Channel flow of Ellis fluid due to cilia motion

Muhammad Bilal Arain; Sidra Shaheen; Kottakkaran Sooppy Nisar; Junhui Hu; Taseer Muhammad

doi:10.1515/nleng-2024-0036

Article Open Access

Channel flow of Ellis fluid due to cilia motion

Muhammad Bilal Arain , Sidra Shaheen , Kottakkaran Sooppy Nisar , Junhui Hu and Taseer Muhammad

Published/Copyright: March 26, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

In this report, a theoretical study is directed at the cilia motion of the Ellis fluid model due to the propagation of an infinite metachronal wave train traveling along the walls of the channel due to the continuous beating of the cilia. The governing equations of continuity and equation of motion for the well-known Ellis fluid model are carried out by taking long-wavelength approximations. The equations were solved with the help of the computational software Mathematica 13.0, and the results were obtained. The influence of physical parameters on pressure rise ( $∆ p$ ), axial velocity $u ( x , y )$ , and stream function $ψ ( x , y )$ are exhibited graphically and discussed briefly. It is observed from graphical results that the velocity field shows a parabolic nature, achieves its highest magnitudes at the mid of the channel, and then reduces drastically at the walls of the channel. Velocity reduces with increasing material constant α and rises with increasing material constant β. It is also noted that pressure increases with rising $α$ and $β$ . The present inquiry is also valid in the treatment of different symptomatic problems and different medicament delivery methods in pharmacological and biomedical engineering.

Keywords: Ellis fluid model; ciliary movement; metachronal wave; physiological propulsion; exact solution

1 Introduction

Cilia were first discovered in 1675, but their role in human body functions became clearer after the advancement of microscopy in the early 1960s. Brennen and Winet [1] provided a thorough overview of early investigations on fluid dynamics of ciliary motility. We will not go into depth on past research on fluid dynamics of ciliary motility. However, concerned readers may find these details in the previous studies [2–5]. Other related studies with heat and mass transfer with application can be found in previous literature [6,7,8,9,10]. The associated cilia normally work in big groups, and one group of cilia beats (swings) in opposite direction (out of phase) with regard to another group of cilia. These synchronized ciliary motions generate a wave known as a metachronal wave, which propagates all along the surface produced by the cilia tips. Many medical therapies make use of electromagnetic effects and heat and mass transfer, such as hyperthermia and radiofrequency therapies, hemodialysis, thermoregulation in testis, oxygenation, X-rays and MRI [11,12]. This phenomenon has significance for the creation of magnetic artificial cilia for application in nanodrug delivery systems, rapid illness diagnostics, and daily health monitoring systems. Another prominent study field in the medical sciences is thermal analysis of biological tissues and different metabolic processes. Furthermore, chemical reactions serve critical roles in all individuals’ lives, creating energy and controlling the body’s growth processes [13].

Ellis fluid model is a mathematical fluid model, which is used in fluid dynamics. It was introduced by American mathematician Richard Ellis in 1971 to generalize the concept of a perfect fluid to include fluids with more complicated properties. In Ellis fluid model, fluid is treated as a collection of particles that interact with each other through forces that depend on local properties of fluid. These forces can be modeled using a stress-energy tensor, which describes the distribution of energy and momentum within the fluid. Unlike a perfect fluid, which has a simple relationship between its pressure, density, and velocity, an Ellis fluid can have more complex relationships between these properties. For example, an Ellis fluid can have a nonlinear equation of state, which means that its pressure may not be proportional to its density. The Ellis fluid model has been used in a variety of applications, including the study of cosmology, where it has been used to model the behavior of dark energy. It has also been used in the study of astrophysical phenomena such as accretion disks and black holes.

The primary reason for this work is the expectation that such a challenge will be useful in various scientific and industrial applications, particularly in the investigation of human infertility issues and the production of micropumps for drug delivery systems. Various studies [14,15,16,17,18,19,20,21,22,23,24] discuss the critical functions of electromagnetic body forces, heat and mass transfers, and chemical reactions on numerous biological fluxes in the human body. Considering the previously mentioned literature, this report examines the channel flow of Ellis fluid caused by cilia motion. Given the previous published literature, no findings are found for the channel flow due to the Ellis fluid model. The governing equations of continuity and equation of motion for Ellis’s fluid model are solved by taking the long wavelength approximation. We used the computational software Mathematica 13.0 to solve the equations and obtain the results. We exhibit and briefly discuss the effects of physical parameters on pressure rise, velocity, and stream function in detail.

1.1 Mathematical formulation

Let us assume an incompressible Ellis’s fluid flow in a channel of finite length $L$ . Assume infinite number of continuously beating cilia are present at the inner walls of channel generating symplectic metachronal wave, which moves toward positive $X$ -axis with wave speed $c$ . As cilia are present at the internal walls of the tube and due to continuous beating of cilia, fluid flow is established. We will use envelope model approach to discuss the transport of mucus by ciliary movement. We propose a Cartesian Coordinates system $X , Y$ in which $X$ -axis is along direction of flow and $Y$ -axis is perpendicular to it (Figure 1).

Figure 1

Schematics flow diagram.

The wall of channel is mathematically expressed by equation below as in the study by Siddiqui et al. [11]:

(1)

$X = F ( X , t ) = X 0 + ε a α 1 sin 2 π λ ( X − c t ) ,$

(2)

$Y = H ( X , t ) = ± a + ε α 1 cos 2 π λ ( X − c t ) = ± L ,$

where $a$ , $ε$ , $X 0 ⁎ , α$ , $λ$ , and $c$ represent mean radius of tube, cilia length parameter, reference position of cilia, eccentricity of ellipse, wavelength, and wave speed of metachronal wave, respectively.

The equation of momentum in frame (fixed) is given by Siddiqui et al. [11]

(3)

$∇ ⋅ V = 0 ,$

(4)

$ρ d V d t = − ∇ P + Div S ,$

where the velocity is denoted by V, the density is denoted by ρ, the material derivative is represented by $d / d t$ , and the hydrostatic pressure is represented by P. Extra stress $S$ for an Ellis fluid is given as [25,26]

(5)

$S = μ 1 + ∏ s τ 0 2 α − 1 A 1 .$

where µ is the viscosity (dynamic), α represents the material constant, $S$ represents the extra stress tensor, $∏ s$ is the second order invariant of stress tensor and $A 1$ denoted the first order Rivlin-Ericksen tensor. The constant $τ 0$ generally represents the shear stress corresponding to half dynamic viscosity. The model (5) converts to Newtonian model for values of α = 1 or 1/ $τ 0 2$ → 0.

Both equations, i.e., momentum and energy in frame (fixed) are given as

(6)

$∂ U ∂ X + ∂ V ∂ Y = 0 ,$

(7)

$ρ U ∂ U ∂ X + V ∂ U ∂ Y = − ∂ P ∂ X + ∂ S X X ∂ X + ∂ S X Y ∂ Y ,$

(8)

$ρ U ∂ V ∂ X + V ∂ V ∂ Y = − ∂ P ∂ Y + ∂ S X Y ∂ X + ∂ S Y Y ∂ Y ,$

where

(9)

$S X X = 2 μ ∂ U ∂ Y 1 + ∏ s τ 0 2 α − 1 ,$

(10)

$S X Y = μ ∂ U ∂ Y + ∂ V ∂ X 1 + ∏ s τ 0 2 α − 1 ,$

(11)

$S Y Y = 2 μ ∂ V ∂ X 1 + ∏ s τ 0 2 α − 1 .$

Following are the non-dimensional variables that are implemented.

(12)

$x = x ¯ λ , y = y ¯ a , u = u ¯ c , v = λ v ¯ a c , p = a 2 p ¯ c λ μ , t = c t ¯ λ , h = L a , Re = ρ c a μ , β 1 = a λ , S = μ c a S ¯ .$

Using Eqs. (6)–(12) and applying the low Reynolds number approximation from lubrication theory, Eqs. (6)–(11) take the following form:

(13)

$∂ u ∂ x + ∂ v ∂ y = 0 ,$

(14)

$∂ p ∂ x = ∂ S x y ∂ y ,$

(15)

$∂ p ∂ y = 0 ,$

where

(16)

$S x x = S y y = 0 , S x y = ∂ u ∂ y 1 + ( β χ ) α − 1 ,$

(17)

$χ = 1 2 ( ( S x x ) 2 + 2 ( S x y ) 2 + ( S y y ) 2 ) 1 2 .$

$β = c a τ 0 2$ is the dimensionless material parameter.

The related boundary conditions (BCs) appear as

$u ( h ) = − 2 π ε α 1 β 1 cos ( 2 π x ) − 1 ,$

$v ( h ) = 2 π ε sin ( 2 π x ) + β 1 ( 2 π ε ) 2 α 1 sin ( 2 π x ) cos ( 2 π x ) at y = h ,$

(18)

$∂ u ∂ y = 0 , at y = 0 .$

Here

$h = 1 + ε cos ( 2 π x ) .$

The stream function $ψ$ is stated as

(19)

$u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x .$

Eqs. (15)–(21) and BC’s (22) in the form of $ψ$ take the subsequent form

(20)

$∂ p ∂ x = ∂ S x y ∂ y ,$

(21)

$∂ p ∂ y = 0 ,$

where

(22)

$S x x = S y y = 0 , S x y = ∂ 2 ψ ∂ y 2 1 + ( δ χ ) α − 1 ,$

(23)

$χ = 1 2 ( ( S x x ) 2 + 2 ( S x y ) 2 + ( S y y ) 2 ) 1 2 .$

The BCs can be stated as follows:

(24)

$ψ = 0 , ∂ 2 ψ ∂ y 2 = 0 at y = 0 ,$

$ψ = F , ∂ ψ ∂ y = − 1 − 2 π ε α 1 β 1 cos ( 2 π x ) ,$

$∂ ψ ∂ x = 2 π ε sin ( 2 π x ) + β 1 ( 2 π ε ) 2 α 1 sin ( 2 π x ) cos ( 2 π x ) , at y = h .$

where the flux is denoted by $F$ and the volumetric flow rate is denoted by $Q$ , and linked to flux by the resulting relation

(25)

$Q = ∫ 0 h ∂ ψ ∂ y + 1 d y = F + h .$

The time mean rate of volumetric flow in frame (fixed) is stated as follows:

(26)

$Q ¯ = 1 T ∫ 0 h ( h + F ) d t ¯ = ∫ 0 1 ( h + F ) d t = F + 1 .$

2 Solution of the problem

From Eq. (25), it is found that $p$ is not a function of $y$ . Thus, only chance is that $p$ is function of $x$ . This suggests that $d p d x$ is only a function of $x$ and can be handled as a constant. While integrating Eq. (24) regarding $y$ and by using BC $∂ 2 ψ ∂ y 2 = 0$ , we obtain

(27)

$S x y = d p d x y .$

Invoking Eq. (27) in Eq. (22), integrating twice, and using BCs we obtain

(28)

$ψ = u ( h ) y + 1 2 d p d x y 3 3 − y h 2 + β α − 1 d p d x α ( α + 1 ) y α + 2 α + 2 − h y α + 1 .$

The abovementioned expression $d p d x$ remains unidentified. On utilizing BC $ψ = F at y = h$ , the subsequent nonlinear algebraic equation in $d p d x$ can be achieved.

(29)

$F = h u ( h ) + 1 2 d p d x h 3 3 − h 3 + β α − 1 d p d x α ( α + 1 ) h α + 2 α + 2 − h 2 α + 1 .$

The hypertension (pressure rise) per wavelength can be achieved by integrating Eq. (29). However, arithmetic value of integral is calculated employing computational package MATHEMATICA 13.0 v.

(30)

$Δ p = ∫ 0 1 d p d x d x .$

3 Results and discussion

The affect of emerging parameters of interest on pressure rise ( $Δ p$ ), velocity axial $u ( x , y )$ , and stream function $ψ ( x , y )$ are shown in graphic form in Figures 2–5. If $α → 1 or β → 0$ , the present model reduces to viscous fluid model and our results precisely match with the existing results [11] for $M = 0$ .

$Figure 2 (a) and (b) Influence of α , β \alpha ,\hspace{.5em}\beta on pressure rise Δ p \text{Δ}p with time mean volumetric flow rate Q ¯ \bar{Q} .$

Figure 2

(a) and (b) Influence of $α , β$ on pressure rise $Δ p$ with time mean volumetric flow rate $Q ¯$ .

$Figure 3 (a) and (b) Influence of α , β \alpha ,\hspace{.5em}\beta on axial velocity u ( x , y ) \text{ }u(x,y) .$

Figure 3

(a) and (b) Influence of $α , β$ on axial velocity $u ( x , y )$ .

$Figure 4 (a)–(c) Influence of α \text{α} on stream function.$

Figure 4

(a)–(c) Influence of $α$ on stream function.

$Figure 5 (a)–(c) Influence of β \beta on stream function.$

Figure 5

(a)–(c) Influence of $β$ on stream function.

3.1 Pumping characteristics

Figure 2a and b shows different values of material constants $α and β$ on pressure rise ( $Δ p$ ) vs $Q ¯ .$ It is viewed from those graphs that there are nonlinear relation betwixt pressure rise ( $Δ p$ ) and rate of volumetric flow $Q ¯ .$ Figure 4a establishes the pressure which increases with ( $α$ ), in zone $− 1 < Q ¯ < − 0.6$ , since this region resisting force due to material constant involves further pressure distinction and opposite actions is followed in range $− 0.6 < Q ¯ < 4$ , whereas it declines with larger values of material constant $β$ in the range $− 1 < Q ¯ < 0.4$ because the viscosity coefficient of the Ellis fluid is high, the fluid will resist deformation and flow more slowly, resulting in a higher pressure drop or pressure rise in the system. Similarly, if the power-law index of the fluid is high, the fluid will exhibit shear-thickening behavior, which can also result in a higher-pressure rise.

3.2 Axial velocity

Figure 3a and b shows the effect of material constants $α and β$ on horizontal velocity $u ( x , y )$ . It is examined from this figure that behavior of velocity is unaffected in center and near the channel wall due to the existence of cilia at the inner side of channel wall. Figure 3a shows that the axial velocity reduces with the improvement in $( α )$ because elasticity of Ellis fluid is high, the fluid will tend to resist deformation and flow more slowly than a fluid with low elasticity. This can result in a decrease in velocity, whereas inverse behavior can be seen in Figure 3b with growth in material constant $β$ as it causes the velocity to increase near the center of the channel.

3.3 Streamlines

Figures 4 and 5 illustrate the influence of material constants $α$ and $β$ on the stream function. Figures 4 and 5 show that the trapped boluses number increase with material constants because of the decrease in fluid velocity, and also, the amplitude of wave increases in both figures since flow decelerates and assists in building-up greater amplitudes produced by metachronal wave motion. When the material constants of an Ellis fluid are increased, the fluid becomes more resistant to flow and more shear-thinning. This can lead to the formation of more and larger boluses, which can become trapped in the fluid and lead to flow blockages.

4 Conclusion

An investigation has been exhibited for transport of an Ellis fluid through ciliated channel. The flow is generated by a metachronal wave, which is created by the synchronized beating of cilia following an oblique path. The governing mathematical equations are modeled using two approaches, i.e., long wavelength approach and small Reynolds’ number approach. The exact solution addressing the transformation from a fixed frame to a wave frame and the non-dimensional boundary value problem has been obtained. The numerical evaluation of pressure rise is accomplished in MATHEMATICA 13.0 v software with approximately 1 hour of CPU time. In the present analysis, if $α → 0$ , then the fluid model will reduce to Newtonian fluid model. The present study will be helpful to examine problem caused by dust particles in the respiratory tract. The key outcome present that:

Velocity profile shows parabolic nature and achieve utmost magnitudes at the center of channel and further decline rapidly at channel walls.
Velocity reduces with increasing values of material constant α and increases with increasing value of $.$ β
Pressure rise is higher with expanding material constants α and β.
A linear behavior is observed for pressure rise and flow rate of volume for increasing values of material constant $α$ .
Nonlinear behavior is reported for different values of material constant $β .$
Number and size of trapped boluses increase with the increase in the numerical values of material parameters.

5 Aims and applications of the current study

This mathematical study has a variety of applications in biological transport processes. Fluid dynamics is gaining popularity because of several applications in chemical engineering, industrial realm, and biological sciences. The current work has dynamic significance in comprehending the transportation of physiological fluids through arteries and nonuniform vessels in creeping phenomena in medical arena. The current study is useful for comprehending the rheological properties of the Ellis fluids model via a channel. Furthermore, the current work serves as a standard for ciliated driven flow of viscoelastic fluid models.

n.sooppy@psau.edu.sa

Acknowledgments

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445). This study was also supported by National Natural Science Foundation of China, grant number 12474454.

Funding information: This study was also supported by National Natural Science Foundation of China, grant number 12474454.
Author contributions: Conceptualization: M.B.A.; formal analysis: K.S.N., J.H., and T.M.; investigation: M.B.A., S.S., K.S.N., and J.H.; methodology: M.B.A.; software: M.B.A., S.S., K.S.N., and T.M.; supervision: J.H.; validation: K.S.N., and T.M.; visualization: M.B.A. and S.S.; writing – original draft: M.B.A., S.S., K.S.N., J.H., and T.M.; writing – review editing: M.B.A., S.S., K.S.N., and J.H. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-02-06

Revised: 2024-04-10

Accepted: 2024-09-20

Published Online: 2025-03-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2024-0036

Keywords for this article

Ellis fluid model; ciliary movement; metachronal wave; physiological propulsion; exact solution

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