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An Insight Into the Dynamics of Carreau-Yasuda Nanofluid Through a Wavy Channel with Electroosmotic Effects: Relevance to Physiological Ducts

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Abstract

Many physiological ducts, like sperm vessels, intestines, common bile ducts, ducts of salivary glands, and ducts in the repertory system, are common ducts in which transport occurs due to the propagation of complex waves on the boundary. Electroosmotic flow phenomena are significant in membrane separation processes, protein and peptide delivery, iontophoresis in drug delivery through physiological ducts, and a lot of application in plant physiology regarding pressure control in xylem and phloem vessels. The aim of this theoretical study is to analyze the thermal characteristics of physiological fluid characterized by the Carreau-Yasuda model in a complex wavy channel. The two agents of flow, i.e., the propagation of complex waves on the walls of the channel and the electric field in the axial direction, are considered. Different zeta potentials at both walls of the asymmetric channel are employed. Basic conservation laws, along with the Poisson-Boltzmann equation, are utilized to model the problem in the Cartesian coordinate system. Theoretical and biological assumptions like greater wavelength, lubrication transport, and Debye-Hückel linearization transform the governing equations of PDE into a coupled system of ODE. The shooting method is used, which is Solve, a built-in function in the computational software Mathematica to compute the stream function, temperature profile, concentration profile, and various flow and heat transfer characteristics. The impact of various penetrating parameters is described through plots. The magnitude of the velocity profile is increased near the center of geometry by increasing the electroosmotic parameter in the lower half of the regime. Then the Sherwood number and the Bejan number show enhanced behavior by increasing the electroosmotic parameter. The current analysis predicts dynamic applications in thermodynamical systems, cooling systems, biochemistry, and drug delivery systems.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

\({e}_{i},i=\mathrm{1,2},\mathrm{3,4}\) :

Amplitude of the peristaltic wave

T′ :

Averaging temperature of electrolyte

\(\overline{X}\) :

Axial coordinate in wave frame

x :

Axial coordinate in dimensionless form

\({\overline{E}}_{\overline{X}}\) :

Axial electric field

\(\overline{U}\) :

Axial velocity component in wave frame

\(u\) :

Axial velocity component in dimensionless form

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

Boltzmann constant

\(Br\) :

Brinkman number

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

Brownian diffusion coefficient

\(Nb\) :

Brownian motion parameter

\({C}_{0},{C}_{1}\) :

Concentration at walls

\(z\) :

Charge balance

\({\rho }_{e}\) :

Density of net ionic energy

\({\lambda }_{d}\) :

Debye-length of electric double layer

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

Dielectric constant

\(\varphi\) :

Dimensionless potential distribution

\(\overline{t}\) :

Time

\(\alpha\) :

Dimensionless fluid parameter

\(Nt\) :

Dimensionless thermophoresis parameter

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Dimensionless concentration

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

Diffusivity of chemical species

\(Ec\) :

Eckert number

\(e\) :

Electronic charge

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

Electric conductivity

\(k\) :

Electro-osmotic parameter

\({E}_{S}\) :

Entropy generation number

\({S}_{i j}, i,j=\mathrm{1,2},3\) :

Extra stress tensor

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

Fluid density

\(Be\) :

Bejan number

\({a}_{1},{a}_{2}\) :

Half widths of the channel

\(S\) :

Joule heating parameter

\({U}_{hs}\) :

Helmholtz-Smoluchowski velocity

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

Mean temperature of fluid

\(\overline{C}\) :

Nanoparticle concentration

\(Nu\) :

Nusselt’s number

\(Pe\) :

Peclet number

\(\Phi\) :

Phase difference

\(n\) :

Power law index

\(\overline{P}\) :

Pressure in wave frame

\(\overline{V}\) :

Radial (transverse) velocity component in wave frame

\(v\) :

Radial (transverse) velocity component in dimensionless form

\(Re\) :

Reynolds number

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

Specific heat capacity of fluid

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

Specific heat capacity of nanoparticles

\(Sh\) :

Sherwood number

\(\psi\) :

Steam function

\(\overline{T}\) :

Temperature

\({T}_{0},{T}_{1}\) :

Temperature at walls

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

Thermophoresis diffusion coefficient

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

Thermal conductivity

\(F\) :

Flow rate in wave frame

\(y\) :

Transverse coordinate in dimensionless form

\(\overline{Y}\) :

Transverse coordinate in wave frame

\(\delta\) :

Wave number

\({H}_{1},{H}_{2}\) :

Lower and upper walls

\({h}_{1},{h}_{2}\) :

Lower and upper walls in dimensionless form

\(We\) :

Weissenberg number

\({\xi }_{1},{\xi }_{2}\) :

Zeta potentials at walls

\({R}_{\xi }\) :

Zeta potential ratio parameter

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Authors and Affiliations

  1. Department of Mathematics, University of Azad Jammu and Kashmir Muzaffarabad, Muzaffarabad, 13100, Pakistan

    A. Abbasi & W. Farooq

  2. Department of Mathematics, Namal University, Mianwali, 42250, Pakistan

    Sami Ullah Khan

  3. Department of Mathematics, Mohi-Ud-Din Islamic University, Nerian Sharif, 12080, AJ&K, Pakistan

    Adnan

  4. Division of Science and Technology, Department of Mathematics, University of Education, Lahore, 54770, Pakistan

    Arshad Riaz

  5. College of Mathematics and Systems Science, Shandong University of Science & Technology, Qingdao, 266590, China

    M. M. Bhatti

  6. Material Science Innovation and Modelling (MaSIM) Research Focus Area, North-West University (Mafikeng Campus), Private Bag X2046, Mmabatho, 2735, South Africa

    M. M. Bhatti

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  1. A. Abbasi
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Contributions

A. Abbasi and W. Farooq provided the concept and edited the draft of the manuscript. Sami Ullah Khan and Adnan conducted the literature review and wrote the first draft of the manuscript. Arshad Riaz and M. M. Bhatti edited the final draft of the manuscript. The authors declare that all data were generated in-house and that no paper mill was used.

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Abbasi, A., Farooq, W., Khan, S.U. et al. An Insight Into the Dynamics of Carreau-Yasuda Nanofluid Through a Wavy Channel with Electroosmotic Effects: Relevance to Physiological Ducts. Braz J Phys 54, 66 (2024). https://doi.org/10.1007/s13538-024-01438-6

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