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Heat transfer analysis of tangent hyperbolic nanofluid in a ciliated tube with entropy generation

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Abstract

In this paper, we analyze the effect of heat transfer on the flow of tangent hyperbolic nanofluid in a ciliated tube (fallopian tube where embryo in blood make the development). This study will be beneficial for the researchers and medical experts in the field of embryology. The nanoparticles are beneficial to remove the cysts from the fallopian tube where development of embryo takes place. To resolves the ciliary flow problems, medical doctors use nanoparticles (drug delivery) that may create a temperature gradient. The heat transfer helps to optimize the energy for which the entropy generation is reduced. Therefore, in this research we discuss the heat transfer effect on tangent hyperbolic nanofluid and entropy generation due to ciliary movement. The governing partial differential equations are solved by HPM and software MATHEMATICA™. Effect of viscoelastic parameter, nanoparticles, cilia length and Brinkman number on the velocity, temperature and entropy generation has been illustrated with the help of graphs. Graphical results show that thermal conductivity of fluid increases by adding nanoparticles. The entropy generation due to nanoparticles will decrease the viscosity near the tube wall and blood through tube will flow with normal pressure.

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Abbreviations

\({\varvec{V}}\) :

Velocity field vector

\(U, W\) :

Velocity components in fixed frame

\(u, w\) :

Velocity components in wave frame

\(R, Z\) :

Cylindrical coordinates of ciliated tube in fixed frame

\(r, z\) :

Cylindrical coordinates of ciliated tube in wave frame

\(P\) :

Pressure in fixed frame

\(p\) :

Pressure in wave frame

\(\tau\) :

Cauchy stress

\({\varvec{S}}\) :

Shear rate

\(c\) :

Wave speed

\(\left( {\rho c_{{\text{p}}} } \right)_{{{\text{nf}}}}\) :

Specific heat capacity of nanofluid

\(\left( {\rho c_{{\text{p}}} } \right)_{{\text{f}}}\) :

Specific heat capacity of base fluid

\(\left( {\rho c_{{\text{p}}} } \right)_{{\text{s}}}\) :

Specific heat capacity of nanoparticles

\(\epsilon\) :

Cilia length

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

Density of fluid

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

Density of nanofluid

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

Density of nanoparticles

\(\varphi\) :

Solid volume fraction

\(k_{{{\text{nf}}}}\) :

Thermal conductivity of nanofluid

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

Thermal conductivity of base fluid

\(k_{{\text{s}}}\) :

Thermal conductivity of solid nanoparticles

\(\alpha\) :

Eccentricity of elliptical path

\(\eta_{\infty }\) :

Infinite shear rate viscosity

\(\eta_{0}\) :

Zero shear rate viscosity

\(\Gamma\) :

Time constant

\({\dot{\gamma }}\) :

Strain rate tensor

\({\varvec{\pi}}\) :

Second order tensor

\(\beta\) :

Wave number

\(m\) :

Power law index

\(A_{1}\) :

Rivlin–Erickson tensor

\(We\) :

Weissenberg number

\(Q\) :

Volume flow rate

\(\overline{Q}\) :

Mean volume flow rate

\(j\) :

Embedding parameter

\(T\) :

Temperature profile

\(T_{0}\) :

Temperature at the center of the tube

\(T_{1}\) :

Temperature at the ciliated wall

\({\text{Re}}\) :

Reynolds' number

\({\text{Br}}\) :

Brinkman number

\(Ns\) :

Entropy generation

\({\text{Be}}\) :

Bejan number

\(\theta\) :

Dimensionless temperature

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan

    S. Shaheen, K. Maqbool & R. Ellahi

  2. Center for Communications and IT Research, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Sadiq M. Sait

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  1. S. Shaheen
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Shaheen, S., Maqbool, K., Ellahi, R. et al. Heat transfer analysis of tangent hyperbolic nanofluid in a ciliated tube with entropy generation. J Therm Anal Calorim 144, 2337–2346 (2021). https://doi.org/10.1007/s10973-021-10681-x

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