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Bioconvection flow of magnetized Williamson nanoliquid with motile organisms and variable thermal conductivity

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Among the fabrication of the nano-biometerials, the bioconvection of nanoparticles attained the utmost importance in this decade. Therefore, this theoretical continuation is performed to utilize the bioconvected flow of Williamson nanofluid caused by an oscillatory stretching surface. The flow is generated due to periodic motion of the sheet. The energy equation is modified by variable thermal conductivity. The significance of present flow problem increases by utilizing the thermophoresis and Brownian movement factors. The available formulated partial differential equations are promoted into non-dimensional structure via similarity variables. The analytical solution is fulfilled using convergent technique. The implications of promising parameters on velocity, temperature profile, nanoparticles volume fraction and microorganisms profile are evaluated graphically. Locally constituted physically expressions such as Nusselt number, Sherwood number and motile density number are treated numerically as well as graphically. The presence of variable thermal conductivity, thermophoresis and Brownian motion effects are more frequent for enhancement of heat transfer. The detected observation can involve the theoretical significance in various engineering processes, bio-fuel cells, solar energy system and enhancement of extrusion systems.

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\(\Gamma\) :

Time constant

\(\rho_{{\text{f}}}\) :

Fluid density

\(\varphi\) :

Porous medium

\(\rho_{{\text{m}}}\) :

Microorganism particles

\(g\) :


\(K\left( T \right)\) :

Thermal conductivity

\(K_{\infty }\) :

Ambient liquid conductivity

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

Brownian diffusion coefficients

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

Thermophoretic diffusion coefficient

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

Microorganism diffusivity

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

Maximum speed of swimming cell

\(C_{{\text{w}}}\) :

Surface concentration

\({\text{We}}\) :

Local Williamson parameter

\(\beta\) :

Combined parameter

\({\text{Nr}}\) :

Buoyancy ratio parameter

\(\Pr = \nu /\alpha_{{\text{f}}}\) :

The Prandtl number

\({\text{Sc}}\) :

The Schmidt number

\({\text{Lb}}\) :

Bioconvected Lewis number

\(\sigma\) :

Microorganism concentration difference parameter

\({\text{Sh}}_{2}\) :

Local Sherwood number

\(\sigma^{ * }\) :

Electrical conductivity

\(k^{ * }\) :

Permeability parameter

\(\rho_{{\text{p}}}\) :

Density of nanoparticles

\(\beta^{ * }\) :

Volume expansion coefficient

\(T\) :


\(C\) :


\(\varepsilon\) :

Thermal dependence conductivity parameter

\(\tau_{1} = \left( {\rho c} \right)_{{\text{p}}} /\left( {\rho c} \right)_{{\text{f}}}\) :

Heat capacitance of nanoparticles to heat capacitance of fluid ratio

\(n\) :

Gyrotactic microorganism density

\(b_{1}\) :

Chemotaxis constant

\(T_{{\text{w}}}\) :

Surface temperature

\(n_{{\text{w}}}\) :

Surface motile organisms

\(S\) :

Oscillation frequency to stretching rate ratio

\(\lambda\) :

Mixed convection parameter


Bioconvected Rayleigh number


Thermophoresis parameter


Brownian motion parameter


Peclet number

Nu1 :

Local Nusselt number

Nn3 :

Local motile organism density number


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Correspondence to Sabir Ali Shehzad.

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Khan, S.U., Shehzad, S.A. & Ali, N. Bioconvection flow of magnetized Williamson nanoliquid with motile organisms and variable thermal conductivity. Appl Nanosci 10, 3325–3336 (2020).

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