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Bioconvective Applications of Unsteady Slip Flow Over a Tangent Hyperbolic Nanoliquid with Surface Heating: Improving Energy System Performance

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Abstract

The gyrotactic microorganisms in nanofluid is necessary in many engineering and industrial systems. Also, stretching surface plays a significant role in the acceleration of thermal energy. Our motive in the current study is to present the numerical simulation of bioconvective flow of tangent hyperbolic nanofluid over a stretching surface with velocity slip and convective boundary condition. The tangent hyperbolic nanoliquid having motile density. The mathematical model for the current flow problem is transformed into a non-dimensional expression using suitable variables. The developed model is formulated for these transformed equations through the Matlab bvp4c method. Engineering interest quantities such as friction factor, heat, mass and microorganism density were obtained against different physical variables. The results have observed that the heat transfer rate and motile microorganism density diminished while increasing the Weissenberg number, the opposite trend is found for mass transfer rate and skin friction. The accuracy of the current result has been shown excellent reliability in contrast to the previous literature.

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Data Availability

The raw data supporting the conclusions of this article will be made available by the corresponding author without undue reservation.

Abbreviations

\(t\) :

Time, Sec

\(\mathrm{We}\) :

Weissenberg number, Dimensionless

\(M\) :

Magnetic parameter, Dimensionless

\(n\) :

Power index number, Dimensionless

\(A\) :

Unsteady parameter, Dimensionless

\(\mathrm{Nb}\) :

Brownian motion parameter, Dimensionless

\(\mathrm{Nt}\) :

Thermophoresis parameter, Dimensionless

\(\mathrm{Pr}\) :

Prandtl number, Dimensionless

\(\mathrm{Bi}\) :

Biot number, Dimensionless

\(\mathrm{Sc}\) :

Schmidt number, Dimensionless

\(\mathrm{Lb}\) :

Bioconvection Lewis number, Dimensionless

\(\mathrm{Pe}\) :

Bioconvection Peclet number, Dimensionless

\({C}_{f}\) :

Skin friction coefficient, Dimensionless

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

Nusselt number, Dimensionless

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

Sherwood number, Dimensionless

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

Microorganism density number, Dimensionless

\(f\) :

Dimensionless velocity profile, Dimensionless

\(\theta \) :

Dimensionless temperature profile, Dimensionless

\(\phi \) :

Dimensionless concentration profile, Dimensionless

\(\psi \) :

Dimensionless microorganism profile, Dimensionless

\(\varpi \) :

Microorganism concentration difference parameter, Dimensionless

\(\eta \) :

Similarity variable, Dimensionless

\(\chi \) :

Stream function, \({\text{m}} {\text{s}}^{ - 1}\)

\(\lambda \) :

Slip parameter, Dimensionless

\(\Gamma \) :

Time material constant, Dimensionless

\(u,v\) :

Components of velocity along \(x, y\)direction, \(\mathrm{m}{ \mathrm{s}}^{-1}\)

\(\tau \) :

Ratio of the effective heat capacity, Dimensionless

\(\upsilon \) :

Kinematic viscosity, \({\mathrm{m}}^{2} {\mathrm{s}}^{-1}\)

\(c\) :

Stretching rate, \({\mathrm{s}}^{-1}\)

\(\rho \) :

Density, \({\text{kgm}}^{ - 1}\)

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

Specific heat, \({\text{J}} {\text{kg}}^{ - 1} {\text{K}}^{ - 1}\)

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

Brownian diffusion coefficient, \({\mathrm{m}}^{2}{ \mathrm{s}}^{-1}\)

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

Thermophoretic diffusion coefficient, \({\mathrm{m}}^{2}{ \mathrm{s}}^{-1}\)

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

Microorganism’s diffusion coefficient, \({\mathrm{m}}^{2}{ \mathrm{s}}^{-1}\)

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

Constant magnetic field, \({\text{kgs}}^{ = 2 } {\text{A}}^{ - 1}\)

\(C\) :

Concentration, \({\text{kgm}}^{ - 3}\)

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

Ambient concentration, \({\text{kgm}}^{ - 3}\)

\({C}_{w}\) :

Surface concentration of nanoparticles, \({\text{kgm}}^{ - 3}\)

\(k\) :

Thermal conductivity, \({\text{m}} {\text{kgs}}^{ - 3} {\text{K}}^{ - 1}\)

\(T\) :

Temperature, \(\mathrm{K}\)

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

Ambient temperature, \(\mathrm{K}\)

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

Fluid temperature, \(\mathrm{K}\)

\(h(t)\) :

Coefficient of heat transfer, W \({\mathrm{m}}^{-2}{\mathrm{K}}^{-1}\)

\(N\) :

Microorganism’s profile, Dimensionless

\({N}_{w}\) :

Surface concentration of microorganisms, \({\text{kgm}}^{ - 3}\)

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

Ambient concentration of microorganisms, \({\text{kgm}}^{ - 3}\)

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

Maximum cell swimming speed, \(\mathrm{m}{ \mathrm{s}}^{-1}\)

\(x,y\) :

Cartesian coordinates, \(\mathrm{m}\)

\(\sigma \) :

Electrical conductivity, \({\text{S}}^{3} {\text{m}}^{2} {\text{kg}}^{ - 1}\)

\({\mathrm{u}}_{\mathrm{w}}\) :

Velocity of the sheet, \(\mathrm{m}{ \mathrm{s}}^{-1}\)

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Authors and Affiliations

  1. Department of Mathematical Sciences, Federal Urdu University of Arts, Sciences & Technology, Gulshan-e-Iqbal, Karachi, 75300, Pakistan

    F. Ali, M. Faizan & A. Zaib

  2. Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan, 303007, India

    K. Loganathan

  3. Research and Development Wing, Live4Research, Tiruppur, 638106, India

    K. Loganathan

  4. Department of Mathematics, Dr. N. G. P. Arts and Science College, Coimbatore, Tamil Nadu, India

    S. Eswaramoorthi

  5. Department of Mathematics, Erode Arts and Science College, Erode, Tamilnadu, India

    E. Prabu

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Ali, F., Loganathan, K., Eswaramoorthi, S. et al. Bioconvective Applications of Unsteady Slip Flow Over a Tangent Hyperbolic Nanoliquid with Surface Heating: Improving Energy System Performance. Int. J. Appl. Comput. Math 8, 276 (2022). https://doi.org/10.1007/s40819-022-01476-1

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