5 = c_1C_\infty \))

\(c_{6}\) :

Dimensionless variable thermal conductivity parameter \((c_6 = c_2C_\infty \))

\(c_{7}\) :

Dimensionless variable mass diffusivity parameter \((c_7 = c_3C_\infty \))

\(c_{8}\) :

Dimensionless variable microorganisms diffusivity parameter \((c_8 = c_4C_\infty \))

\(c_p\) :

Specific heat \(\left( \frac{\mathrm{J}}{\mathrm{kg K}}\right) \)

\(e_{ij}\) :

Rate of strain tensor \(\left( \frac{1}{\mathrm{s}}\right) \)

\(f(\eta )\) :

Dimensionless stream function

k(c):

Variable thermal conductivity \(\left( \frac{\mathrm{W}}{\mathrm{m K}}\right) \)

\(k^*\) :

Mean absorption coefficient

\(k_{\infty }\) :

Constant thermal conductivity \(\left( \frac{\mathrm{W}}{\mathrm{m K}}\right) \)

m :

Stream wise pressure gradient or Falkner–Skan power-law index

\(p_{y}\) :

Yield stress \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(q_N\) :

Local density number of motile microorganisms \(\left( \frac{\mathrm{mol}}{\mathrm{kg}}\right) \)

\(q_w\) :

Local surface heat flux (K)

\(u_e\) :

Wedge velocity at the free stream \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(u_w\) :

Stretching wedge velocity \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

u, v :

Velocity components in the x and y directions respectively \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(v_w\) :

Suction/injection velocity component in the y direction \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

x :

Distance along the surface \(\left( \mathrm{m}\right) \)

y :

Distance normal to the surface \(\left( \mathrm{m}\right) \)

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    I. S. Oyelakin & P. Sibanda

  2. Department of Mathematics, Amity University, Kolkata, Newtown, West Bengal, 700135, India

    S. Mondal

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Oyelakin, I.S., Mondal, S. & Sibanda, P. Nonlinear Radiation in Bioconvective Casson Nanofluid Flow. Int. J. Appl. Comput. Math 5, 124 (2019). https://doi.org/10.1007/s40819-019-0705-0

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