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Significance of entropy generation and heat source: the case of peristaltic blood flow through a ciliated tube conveying Cu-Ag nanoparticles using Phan-Thien-Tanner model

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Abstract

The present speculative investigation is concentrated to analyze the entropy generation and heat transfer phenomena in ciliary induced peristalsis of blood with the suspension of hybrid nanoparticles in a tube with heat source impact. The blood is assumed to contain copper (Cu) and silver (Ag) nanoparticles (NPs). The ciliary inner wall of the tube has been considered with small hair-like structures. The Phan-Thien-Tanner (PTT) fluid model is employed to describe the non-Newtonian rheological characteristics of blood. The conservative equations are normalized and simplified by utilizing scaling analysis with the assumption of low Reynolds number and large wavelength approximations. The analytical inspection exposes that the total entropy generation gets a decrement for mounting values of cilia length, while reversed impact is detected for an increment in heat source parameter. Hybrid nano-blood exhibits a greater total entropy number than mono nano-blood. This research study may be beneficial to medical experts and researchers in the field of embryology. Cysts in the ciliated fallopian tube, where embryos develop, are removed by using nanoparticles (nano-drug delivery).

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Abbreviations

\(\tilde{a}\) :

Mean radius of tube

Be :

Bejan number

Br :

Brinkman number

c :

Metachronal wave speed

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

Specific heat

\(E_{\text{g}}\) :

Characteristic entropy generation rate

F :

Mean flow rate

h :

Ciliated wall

k :

Thermal conductivity

\(N_{\text{S}}\) :

Non-dimensional entropy generation rate

\(\tilde{P}\) :

Pressure in the laboratory frame

p :

Pressure in wave frame

Q :

Volume flow rate

\(Q_0\) :

Internal heat source coefficient

Re :

Reynolds number

t :

Non-dimensional time parameter

\(\tilde{t}\) :

Dimensional time parameter

\(\tilde{T}\) :

Blood temperature

\(\tilde{T}_0\) :

Temperature at tube wall

(u, w):

Non-dimensional velocity components in (r, z)

\((\tilde{u}, \tilde{w})\) :

Moving frame velocity components in \((\tilde{r} ,\tilde{z})\)

\((\tilde{U},\tilde{W})\) :

Fixed frame velocity components in \((\tilde{R} ,\tilde{Z})\)

We :

Weissenberg number

\(\tilde{Z}_0\) :

Reference particle position

\(Z^*\) :

Heat transfer coefficient

\(\alpha\) :

Eccentricity due to elliptical movement

\(\beta\) :

Wave number

\(\delta\) :

Dimensional cilia length

\(\gamma\) :

Heat source parameter

\(\lambda\) :

Metachronal wavelength

\(\Lambda\) :

Relaxation time

\(\mu\) :

Constant viscosity coefficient

\((\phi _1, \phi _2)\) :

Solid Volume fractions of Cu and Ag-NPs

\(\psi\) :

Stream function

\(\rho\) :

Blood density

\(\tilde{\tau }\) :

Extra stress tensor

\(\tau _{ij}\) :

Component of stress tensor

\(\theta\) :

Non-dimensional blood temperature

\(\tilde{\theta }_0\) :

Temperature difference

\(\varepsilon\) :

Elongation parameter

\(\zeta\) :

Temperature difference parameter

\(s_1\) :

Copper nanoparticles (Cu-NPs)

\(s_2\) :

Silver nanoparticles (Ag-NPs)

f :

Base fluid (blood)

nf :

Cu-blood nanofluid

hnf :

Cu-Ag/blood hybrid nanofluid

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Acknowledgements

The authors are very grateful to anonymous reviewers for their fruitful comments and constructive suggestions to improve our manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Bajkul Milani Mahavidyalaya, Purba Medinipur, 721 655, India

    Asgar Ali

  2. Department of Applied Mathematics, Vidyasagar University, Midnapore, 721 102, India

    R. N. Jana

  3. Department of Mathematics, University of Gour Banga, Malda, 732 103, India

    Sanatan Das

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  1. Asgar Ali
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Ali, A., Jana, R.N. & Das, S. Significance of entropy generation and heat source: the case of peristaltic blood flow through a ciliated tube conveying Cu-Ag nanoparticles using Phan-Thien-Tanner model. Biomech Model Mechanobiol 20, 2393–2412 (2021). https://doi.org/10.1007/s10237-021-01515-8

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