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Scrutinization of marangoni convective flow of dusty hybrid nanofluid with gyrotactic microorganisms and thermophoretic particle deposition

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Abstract

The objective of the present work is to study the influence of heat source on steady MHD Marangoni convective flow of incompressible hybrid nanofluid with deferment of dust particles over a sheet. The thermophoretic particle deposition, gyrotactic microorganisms, are taken into account along with heat source (temperature and exponential dependent) and mixed convection. The main goal of this study is to determine the thermal mobility of nanoparticles with water base fluid. The elements of \({{{\text{Al}}}_{2}{\text{O}}}_{3}\) and \({\text{Cu}}\) are adept for the thermal analysis. We combined dust with microorganisms in this work to promote the heat and mass transport phenomena. It is vital for improving cooling systems in electronics, enhancing industrial heat transfer, and optimizing microfluidic devices. Additionally, the research aids environmental science by understanding pollutant dispersion and removal. It also finds applications in biomedical research, supporting drug delivery systems and medical diagnostics. Overall, this study promises advancements in technology, environmental conservation, and healthcare. The PDEs, which result from the conservation of concentration, energy, momentum and density of microorganisms in both hybrid nanofluid and dusty phases. Appropriate similarity transformations have been used to obtain the ODEs (ordinary differential equations). The resulting problem is numerically solved by a shooting technique based on the RKF-45th order. The results showed that the velocity profiles of the dust and fluid phases rise as the Marangoni convection parameter increases, but the microorganisms, concentration, and temperature profiles deteriorate in both phases.

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Data availability

No data were used in this paper.

Abbreviations

\(\left(x ,y\right)\) :

Cartesian coordinates

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

Grashof number

\(\left(u ,v\right)\) :

Velocity fields of fluid

\(K=6\pi \mu r\) :

The coefficient of drag stokes

\(\left({u}_{{\text{p}}} ,{v}_{{\text{p}}}\right)\) :

Velocity fields of particle phase

\(r\) :

Radius of the dust particle

\({q}_{{\text{r}}}\) :

Radiative heat flux

\(M\) :

Magnetic parameter

\({Q}_{{\text{e}}}\) :

Exponential dependent heat source.

\({W}_{{\text{c}}}\) :

Maximum cell swimming speed

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

Eckert number

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

Thermal relaxation time

\(\psi \left(x,y\right)\) :

Streams functions of fluid phase

\(\Psi \left(x,y\right)\) :

Streams functions of particle phase

\({\beta }_{{\text{m}}}\) :

Fluid-particle interaction parameter for bio-convection

\({\Phi }_{1}\) :

Volume fraction of \({{{\text{Al}}}_{2}{\text{O}}}_{3}\)

\({\beta }_{{\text{c}}}\) :

Parameter for fluid-particle interaction for concentration

\({\gamma }_{{\text{C}}}\) :

Surface tension coefficients for concentration

\({C}_{{\text{p}}}\) :

Specific heat of the fluid

\(\sigma\) :

Electrical Conductivity

\({K}^{*}\) :

Chemical reaction co-efficient

\({\mu }_{{\text{f}}}\) :

Dynamic viscosity (kg/ms)

\(l\) :

Reference length

\({\text{Pr}}\) :

Prandtl number

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

Fluid ambient temperature

\({\text{Le}}\) :

Lewis number

\({k}_{{\text{f}}}\) :

Thermal conductivity of the fluid

\({B}_{0}\) :

Uniform magnetic field

\(T\) :

Fluid temperature \(({\text{k}})\)

\({N}^{*}\) :

Dimensions of dust particle density

\({K}_{1}\) :

Permeability of the porous medium

\({\text{Pe}}\) :

Bioconvection Peclet number

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

Bio-convection Lewis number

\({q}_{{\text{n}}}\) :

Motile microorganism’s flux

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

Particle temperature

\(G{\text{n}}\) :

Bioconvection mixed convection parameter

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

Concentration mixed convection parameter,

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

Local density of motile microorganisms

\(\Omega\) :

Microorganisms concentration difference parameter

\({q}_{{\text{w}}}\) :

Heat flux

\({\text{Rd}}\) :

Radiation parameter

\({C}_{{\text{fx}}}\) :

Skin friction

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

Marangoni ratio parameter

\({k}^{*}\) :

Mean absorption coefficient

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

Temperature dependent heat source

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

Nusselt number

\({C}_{{\text{p}}}\) :

Concentration species (particle phase)

\({\rho }_{{\text{P}}}\) :

Particle density

\({\tau }_{{\text{v}}}\) :

Momentum relaxation time

\({\rho }_{{\text{f}}}\) :

Fluid density

\({\beta }_{{\text{v}}}\) :

Fluid-particle interaction parameter

\({\beta }_{{\text{c}}}^{*}\) :

Concentration expansion coefficient

\({\upepsilon }_{2}\) :

Variable thermal conductivity parameter

\(C\) :

Species of the fluid phase concentration

\({\epsilon }_{3}\) :

Variable mass diffusivity parameter

\({N}_{{\text{p}}}\) :

Density particle phase

\({\nu }_{{\text{f}}}\) :

Kinematic viscosity

\({\sigma }_{1}\) :

Surface tension

\({\tau }_{{\text{v}}}\) :

Relaxation time of the dust particles

\({\Phi }_{{\text{p}}}\) :

Volume fraction of dust particle

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

Surface tension coefficients for temperature

\({\sigma }^{*}\) :

Stefan-Boltzmann constant

\({D}_{{\text{m}}}\) :

Mass diffusivity coefficient

\({\Phi }_{2}\) :

Volume fraction of \({\text{Cu}}\)

\({D}_{{\text{n}}}\) :

Diffusivity of microorganisms

\(\gamma\) :

Specific heat ratio

\({\tau }_{{\text{w}}}\) :

Surface shear stress

\({\sigma }_{0}\) :

Surface tension

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

Thermal dust parameter

\({\tau }_{{\text{C}}}\) :

Time required by a dust particle to adjust its relative concentration to the fluid

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

Thermophoretic velocity

\(N\) :

Density of motile microorganism

\(k\) :

Thermophoretic coefficient

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

Reference temperature

\(\beta\) :

Thermal expansion coefficient

\(\tau\) :

Thermpopherotic parameter

\({\beta }_{{\text{n}}}^{*}\) :

Bioconvection expansion coefficient

\({C}_{{\text{m}}}\) :

Specific heat of the dust particle

\({T}_{0}\) :

Constants

\({r}_{{\text{P}}}\) :

Radius of dust particles

\({\epsilon }_{1}\) :

Viscosity variation exponent

\(L\) :

Dust particle mass concentration

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

Sherwood number

\(m\) :

Mass of dust particles

\({\text{Mn}}\) :

Marangoni number

\({q}_{{\text{m}}}\) :

Mass flux

\({\tau }_{{\text{m}}}\) :

Time required by the motile organisms

\(f\) :

Base fluid

\({\prime}\) :

Derivative with respect to \(\eta\)

\(\infty\) :

Ambient

\({\text{hnf}}\) :

Hybrid nanofluid

\(0\) :

Surface

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Authors and Affiliations

  1. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan

    Munawar Abbas & Nargis Khan

  2. Department of Mathematics, The Government Sadiq College Women University, Bahawalpur, Pakistan

    M. S. Hashmi

  3. Department of Mathematics, Firat University, 23119, Elazig, Turkey

    Mustafa Inc

  4. Department of Computer Engineering, Biruni University, 34010, Istanbul, Turkey

    Mustafa Inc

  5. Department of Medical Research, China Medical University, 40402, Taichung, Taiwan

    Mustafa Inc

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  1. Munawar Abbas
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Abbas, M., Khan, N., Hashmi, M.S. et al. Scrutinization of marangoni convective flow of dusty hybrid nanofluid with gyrotactic microorganisms and thermophoretic particle deposition. J Therm Anal Calorim 149, 1443–1463 (2024). https://doi.org/10.1007/s10973-023-12750-9

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