key: cord-0314338-881xynf8
authors: Obembe, Abiola D.; Roostaie, Mohammad; Boudreault, Richard; Leonenko, Yuri
title: On the effect of boundary vibration on Mucus Mobilization
date: 2022-03-13
journal: nan
DOI: 10.1016/j.ijnonlinmec.2022.104019
sha: 9b1b4f96db2c2618b017d1e14376f78ced2ded7a
doc_id: 314338
cord_uid: 881xynf8

In this study, we investigate the mobilization of mucus in a cylindrical tube of constant cross-section subjected to a Small Amplitude Oscillatory Shear (SAOS) assuming the viscoelastic behavior of mucus is described by the Oldroyd-B constitutive equation. The Laplace transform method was adopted to derive expressions for the velocity profile, average velocity, instantaneous flowrate and mean flowrate (i.e., mobilization) of the mucus within the tube. Additionally, a 2D finite element model (FEM) was developed using COMSOL Multiphysics software to verify the accuracy of the derived analytical solution considering both a Newtonian and an Oldroyd-B fluid. Furthermore, sensitivity studies were performed to evaluate the influence of the vibration frequency, vibration amplitude, mean relaxation time, and zero-shear viscosity on mucus mobilization. Results prove that the analytical and numerical results are within acceptable error tolerance and thus in excellent agreement. Besides, the parametric studies indicate a 48%, 57%, and 343% improvement in mucus mobilization when the mean relaxation time, vibration amplitude, and vibration frequency, respectively were increased by a factor of 6. Conversely, the mucus mobilization decreased by 25% when the zero-shear viscosity was increased by a factor of 6. In general, this study confirms that mucus mobilization in a tube can be improved by increasing the magnitude of vibration amplitude and vibration frequency. Similarly, the larger the magnitude of mucus mean relaxation time the better the mucus mobilization when the tube is subjected to boundary vibrations. Finally, mucus mobilization decreases as the magnitude of mucus viscosity increases.

The effect of mechanical oscillations on the flow of viscoelastic and non-Newtonian fluids has garnered the attention of a plethora of researchers due to its numerous application in different industrial processes including pipe flow, secondary oil recovery, filtration, and fluid pumping [1] [2] [3] [4] [5] [6] [7] . Adequate understanding of the complex flow dynamics in these industrial processes is key to preventing operational challenges and optimizing flow processes (i.e., reducing total power requirements, and the detrimental effects induced by sudden start-up or shut down during pipe flow). It is argued that varying the pump or valve pulse frequencies in pipes helps alleviate problems related to sudden start-up or shut down of pipe flows [8] [9] [10] [11] . Other important applications of oscillatory fluid flow comprise acoustics propagation in biological organs (i.e., in the respiratory and circulatory systems of living beings).

Understanding the biological-fluid dynamics, including heartbeat, stenosis, and mucus movement, is beneficial in the diagnosis and treatment of diseases, including cardiovascular and respiratory diseases, and the design of new mechanical systems associated with the biological systems in our bodies [12] . A potentially detrimental example of cardiovascular disease is the sudden blockage resulting from blood clots increasing blood pressure. The resulting increase in pressure typically leads to intracerebral hemorrhage due to the high pressure and shear stress on the vessels' walls upstream of the blockage site and the oxygen deprivation on the downstream side [13] . Therefore, an in-depth understanding of the atherosclerotic-plaques formation in carotid arteries and the associated shear stress in blood vessels is crucial for medical research, diagnosis, and treatment planning [14] [15] [16] . Similarly, Mucus is acknowledged to exhibit non-Newtonian viscoelastic properties, and numerous studies have been devoted to the application of acoustic waves and mechanical oscillations to address respiratory airway clearance. Specifically, the mucus adherence and accumulation in the alveoli of COVID-19 patients exacerbate the already declining lung function by airflow reduction, airway obstruction, and respiratory failure, initiating COVID-19-induced mortality [17] [18] [19] . Some of the devices created to achieve improved airway clearance have also been successfully deployed to treat cystic fibrosis, pulmonary, neurological, and neuromuscular diseases [20] [21] [22] .

In general, non-Newtonian fluids may exhibit either finite yield stress, or shear-thinning, or thixotropy, or a combination of all these characteristics [23] [24] [25] . It has been presented in numerous studies that the shear modulus of some of these fluids is frequency-dependent and increases to a plateau as the oscillation frequency increases [26] [27] [28] . Stress application rate and its continuity, triggered by acoustic waves, alter the non-linear stress-strain curve of non-Newtonian fluids and their elasticity, resulting in an improved displacement of these fluids [29] . The mathematical literature in this field can be categorized into either analytical or numerical approaches. Analytical methods provide for rapid prototypes and better mechanistic/physical insight into the phenomena.

Alternatively, numerical methods provide for a more comprehensive and robust investigation of physical problems since they require fewer assumptions in their development.

In 1963, Ting [30] derived the first exact solution of the non-Newtonian second-grade fluids flow in a cylinder. Subsequently, Srivastava [31] , and Waters and King [32] provided the exact solutions, respectively for Maxwell and Oldroyd-B fluids. The Maxwell constitutive relationship was presented as a simple mechanical model for describing viscoelastic materials that comprise a linear spring and a linearly viscous dashpot arranged in series [33, 34] . Similarly, the Oldroyd-B model is popular in modeling non-linear fluids and is a three-dimensional generalization of the Maxwell model. This model takes into account the fluid incompressibility and its response when satisfying appropriate invariance requirements [35] . Rajagopal [36] presented the exact solutions for both an incompressible and homogeneous second-grade fluid and studied the effects of torsional and longitudinal oscillations in an infinite rod. Rajagopal and Bhatnagar [37] extended their original solutions for an Oldroyd-B fluid. Penton [38] studied the flow of a viscous fluid initiated by an oscillating plate and provided the first closed-form transient solution for this problem. Erdogan [39] using the Laplace transformation technique derived two starting solutions for the flow of a linearly viscous fluid based on the cosine and sine oscillations of a flat plate.

Their solution was further extended to non-Newtonian fluids using the second-grade fluid, Maxwell, and Oldroyd-B models in the following literature just to name a few [37, [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] .

Theoretical predictions show that perturbation methods on viscometric flows (or nearly viscometric flows) can increase the flow rate as a function of frequency and amplitude of oscillations [4, 5, [52] [53] [54] [55] [56] [57] [58] [59] . Mena et al. [4] using numerical and experimental procedures showed that a longitudinal vibration of the pipe wall strongly affects the flow in non-Newtonian fluids. The viscoelastic properties, shear-thinning behavior, and amplitude of the superimposed oscillations are the main parameters of this phenomenon with elasticity as a secondary parameter. Herrera-Velarde et al. [57] investigated the temperature change of a non-Newtonian fluid due to the oscillation of the pipe wall. Their results illustrate peaks in flow rate enhancement at specific wall shear stresses. Fetecau et al. [60] adopted the Fourier transform method to analytically investigate the effect of sidewalls on the flow of a viscous fluid over an oscillating infinite plate. Chen et al. [61] employed experimental and analytical methods to investigate the effect on the flow behavior of a cylindrical rod vibrating in a cylindrical shell filled with viscous fluid. Duarte et al. [62] investigated the flow of a viscoelastic fluid both in pulsating and start-up tests using upperconvected Maxwell and Oldroyd-B fluid models. Kazakia and Rivlin [3, 63] in two consecutive analytical studies investigated the effects of longitudinal, transverse, and rotational vibrations and their combination on non-Newtonian fluid flow in a pipe. They showed that when the fluid viscosity decreases due to the increase in the shear rate, the discharge rate of non-Newtonian fluid may increase with the vibration frequency and amplitude. The fluid flow in their works is assumed to be laminar. Phan-Thien [2] also provided an analytical method to investigate the effect of vibration on fluid flow. On the other hand, recent studies suggest employing specific fluid models, such as upper-convected Maxwell and Oldroyd-B, for non-Newtonian fluids. Hullender [7] studied analytically the pre-transient and turbulent flow of an Oldroyd-B fluid in a circular pipe.

Surprisingly, the effects of vibration on the mucus movement, whether it is in conjunction with airflow or not, have not still been closely examined. Chang et al. [64] in their experimental and theoretical studies about mucus-air movement in airways demonstrated the significant effect of shear stress at the air-mucus interface on mucus velocity. Button et al. [65] reported the effectiveness of cyclic compressive stress, chest percussion, and mechanical therapeutic devices in lung clearance of cystic fibrosis patients. In 1982, Radford et al. [66] reported altered mucus flow rates by applying percussion energy to dogs' and humans' chest walls with the optimal frequency range at 25-35 Hz. In 1983, King et al. [67, 68] studied the mucus clearance in the trachea of nine anesthetized dogs using high-frequency chest wall compression. In their work, the tracheal mucus clearance rate increased with the most pronounced clearance in the range of 11-15 Hz oscillation, peaking at 13 Hz. Then, Gross et al. [69] also reported improvements in peripheral mucociliary clearance by applying high-frequency chest wall oscillation at a 13 Hz and a similar spontaneously breathing population and measurement technique in anesthetized dogs. Rubin et al. [70] experimentally found that a low-energy chest wall oscillator (13 Hz) 

In the following, we present the theoretical description of the transient flow problem of mucous within a vibrated tube where the background flow is driven by a constant pressure gradient.

Typically, such vibrations at the pipe wall, defined herein by ( ), induce mechanical oscillation in the bulk fluid and micro-structures encountered in such rheological complex fluids. In reality, wall vibration in human flow passages may be triggered through a mechanical solicitation (e.g., clapping of the chest walls) [74] or mechano-acoustic treatment devices (i.e. acoustic percussion) [29] , to mention but a few. Figure The Navier-Stokes and continuity equations in the vectorial form are expressed as follows:

where represents pressure and is the mucous density and = 9.81 m/s 2 is the gravitational acceleration.

Therefore, considering the cylindrical geometry of the pipe, the governing equations describing the laminar motion of viscoelastic gel under unsteady-state condition comprise [7] :

where is the axial velocity, = + − or = for flow driven purely by gravity is the net constant pressure gradient, , and define the axial velocity, net pressure gradient, stress tensor, and fluid density respectively.

Besides, is a function of , is a function of and ; however, is assumed to be uniform at any specific . Additionally, to aid the mobilization of mucous within the tube, longitudinal oscillations are introduced at the pipe wall. The velocity of the wall is given by the following relation:

where ( ) is a general function of time that allows for easy Laplace transformation.

Specifically, we consider two different expressions for the wall velocity in this research:

Here, 0 is the acceleration coefficient, 1 is the phase angle, is the displacement amplitude and = 2 is the angular frequency of the vibration, and is the frequency of vibration. The choice of including the constant 1 (i.e., phase shift) in Eq. (7) is to cater for periodic functions or acoustic waves that do not necessarily start at the sinusoidal axis or a maximum or a minimum. Specifically, this allows periodic functions with the same amplitude and angular frequency to exhibit different starting points.

Finally, we assume the stress strain-rate relationship for the fluid is described by the Oldroyd-B tensor model [75] expressed as follows:

Here 0 is the zero-shear viscosity which quantifies the viscous contribution, 1 is the relaxation 

In this section, the semi-analytical solution for the mathematical model developed in Section 3.1 is detailed in the Laplace domain. Interested readers may refer to Appendix A for complete derivation.

The following assumptions are considered to arrive at an analytical solution for laminar flow of mucus in a vertical smooth tube with a uniform cross-sectional area: a) Flow is axisymmetric, and the fluid is assumed to be incompressible. e) The mucus viscosity is assumed to be constant i.e., shear-thinning viscosity is neglected.

The transient flow field (i.e., velocity profile) within the tube in the Laplace domain given by Eq. (A-25) in Appendix A is:

Where the coefficient 1 is defined in Eq. (A-26) as:

Substitution of Eq. (10) in Eq. (9) leads to:

where ̂ is defined as the Laplace transform of .

Assuming the flow passage is a circular tube with radius, , the average velocity is calculated by the expression:

Substituting Eq. (9) in Eq. (12) results in

Adopting integration by parts, Eq. (13) reduces to

For convenience, Eq. (14) can be expressed as

We define the instantaneous flowrate in the tube by the expression:

Similarly, the substitution of Eq. (9) in Eq. (16) yields

Performing integration by parts, the instantaneous flowrate is governed by the expression:

Similarly, we define the mean flowrate as the average of the instantaneous flowrate over a period of oscillation in the real-time domain by the expression [76] :

Where is the period of oscillation.

Applying the Laplace transform operator (ℒ) on Eq. (19) leads to:

Where ̂ is the instantaneous volumetric flowrate in Laplace space derived from Eq. (18).

The expressions for velocity, average velocity, and instantaneous flowrate have been converted from the Laplace-domain to the real-time domain using the Sthefest algorithm for Laplace transform inversion is computed by the expression [77] :

And the weighting coefficients are defined by

All computations in this research were implemented using MATLAB scientific programming language.

A 2D numerical model was simultaneously implemented using the viscoelastic flow module in the COMSOL Multiphysics software (version 5.6) to validate the herein derived semi-analytical solution. The tube is modeling using the 2D axisymmetric geometry (see Fig. 2 ) and the fluid is treated as a viscoelastic gel. Additionally, the tube wall is specified with the appropriate condition, and the pressure at the inlet and outlet of the tube is specified as and , respectively. The program solves the following coupled equations [78] :

∇. = 0 (24)

= ∑

Where is the number of modes/branches of the Oldroyd-B model.

It is important to highlight that the FEM model easily accommodates the shear-thinning viscosity of the gel (i.e., power-law model, Carreau-Yasuda, etc.)

In this paper, two different solutions for the flow of a viscoelastic fluid within a vibrated tube are presented. Both solutions neglect the shear-thinning property of the viscosity; however, more nonlinear viscoelastic models such as the Giesekus model can easily be implemented using a FEM model to cater to the shear-thinning property of viscosity. Table 1 provides an overview of the key attributes of both solutions presented in this research. 

The results and discussion section comprises of the following: In Section 4.1, the derived semianalytical solution is validated against the FEM solution obtained from the COMSOL Multiphysics software for steady-state and transient flow synthetic problems. Additionally, the remainder section (Section 4.2) which consists of four sub-sections presents the results of the parametric analysis conducted to illustrate the effects of the vibration frequency, vibration amplitude, mean relaxation time, and zero-shear viscosity on the flow and mucus mobilization within the vibrated tube.

Herein, the accuracy of the derived analytical solution for flow field in the vibrated cylindrical tube was verified against the numerical model from COMSOL considering both the static and dynamic flow configurations. First, a stationary analysis is computed with no movement of the pipe wall considering both a vicious and viscoelastic fluid. Furthermore, a time-dependent analysis for a viscoelastic fluid is solved where the tube wall velocity is prescribed according to Eq. (6) . Besides, to establish the accuracy of the analytical solution we neglect the shear-thinning of viscosity in the FEM model. Table 2 enumerates the synthetic base-case input data utilized for validating the analytical and numerical solutions. 

The steady-state case was first solved considering a viscous fluid and a viscoelastic fluid for the limiting case of no wall movement (i.e., = 0). It is important to note that setting 1 = 2 , the Oldroyd-B constitutive equation reduces to a Newtonian fluid with a viscosity equal to the solvent viscosity . Noting that the zero-shear viscosity is the addition of the polymer viscosity and implies that = 0 for a Newtonian fluid. Furthermore, to arrive at the steady-state solution, we set → ∞ for the analytical solution. The velocity profiles generated from both solutions considering a viscous fluid and a viscoelastic gel are shown in Fig. 3 . Each data point in the plot corresponds to the velocity at a given position along the pipe radius at steady-state conditions. As expected, Fig. 3(b) indicates that the velocity profile for the viscoelastic elastic fluid is ~ 10 4 order of magnitude smaller than that of the Newtonian fluid in Fig. 3(a) . Besides, an inspection of Fig. 3 reveals that there is reasonable overlap between both curves for both fluids providing a first-step verification of the derived analytical solution and numerical model. Additionally, the average velocity and volumetric flow rate predicted from both solutions are in close agreement with a relative error of less than 1% as shown in Table 3 . 

The accuracy of the derived analytical solution was further verified against the numerical solution by performing a time-dependent analysis for an Oldroyd-B fluid where the tube wall velocity is defined as a linear function of time i.e., Eq. (6). Applying the Laplace transform operator on the Eq. (6) leads to:

Adopting a linear function of time for the wall velocity, the flow behavior (i.e., velocity and flow rate) of the viscoelastic fluid predicted from both solutions are displayed in Fig. 4 . 

Herein, we investigate the transient flow behavior of mucus through a vibrated tube where the flow is driven by a combination of pressure and gravity. In addition, the axial velocity at the wall of the tube is an oscillating function of time as defined in Eq. (7) . The corresponding sinusoidal wall velocity in the Laplace domain is given by the expression:

Additionally, to capture the viscoelastic behavior of human sputum, rheometric data representative of the healthy range of human lung mucus extracted from the literature was utilized as input data [79] . Table 4 enumerates the data and corresponding fit using five Upper Convected Maxwell modes. Accordingly, from the data listed in Table 4 , the zero-shear viscosity and mean relaxation time can be estimated from the following expressions [80] :

The contribution of the Giesekus parameter would be neglected in this study since the Oldroyd-B constitutive equation assumes the mucus viscosity is constant (i.e., shear-rate-independent). Besides, only the smallest relaxation time would be utilized to characterize the mucus. Additional data presented in Table 5 would be utilized to quantitatively establish the effects of vibration frequency, vibration amplitude, relaxation time, and zero-shear viscosity on flow quantities (i.e., velocity and instantaneous flow rate, and mean flow rate). To ensure a laminar flow regime, the parameters were chosen such that the vibrational Reynolds number ( = ) was always less than 30 [81] . Finally, the transient vibrational simulation was conducted over 4 periods of the cosine wave (i.e., from 0 = 0 to = 8 ).

The magnitude of frequency denotes the number of vibrations (i.e., pressure peaks) perceived per second. The effect of frequency on the vibrational flow of mucus was investigated by independently varying the frequency between 15 Hz to 90 Hz while keeping other inputs at their base level enumerated in Table 5 . As shown in Fig. 5 , increasing the magnitude of vibration frequency leads to axial velocity curves shifted upwards (i.e., larger axial velocity values).

Thus, Fig. 5 suggests that mucus transport within the tube improves as the magnitude of vibration frequency increases. To further establish this hypothesis, the average velocity, and instantaneous flow rate as a function of vibration frequency after four periods of the cosine wave are exhibited in Fig. 6 .

Evidently, Fig. 6 indicates that that the vibration frequency has a positive relationship with both the average velocity and instantaneous flow rate, respectively (i.e., the average velocity and instantaneous flow rate increase with vibration frequency). For instance, the instantaneous flow rate is 7.36 × 10 −9 m 3 /s, and 1.61 × 10 −8 m 3 /s, when = 15 Hz, and 90 Hz, respectively. Furthermore, the effect of vibration frequency on the mean flow rate (i.e., mucus mobilization) was quantified and summarized in Table 6 . Of interest, Table 6 confirms that for a = 1.6 × 10 −3 m, when the vibration frequency increased by a factor of 2 (i.e., from 15 Hz to 30 Hz) and a factor of 6 (i.e., from 15 Hz to 90 Hz), the mucus mobilization improved by 68% and 343%, respectively. Consequently, we argue that a linear viscoelastic constitutive model can predict a 68% improvement in mucous mobilization by only increasing the vibration frequency by a factor of 2 i.e., from 15 Hz to 30 Hz.

The influence of the amplitude of vibration on the flow behavior including mucus mobilization was quantified by varying the amplitude values up to 1.6 × 10 −3 m while keeping other parameters fixed at their base values listed in Table 5 . As shown in Fig. 7 , the axial velocity curves are shifted upward as the vibration amplitude increases.

Albeit it is important to specify that the curves are less spaced when compared to the plot generated in Fig. 5 . Inspection of Fig. 7 suggests that higher vibration amplitude leads to larger axial velocity values along the radial cross-section. The average velocity and instantaneous flowrate as a function of vibration amplitude displayed in Fig. 8 were then estimated to validate this hypothesis. Fig. 8(b) , the instantaneous flow rate is 7.36 × 10 −9 m 3 /s, 8.58 × 10 −9 m 3 /s, 9.92 × 10 −9 m 3 /s, 11.46 × 10 −9 m 3 /s, 13.39 × 10 −9 m 3 /s, and 16.10 × 10 −9 m 3 /s, for = 0.1 × 10 −3 m, 0.2 × 10 −3 m, 0.4 × 10 −3 m, 0.6 × 10 −3 m 1.0 × 10 −3 m, and 1.6 × 10 −3 m, respectively. Finally, the mucus mobilized as a function of vibration amplitude is enumerated in Table 7 . Table 7 confirms that at = 15 Hz, increasing the vibration amplitude by a factor of 2 (i.e., 0.1 × 10 −3 m to 0.2 × 10 −3 m) and 6 (i.e., 0.1 × 10 −3 m to 0.6 × 10 −3 m) leads to 11% and 57% respectively, improvement in mucus mobilization. Therefore, we argue that a significant increase in the amplitude of vibration is required for significant improvement in mucus mobilization.

For viscoelastic fluids, the relaxation time characterizes the time taken for the fluid to return to an unperturbed (i.e., equilibrium) state following a deformation. Typically, large values of relaxation time indicate elastic response while small values imply a viscous response. The role of ̅ 1 on the flow behavior and mobilization of mucus due to vibration was established by independently varying the magnitude of ̅ 1 between 0.089 s to 2.67 s (i.e., ̅ 1 < 3 s) over 4 periods of the cosine wave. The plot depicted in Fig. 9 indicates that when boundary vibrations are introduced at the tube wall, the curves representing the axial velocity are shifted upwards as ̅ 1 increases.

Oscillation at the boundary wall leads to larger values of axial velocity in the tube as the elastic contribution to the overall material response increases. For example, the axial velocity at the pipe centerline equals 1.11 × 10 −3 m/s, 1.70 × 10 −3 m/s, and 5 × 10 −3 m/s for ̅ 1 = 0.089 s, 0.89 s, and 2.67 s, respectively. Therefore, neglecting the elastic contribution from a viscoelastic fluid will lead to underpredicting the axial flow velocity. The relationship between mucus average velocity and instantaneous flow rate as a function of ̅ 1 are exhibited in Fig. 10 . Table 8 summarizes the mucus mobilized in the vibrated tube for the given values of ̅ 1 . Table 8 indicates that for a given vibration frequency and amplitude (i.e., 15 Hz and 1.6 × 10 −3 m), a larger value of mucus is mobilized within the tube as the magnitude of ̅ 1 increases. Specifically, a 9% and 48% improvement in the mean flow rate was observed when ̅ 1 was increased from 0.089 s to 0.178 s and from 0.089 s to 0.534 s, respectively. Therefore, we argue that as the elastic contribution of mucus increases due to chronic lung infection or inflammation, the mucus clearance efficiency can be improved via introducing boundary oscillations.

Viscosity is a measure of the resistance to deformation of fluid at the flow condition and it quantifies material response for viscous fluids. It is important to recall that for the linear viscoelastic theory, the viscosity is independent of shear rate and is quantified by the zero-shear viscosity. The axial velocity profile as a function of radial distance after 4 periods of the cosine wave is illustrated in Fig. 11 . Table 8 . Table 9 indicates that for a given value of vibration frequency of 15 Hz and amplitude of 1.6 × 10 −3 m, a 15% and 26% decrease in mucus mobilization were estimated by increasing the viscosity by a factor of 2 (i.e. 11 Pa.s to 22 Pa.s) and 6 (i.e., 11 Pa.s to 66 Pa.s), respectively.

In this paper, a semi-analytical solution for the Poiseuille flow of mucus in a vibrated cylindrical tube was derived by adopting the Laplace transform method on the governing partial differential equations. The derived expressions for the velocity field, average velocity, instantaneous flowrate, and mean flowrate were subsequently utilized to examine the transient flow response and mobilization of mucus due to the sinusoidal oscillations at the tube wall. It is important to highlight some of the key advantages of the proposed analytical solution includes its convenience for rapid prototyping due to speed, stability, and accuracy since numerical solutions for viscoelastic flows sometimes fail to converge especially for high Weissenberg numbers. Besides, the presented analytical solution provides a convenient tool for experimenting with different wall velocity expressions (i.e., sinusoidal, non-sinusoidal, and other novel functions). The only condition is that the proposed wall velocity function satisfies the conditions for applicability of Laplace transform (i.e., the function must be locally integrable for the interval [0, ∞]). Furthermore, the analytical solution allows for studying viscoelastic flows driven by timedependent pressure gradients which may stem from airflow due to coughing [82] , and pulsating pressure gradients. The main findings from this research are outlined as follows:

1. The vibration frequency exhibits a positive relationship with both the average velocity and instantaneous flow rate. Analysis indicated that the mucus mobilized in the tube improved by 68% and 343% when the vibration frequency was increased by a factor of 2 (i.e., 15 Hz to 30 Hz) and 6 (i.e., 15 Hz to 90 Hz), respectively. 2. The vibration amplitude displays a positive trend with the average velocity and instantaneous flowrate. Besides, mucus mobilization increased by 11% and 57% when the vibration amplitude by a factor of 2 (i.e., 0.1 × 10 −3 m to 0.2 × 10 −3 m) and 6 (i.e., 0.1 × 10 −3 m to 0.6 × 10 −3 m), respectively. 3. The mean relaxation time depicts a positive correlation with the average velocity and instantaneous flow rate. Results indicated that for a frequency of 15 Hz and vibration amplitude of 1.6 × 10 −3 m, mucus mobilization improved by 9% and 48% when the value of ̅ 1 was increased by a factor of 2 (i.e., 0.089 s to 0.178 s) and a factor of 6 (i.e., 0.089 s to 0.534 s), respectively. 4. The average velocity and instantaneous flow display a negative trend with the zero-shear viscosity. Numerical experiments indicated that mucus mobilization decreased by 15% and 26% when the zero-shear viscosity was multiplied by a factor of 2 (i.e. 11 Pa.s to 22 Pa.s) and 6 (i.e., 11 Pa.s to 44 Pa.s), respectively.

The authors would like to acknowledge the support provided by Dymedso Inc, Natural Sciences and Engineering Research Council of Canada and by a grant in aid of research from the Mitacs Accelerate Fellowship.

The amplitude of wall displacement, m Where 1 and 2 are coefficients to be obtained from the specified boundary conditions, and is the notation for √−1, I 0 (. ) and K 0 (. ) are the modified Bessel functions of the first and second kind with order 0. 

Annular effect in viscoelastic fluids

The effects of random longitudinal vibration on pipe flow of a non-Newtonian fluid

The influence of vibration on Poiseuille flow of a non-Newtonian fluid, I

Complex flow of viscoelastic fluids through oscillating pipes. Interesting effects and applications

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Exact solutions for some simple flows of an Oldroyd-B fluid

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Figure Caption 1. Fig. 1 Schematic of the flow geometry 2. Fig. 2 Meshed 2D axisymmetric geometry of the vibrated tube 3. Fig. 3 Velocity profile solution in the stationary regime for (a) Newtonian fluid and (b) Viscoelastic fluid

Model validation for dynamic wall (a) velocity profile at = 0.005 s, (b) average velocity history, and (c) volumetric flowrate history 5

Effect of vibration frequency on (a) average velocity and (b) Instantaneous flowrate

Effect of vibration amplitude on the axial velocity profile

Effect of vibration amplitude on (a)average velocity and (b) instantaneous flowrate

Effect of mean relaxation time on axial velocity 10. Fig. 10 Effect of mean relaxation time on (a) average velocity and (b) instantaneous flowrate 11. Fig. 11 Effect of zero-shear viscosity on the axial velocity profile 12