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Cilia-assisted flow of viscoelastic fluid in a divergent channel under porosity effects

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Abstract

Cilia-driven laminar flow of an incompressible viscoelastic fluid in a divergent channel has been conducted numerically using the BVP4C technique. The non-Newtonian Jeffrey rheological model is utilized to characterize the fluid. The flow equations are formulated in a curvilinear coordinate system, and the porosity effects are simulated with a body force term in the Navier–Stokes equation. The flow equations are transformed into a wave frame from a fixed frame of reference using a linear mathematical relationship. A biological approximation of creeping phenomena and the long-wavelength assumption is used in the flow analysis. The flow analysis is carried out by using a complex (wavy) propulsion of cilia beating. The two-dimensional flow is controlled by physical parameters—Darcy’s number, curvature parameter, viscoelastic parameter, phase difference, cilia length, and divergent parameter. They also examined the ciliated pumping and bolus trapping in their flow analysis. The boundary layer phenomena in the velocity profile are noticed under more significant porosity and time relaxation effects. The bolus circulations are reduced for a larger porosity medium and larger numeric values of the time relaxation parameter.

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Abbreviations

\(B\) :

Half-width of the channel (unit: meter)

\(\overline{\eta }\) :

Axial component

\(C\) :

The wave speed (unit: meter per sec)

\(\overline{\zeta }\) :

Radial component

\(\left( {\overline{U},\overline{V},0} \right)\) :

Two-dimensional velocity field (unit: meter per sec)

\(\overline{z}\) :

Orthogonal component of flow geometry

\(\overline{t}\) :

Time (unit: sec)

\(\overline{U}\) :

Radial velocity (unit: meter per sec)

\(\overline{ \in }_{i} \left( {i = 1 - 3} \right)\) :

Different wave amplitudes

\(\overline{V}\) :

Axial velocity (unit: meter per sec)

\(\overline{\alpha }_{j} \left( {j = 1 - 3} \right)\) :

Wave parameters

\(\in\) :

Wavelength

\(\left( {\overline{\eta },\overline{\zeta },\overline{z}} \right)\) :

Curvilinear coordinates

\({\overline{\Omega }}\) :

Dimensional radius of curvature

\(\left( {\overline{u}, \overline{v}} \right)\) :

Velocity field in the wave frame (unit: meter per sec)

\(Re\) :

Reynolds number

\(Da\) :

Darcy number

\(\delta\) :

Wave number

\(k\) :

Dimensionless radius of curvature

M :

Dimensionless divergent parameter

\(\overline{N}\) :

Divergent parameter

\(\overline{\theta }\) :

Phase difference

\(\kappa\) :

Eccentricity of the elliptical motion

\(\overline{\eta }_{0}\) :

Reference position of the particle

\(\overline{p}\) :

Pressure term (unit: Pa)

\(\overline{I}\) :

Identity tensor

\({\overline{\Upsilon }}\) :

Extra-stress tensor

\(\mu\) :

Absolute viscosity

\(\mathbf {\dot{{\beta }}}\) :

Rate of strain

\(\mathbf {\dot{\beta }}\) :

Slope of \(\dot{\beta }\)

\(\overline{\nabla }{\overline{\text{W}}}\) :

Gradient of the velocity vector

T:

Transpose of the gradient vector

\(\rho\) :

Density (unit: kilogram per meter-cube)

\(K\) :

Permittivity of medium

\(\upsilon\) :

Kinematic viscosity (unit: meter per sec)

\({\overline{\Upsilon }}_{{\overline{\eta }\overline{\eta }}} , {\overline{\Upsilon }}_{{\overline{\eta }\overline{\zeta }}} , {\overline{\Upsilon }}_{{\overline{\zeta }\overline{\zeta }}}\) :

Components of extra-stress tensors

\(\eta\) :

Dimensionless axial component

\(\zeta\) :

Dimensionless radial component

\(\left( { \in_{1} , \in_{2} } \right)\) :

Dimensionless symbols of different amplitudes of complex peristaltic waves

\(\overline{{\lambda_{i} }} \left( {i = 1,2} \right)\) :

Retardation and relaxation time parameters

\(\overline{\nabla }\) :

Nabla operator

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Authors and Affiliations

  1. Department of Mathematics, Northern University, Wattar-Walli Road, Nowshera, 24110, KPK, Pakistan

    Khurram Javid & Touqeer Ahmad

  2. College of Engineering, Department of Mechanical Engineering, Prince Sattam bin Abdulaziz University, Alkharj, 16273, Saudi Arabia

    Umar F. Alqsair

  3. Department of mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, 54000, Pakistan

    Mohsan Hassan

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

    M. M. Bhatti

  5. Department of Medical and Surgical Specialties, Faculty of Medicine, Transilvania University of Brasov, Brasov, Romania

    Elena Bobescu

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  1. Khurram Javid
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  2. Umar F. Alqsair
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  3. Mohsan Hassan
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  4. M. M. Bhatti
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  5. Touqeer Ahmad
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Correspondence to M. M. Bhatti.

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Javid, K., Alqsair, U.F., Hassan, M. et al. Cilia-assisted flow of viscoelastic fluid in a divergent channel under porosity effects. Biomech Model Mechanobiol 20, 1399–1412 (2021). https://doi.org/10.1007/s10237-021-01451-7

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