Computer Methods in Biomechanics and Biomedical

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Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube a

b

D. Tripathi & O. Anwar Bég a

Department of Mathematics, National Institute of Technology – Delhi, Dwarka, Delhi 110077, India b

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Mathematical Modelling, Gort Engovation (Propulsion and Biomechanics Research), Bradford BD73NU, UK Published online: 25 Jul 2014.

To cite this article: D. Tripathi & O. Anwar Bég (2015) Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube, Computer Methods in Biomechanics and Biomedical Engineering, 18:15, 1648-1657, DOI: 10.1080/10255842.2014.940332 To link to this article: http://dx.doi.org/10.1080/10255842.2014.940332

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Computer Methods in Biomechanics and Biomedical Engineering, 2015 Vol. 18, No. 15, 1648–1657, http://dx.doi.org/10.1080/10255842.2014.940332

Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube D. Tripathia and O. Anwar Be´gb* a

Department of Mathematics, National Institute of Technology – Delhi, Dwarka, Delhi 110077, India; bMathematical Modelling, Gort Engovation (Propulsion and Biomechanics Research), Bradford BD73NU, UK

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(Received 16 August 2013; accepted 27 June 2014) This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo’s definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls. Keywords: biophysics; peristaltic transport; fractional second-grade model; inclined tube; uretral hydrodynamics; Froude number; Mathematica software

1.

Introduction

Physiological fluids are transported from one part to another part of body by continuous process of muscle contraction and relaxation. This process is designated as peristaltic transport. Peristalsis is produced by successive waves of contraction in elastic, tubular structures which push their fluid or fluid-like contents forward. An interesting fact is that in the oesophagus where the movement of masticated food is due to peristalsis, food moves from the mouth to the stomach even when the body is inverted. The vasomotion of blood, food mixing, chyme movement in intestine, transport of bile in bile ducts and transport of spermatozoa in cervical canal are some important physiological examples where peristalsis is prevalent. In the urinary system, peristalsis is due to involuntary muscular contractions of the ureteral wall which drives urine from the kidneys to the bladder through the ureters. The periodic lateral movements of these vessels are induced by electro-chemical reactions taking place in the body. Although peristalsis is a familiar mechanism in physiological sciences, in applied mechanics, investigations of this complex flow phenomenon were initiated quite recently. Latham (1966) stimulated interest in peristalsis modelling in the mid-1960s with a seminal study (analytical and experimental) on biomechanical

*Corresponding author. Email: [email protected] q 2014 Taylor & Francis

peristaltic pumping. Fung and Yih (1968) theoretically verified many of the phenomena described in Latham (1966). Shapiro et al. (1969) further theoretically examined the peristaltic flow of viscous fluid induced by sinusoidal wall propagation, confirming fundamental findings in Latham (1966). They performed the analysis under long wavelength assumption and discussed the phenomena of reflux and trapping during peristalsis. Seminal studies of Newtonian peristalsis were further presented by Fung (1971), Yin and Fung (1971), Jaffrin and Shapiro (1971), Weinberg et al. (1971) and Weinberg (1970). More recent reviews of the subject include Grotberg and Jensen (2004) and Roy et al. (2011). The studies described in Latham (1966), Fung and Yih (1968), Shapiro et al. (1969), Fung (1971), Yin and Fung (1971), Jaffrin and Shapiro (1971), Weinberg et al. (1971), Weinberg (1970), Grotberg and Jensen (2004) and Roy et al. (2011) have been confined to Newtonian viscous flows, i.e. employed simplifications of the classical Navier –Stokes equations. The rheological characteristics of fluids involved in peristaltic transport (e.g. bile, embryological fluids and mucus), however, have been clearly identified for many decades (Mirizzi 1942; Denton 1963; Vander et al. 1975; Mahrenholtz et al. 1978; Macagno and Christensen 1980; Roselli and Diller 2011;

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Computer Methods in Biomechanics and Biomedical Engineering Tam et al. 1980; Wolf et al. 1977). Such fluids have been shown to exhibit strongly viscoelastic properties. In parallel with the physiological studies, numerous mathematically orientated simulations of peristaltic viscoelastic flow have also been communicated, using a diverse range of rheological formulations. Interesting studies in this regard include peristaltic modelling of third-order fluids (Siddiqui and Schwarz 1993), Johnson –Segalman fluids, Oldroyd fluids (Elshehawey and Sobh 2001; Haroun 2011), Maxwell fluids (Pandey and Tripathi 2010a), Jeffrey fluids (Pandey and Tripathi 2010b), Casson fluids (Pandey and Tripathi 2010c) and Phan – Thien – Tanner liquids (El Hakeem and El Naby 2009). The influence of inclination in peristaltic flows is also of significance, in particular in reproductive hydrodynamics. In uterine peristalsis, Eytan and Elad (1999) have developed a Stokes flow model of peristalsis in a twodimensional channel based on lubrication theory, showing that flow dynamics is markedly influenced by the phase shift between the contracting walls. Several researchers have investigated viscoelastic peristaltic transport in inclined geometries. Nadeem and Akbar (2010) used the Walters-B model to simulate peristaltic flow in an inclined conduit. Further studies have utilized the Eringen micropolar model (accounting for micro-inertia and rotary motions of bio-micro-elements in the fluid) (Maruthi Prasad et al. 2010), the Herschel – Buckley fluid (Vajravelu et al. 2005) and, very recently, the Jeffrey fluid model with partial slip effects. Fractional calculus has encountered much success in the description of viscoelasticity. The starting point of the fractional derivative model of non-Newtonian model is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the so-called Riemann –Liouville fractional calculus operators. This generalization allows one to define precisely noninteger order integrals or derivatives. The fractional secondgrade model is a model of viscoelastic fluid. In general, the fractional second-grade model is derived from the wellknown second-grade model by replacing the ordinary time derivatives to fractional-order time derivatives, and this plays an important role to study the valuable tool of viscoelastic properties. A number of recent studies (Friedrich 1991; Tan et al. 2003; Hayat et al. 2004; Nadeem 2007; Qi and Xu 2007; Jia and Hua 2008) have investigated flows of viscoelastic fluids with the fractional Maxwell model, fractional generalized Maxwell model or fractional Oldroyd-B model, for various wall geometries. These investigations have developed solutions for the velocity field and the associated shear stress using an extensive range of mathematical functions including Laplace, Fourier, Weber, Hankel and discrete Laplace transforms. Recently, Tripathi et al. (2010) have studied the peristaltic flow of fractional Maxwell fluids through a channel under long wavelength and low Reynolds number approximations

1649

with both the homotopy perturbation method and the Adomian decomposition method. Motivated by the complex applications in physiological flows, the study in Tripathi et al. (2010) has been considerably extended for other viscoelastic fluids and different geometrical configurations by Tripathi (2010, 2011a, 2011b, 2011c, 2011d). Other important studies addressing fractional models for viscoelastic fluids include Yulita Molliq et al. (2009), Jang (2014), Liu et al. (2011), Shah and Qi (2010), Yulita Molliq and Batiha (2012) and Guo et al. (2013). In the present paper, the peristaltic transport of secondgrade fluid with fractional model through a cylindrical tube with inclination under the assumptions of long wavelength and low Reynolds numbers is presented. The second-grade model is a special case of the general nth-order Reiner – Rivlin ‘differential’ fluid model. This rheological model and the third-grade model have previously been rigorously investigated in the context of multi-physical boundary layer flows of polymers and biotechnological fluids by Anwar Be´g et al. (2001, 2004, 2008, 2011). Other recent investigations implementing viscoelastic models in peristaltic transport include Akbar and Nadeem (2013) who applied a Johnson –Segalman model, Akram and Nadeem (2013) who used a Jefferys model and also Akbar and Nadeem (2013) who implemented a Jefferys six-constant elasto-viscous model. In this study, Caputo’s definition is applied to determine approximate analytical solutions of inclined tube peristaltic flow of a fractional second-order biofluid. Numerical results of the problem for different cases are depicted graphically. The effects of fractional parameter, material constant, inclination, Reynolds number, Froude number and amplitude on the pressure difference and frictional force across one wavelength are discussed.

2.

Caputo’s definition

The following definition (Miller and Ross 1993; Samko et al. 1993; Podlubny 1999; West et al. 2003) is used for solving the fractional differentiations. Caputo’s definition of the fractional-order derivative is specified as D a* f ðtÞ ¼

1 Gðn 2 a*Þ

ðt

f n ðtÞ dt a*þ12n b ðt 2 tÞ

ð1Þ

ðn 2 1 , Re ða*Þ # n; n [ NÞ; where the parameter a* is the order of derivative and is allowed to be real or even complex, and b is the initial value of function f . For Caputo’s derivative, we have

D a* t b ¼

8 0 >
: Gðb 2 a* þ 1Þ t

ðb # a* 2 1Þ; ðb . a* 2 1Þ:

ð2Þ

D. Tripathi and O. Anwar Be´g

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~ g_ and l~1 , is the time, shear stress, rate of shear where ~t; S; strain and material constant, respectively, m is the viscosity and b is the fractional parameter such that 0 , b # 1. This model reduces to second-grade model with b ¼ 1, and the classical Navier –Stokes (Newtonian viscous) model is obtained by substituting l~1 ¼ 0. The governing equations of the motion of secondgrade fluid with fractional model for inclined tubular flow may be specified as follows:

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3. Mathematical flow model The regime studied is illustrated in Figure 1 and comprises a uniform inclined tube, orientated at angle, a, to the horizontal. When the wall of the tube is brought under the influence of a periodic radial contraction wave, a segment of the wall begins to contract initially at the inlet, relaxes and the portion located immediately ahead of this begins to contract, thus resulting in the propagation of contraction wave towards the outlet. Relaxation culminates at the

     9 b D~u ›p~ 1› ›u~ ›2 u~ b › ~ > ¼ 2 þ m 1 þ l1 b r~ r þ 2 þ rg sin a; > > > D~t r~ ›r~ ›x~ ›~t ›r~ ›x~ > > > >       = b 2 v D~v ›p~ › 1 › › ~ b › ¼ 2 þ m 1 þ l~1 b ð~rv~ Þ þ 2 2 rg cos a; r > D~t ›r~ r~ ›r~ ›r~ ›~t ›x~ > > > > > ›u~ 1 ›ð~rv~ Þ > > þ ¼ 0; ; ›x~ r ›r~

where D=D~t ; ð›=›~tÞ þ u~ ð›=›x~ Þ þ v~ ð›=›r~Þ and r, r~; u~ ; v~ ; p~ ; g; a stand for fluid density, radial coordinate, axial velocities, radial velocities, pressure, acceleration due to gravity and inclination angle of tube, respectively. For carrying out further analysis, we introduce the following non-dimensional parameters:

natural boundary without expanding further beyond it. This process continues until complete transportation takes place. The geometry of wall of the tube (Figure 1) is mathematically simulated with the following equation:   ~ x; ~tÞ ¼ a 2 0:5f~ 1 þ cos 2p ð~x 2 c~tÞ ; hð~ l

ð3Þ

9 b x~ r~ u~ v~ > c~t cl~1 > b x ¼ ; r ¼ ; t ¼ ; l1 ¼ ; u¼ ; v¼ ; > > a c cd = l l l p~ a 2 a f~ c2 > > d¼ ; f¼ ; p¼ ; Re ¼ r cad=m; Fr ¼ : > > l a m cl ga ;

where x~ ; ~t; a; f~; l; c and h~ are, respectively, axial coordinates, time, radius of the tube, amplitude of wave, wavelength, wave-speed and radial displacement of the walls from the centre line. The constitutive equation for a second-grade fluid with the fractional model is given by   b b › S~ ¼ m 1 þ l~1 b g_ ; ›~t

ð6Þ where d is defined as the wave number, Re is the Reynolds number and Fr is the Froude number. Introducing non-

ð4Þ

Distensible walls

x

Gravity, g r

Viscoelastic biofluid α Inclined tube Peristaltic wave, celerity c

Figure 1.

Physical model and coordinate system.

ð5Þ

Computer Methods in Biomechanics and Biomedical Engineering dimensional parameters, Equation (3) reduces to hðx; tÞ ¼ 1 2 f cos 2 pðx 2 tÞ:

ð7Þ

Implementing the classical ‘long wavelength’ and ‘low Reynolds number approximations’, following Fung (1971), Equation (5) reduce to the following simplified form: 9   2  b ›p › u 1 ›u Re > b › > sin a; > ¼ 1 þ l1 b þ þ > 2 > r ›r Fr ›x ›t ›r > > > = ›p ð8Þ ¼ 0; > ›r > > > > ›u 1 ›ðrvÞ > > þ ¼ 0: > ; ›x r ›r

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Boundary conditions are given by

›u ¼ 0 at r ¼ 0; u ¼ 0 at r ¼ h: ›r

ð9Þ

Integrating Equation (8) with respect to r, and using the first condition of Equation (9), the velocity gradient is obtained as follows:     b ›u r ›p Re b › sin a : ¼ 2 1 þ l1 b ð10Þ ›t ›r 2 ›x Fr Further integrating Equation (10) from 0 to r, and using the second condition of Equation (9), we arrive at the axial velocity:     ›b 1 ›p Re 2 sin a : ð11Þ 1 þ lb1 b u ¼ ðr 2 2 h 2 Þ 4 ›t ›x Fr Ðh The volume flow rate is defined as Q ¼ 0 2ru dr, which, by virtue of Equation (11), reduces to     b h 4 ›p Re b › 1 þ l1 b Q ¼ 2 2 sin a : ð12Þ ›t 8 ›x Fr The transformations between the wave and the laboratory frames, in the dimensionless form, are given by X ¼ x 2 t; V ¼ v;

R ¼ r;

U ¼ u 2 1;

ð13Þ

q¼Q2h ; 2

where the left-hand side parameters are in the wave frame and the right-hand side parameters are in the laboratory frame. The following are the existing relations between the averaged flow rate, the flow rate in the wave frame and that in the laboratory frame: 2

2

 ¼ q þ 1 2 f þ 3f ¼ Q 2 h 2 þ 1 2 f þ 3f : ð14Þ Q 8 8

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Equation (12), in view of Equation (14), gives   b ›p Re 8 b ›  þ h2 2 1 sin a 2 4 1 þ l1 b ðQ ¼ h ›X Fr ›t þ f 2 3f 2 =8Þ:

ð15Þ

Using Caputo’s definition in Equation (15), we get

›p Re sin a ¼ ›X Fr

   þ h 2 2 1 2 f þ 3f 2 =8Þ 8ðQ tb b 2 1 þ l1 : h4 Gð1 2 bÞ ð16Þ

The pressure difference ðDpÞ and frictional force ðFÞ across one wavelength are, respectively, defined as Dp ¼

F¼

ð1

›p dX; › 0 X

 ð1 ›p 2h 2 dX: ›X 0

ð17Þ

ð18Þ

From Equation (11) and using the transformations of Equation (13), the stream function in the wave frame (obeying the Cauchy – Riemann equations, U ¼ ð1=RÞ  ð›c=›RÞ and V ¼ 2ð1=RÞð›c=›XÞ) takes the form:    þ h 2 2 1 þ f 2 ð3f 2 =8Þ Q c¼2 2h 4 ð19Þ R2 4 2 2 : ðR 2 2R h Þ 2 2 From Equation (19), it is clear that the stream function is independent of the fractional parameter ðbÞ, the material constant ðl1 Þ, the Reynolds number (Re) and the Froude number (Fr), but dependent on amplitude ðfÞ.

4. Numerical results and discussion The flow dynamics in the peristaltic viscoelastic regime examined is effectively controlled by six dimensionless hydromechanical and geometric parameters: the amplitude ðfÞ (determined by the geometry of the problem); the fractional parameter ðbÞ, the material constant ðl1 Þ, the Reynolds number (Re) and the Froude number (Fr), which are defined from the governing equations. A further important geometric parameter is the inclination angle of the conduit ðaÞ. In this section, pressure difference across one wavelength, i.e. pressure ðDpÞ, and frictional force across one wavelength (F) are calculated for various values of the above parameters. Numerical results of the problem under consideration are expressed via graphical

D. Tripathi and O. Anwar Be´g

1652 40

β = 1/5 β = 2/5 β = 3/5 β = 4/5

Δp 30

α = π/2 α = π/3 α = π/4 α = π/5

40 Δp

20

30 20

10

Q

10

Q

0.2 0.2

0.4

0.6

0.8

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Figure 2. Pressure versus averaged flow rate for various values of b at t ¼ 0:5, f ¼ 0:4, a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1.

plots which have been generated using the Mathematica software (Wolfram Research, Champaign, IL) (Hollis 2012). Figures 2 – 7 depict the variation of pressure ðDpÞ with  for different values of fractional the averaged flow rate Q parameter, material constant, inclined angle, Reynolds number, Froude number and amplitude. These plots indicate that the relationship between pressure and averaged flow rate is linear, and it is observed that an increase in the flow rate reduces the pressure. Therefore, maximum flow rate is achieved with zero pressure and vice versa. Figure 2 shows the relationship between pressure and averaged flow rate for different values of b at t ¼ 0:5, f ¼ 0:4, a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1. It is found that the pressure decreases with increase in fractional parameter, b. We note that a low Froude number (Fr ¼ 0.2) peristaltic waves will propagate upstream, i.e. away from the entry point in the inclined tube. For Fr . 1, this situation will be reversed as elucidated by Weinberg (1970). We therefore consider only small Froude numbers in the present study. The Dp profiles are all found to  , 0:85; thereafter the increase in fractional converge at Q parameter,b, is found to exert the opposite effect, i.e. enhance pressure difference. λ1=1 λ1=2 λ1=3 λ1=4

80 Δp

60

0.4

0.6

0.8

1

1

Figure 4. Pressure versus averaged flow rate for various values of a at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, Re ¼ 0:1, Fr ¼ 0:01 and l1 ¼ 1.

The variation of pressure with the averaged volume flow rate for different values of l1 at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1 and Fr ¼ 0:1 is presented in Figure 3, and from this figure, it is observed that the pressure increases with increasing l1 . Again, we observe the convergence of pressure difference profiles towards  , 0:85; thereafter the increase in material parameter Q induces a reduction in pressure difference. Figure 4 illustrates the influence of conduit inclination  in the interval, angle on the variation of Dp with Q p=5 # a # p=2. For this plot, we use a much lower Froude number (Fr ¼ 0.01), with t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, Re ¼ 0:1 and l1 ¼ 1. It is evident that as we change the orientation of tube from weak inclination (a ¼ p/5) to greater inclinations (a ¼ p/4, p/3) and finally to the vertical tube case (a ¼ p/2), the pressure is significantly accentuated. This observation physically concurs with the experimental findings of Weinberg (1970) for Newtonian fluids, El Hakeem and El Naby (2009) for a Walters-B elasto-viscous fluid and also Vajravelu et al. (2005) for a Herschel –Buckley fluid. Confidence in the validity of the present computations is therefore high. The linear decay in pressure difference from zero to maximum averaged flow rate, for any inclination of the conduit, is also clearly visible in Figure 4.

Re = 0.1 Re =0.3 Re =0.5 Re =0.7

40 Δp 30 20

40 Q

10

Q

20 0.2

0.4

0.6

0.8

1

–20

Figure 3. Pressure versus averaged flow rate for various values of l1 at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1 and Fr ¼ 0:1.

0.2

0.4

0.6

0.8

1

Figure 5. Pressure versus averaged flow rate for various values of Re at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Fr ¼ 0:1 and l1 ¼ 1.

Computer Methods in Biomechanics and Biomedical Engineering Fr=0.01 Fr=0.02 Fr=0.05 Fr=0.1

40 Δp

30 20

Q

10 0.2

0.4

0.6

0.8

1

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Figure 6. Pressure versus averaged flow rate for various values of Fr at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1 and l1 ¼ 1.

Figure 5 illustrates that the influence of Reynolds number on pressure at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Fr ¼ 0:1 and l1 ¼ 1, and it is noted that the pressure increases with increase in Reynolds number. Momentum increase in the peristaltic motion relative to viscous effects will exacerbate the build up of pressure in the inclined tube. Therefore, even at relatively low Reynolds numbers (Re ¼ 0.7 implies the inertial force is 70% that of the viscous hydrodynamic force), momentum exerts a significant effect on the pressure accumulation in the tube. Eytan et al. (2001) have in fact indicated that even at Reynolds numbers as low as 0.001, as characterized by uretral peristaltic hydrodynamics, a slight increase in Reynolds number can induce significant pressure escalations in the human cervical canal. Reynolds numbers can also rise by an order of magnitude in such physiological processes as also highlighted by Fung (1971) and Fauci and Dillon (2006). We further note that in Figures 4 and 5 (a and Re effects, respectively), the cross-over of profiles is absent, i.e. the profiles remain consistent for all values of averaged flow rate, unlike in Figures 2 and 3 where a reversal in non-Newtonian material parameter effects (b, l1) is sustained at high average flow rate. Figure 6 depicts the variation of pressure difference with average flow rate for various Froude numbers (Fr). Fr

80 Δp

is a dimensionless quantity used to evaluate the influence of gravity on fluid motion. It arises generally in all situations where body waves are present, e.g. surface water waves and laminar diffusion flames, and is an important parameter in peristaltic flows. In these various applications, the wave propagation velocity is known as celerity. For confined flows, Fung (1971) has emphasized that very low Froude numbers only are realistic and the socalled critical Froude number associated with hydrodynamic or hydraulic jumps in water wave mechanics is not feasible. The analysis herein is therefore constrained to very small Froude numbers, which ensure that peristaltic waves progress along the inclined tube away from the entry location (the reverse is caused for Froude numbers in excess of unity, an intangible case in closed conduit peristaltic flows). The axial pressure gradient decreases on increasing Froude number, an effect which has also been computed in other viscoelastic peristaltic studies, e.g. Siddiqui and Schwarz (1993) and Elshahed and Haroun (2005) and, of course, Fung (1971) (for Newtonian viscous flows). Pressure difference is therefore strongly decreased with increasing Froude number, and the effect is the opposite to increasing Reynolds number. The present computations also concur with the simulations in Siddiqui and Schwarz (1993) and Elshahed and Haroun (2005) in this regard, although fractional second-grade fluids were not studied in these references. Figure 7 provides the distributions of pressure with averaged flow rate for different values of peristaltic wave amplitude (f) at t ¼ 0:5, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1. We observe that pressure increases with increasing amplitude; the profiles also exhibit a convergence at higher averaged flow rates, but do not intersect the horizontal axis (as with Figures 2 and 3). Furthermore, in Figures 2 –7, it is found that maximum averaged flow rate is unique for different values of b and l1 , while it is different for various values of a, Re, Fr and f. Figures 8– 13 show the variations of frictional force with the averaged flow rate under the influence of all parametersb, l1 , a, Re, Fr and f. The relation between

5

φ=0.2 φ=0.3 φ=0.4 φ=0.5

60

1653

F 0.2

0.4

0.6

0.8

1 Q

–5

40 Q

20

–10

β = 1/5 β = 2/5 β = 3/5 β = 4/5

–15 0.2

0.4

0.6

0.8

1

Figure 7. Pressure versus averaged flow rate for various values of f at t ¼ 0:5, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1.

Figure 8. Frictional force versus averaged flow rate for various values of b at t ¼ 0:5, f ¼ 0:4, a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1.

D. Tripathi and O. Anwar Be´g

1654

5

10 F 0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

–5

–10

Q

Q –10

λ1=1 λ1=2 λ1=3 λ1=4

–20 –30

Re=0.1 Re=0.3 Re=0.5 Re=0.7

–15 –20

–40

Figure 9. Frictional force versus averaged flow rate for various values of l1 at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1 and Fr ¼ 0:1.

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0.2

F

frictional force and averaged flow rate is found to be linear and grows from the zero flow rate case to the maximum flow rate case. Frictional force effectively increases in magnitude with decrease in averaged flow rate. Hence with increasing flow rates, the frictional force is generally significantly decreased. With regard to the inclination effect, Figure 10, in which t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, Re ¼ 0:1, Fr ¼ 0:01 and l1 ¼ 1, clearly demonstrates that with increasing tube inclination from a ¼ p/5 through a ¼ p/4, p/3 to the vertical tube scenario of a ¼ p/2, friction force is weakly decreased. Stronger friction in the peristaltic wave motion therefore is associated with weaker tube inclinations. Comparison of these figures with Figures 2 – 7 clearly establishes that material and geometric parameters consistently exert the opposite effect on the frictional forces relative to the influence on the pressure profiles. Figure 14(a) – (e) depicts visualizations of the streamline patterns and trapping for a prescribed averaged flow  ¼ 0:6Þ with incremental increase in wave rate ðQ amplitude ðfÞ values. In these plots, the ordinate is the transformed radial coordinate (R) and the abscissa is transformed longitudinal coordinate (X), as defined in Equation (13). The streamlines on the centre line in the wave frame of reference are observed to divide and encapsulate the viscoelastic bolus of fluid particles

Figure 11. Frictional force versus averaged flow rate for various values of Re at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Fr ¼ 0:1 and l1 ¼ 1.

5 F 0.2

0.4

0.6

0.8

1

–5

Q

–10 Fr=0.01 Fr=0.02 Fr=0.05 Fr=0.1

–15 –20

Figure 12. Frictional force versus averaged flow rate for various values of Fr at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, a ¼ p=3, Re ¼ 0:1 and l1 ¼ 1.

circulating along closed streamlines under specific conditions, a characteristic termed trapping. This bolus is evidently trapped by the propulsive peristaltic wave, and transported with the same speed as that of the wave. As we progress from Figure 14(a) –(e), the averaged flow rate is invariant and the wave amplitude increases from 0.5, through 0.6, 0.7, 0.8 to 0.9. The clearly results in a progressive and sustained enlargement in bolus size up to (f ¼ 0.8). However, a dramatically different response is computed in Figure 14(e), where for the 10

0.2 F

0.4

0.6

0.8

1

F

–5

0.2

0.4

0.6

0.8

1

Q –10 –15 –20

Q

–10 α = π/2 α = π/3 α = π/4 α = π/5

–25

Figure 10. Frictional force versus averaged flow rate for various values of a at t ¼ 0:5, f ¼ 0:4, b ¼ 1=5, Re ¼ 0:1, Fr ¼ 0:01 and l1 ¼ 1.

–20

φ=0.2 φ=0.3 φ=0.4 φ=0.5

Figure 13. Frictional force versus averaged flow rate for various values of f at t ¼ 0:5, b ¼ 1=5,a ¼ p=3, Re ¼ 0:1, Fr ¼ 0:1 and l1 ¼ 1.

Computer Methods in Biomechanics and Biomedical Engineering (a) 1.5

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(b) 1.5

1

R

R

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5

–1.5 –1

–0.5

0

0.5

1

–1

–0.5

0

0.5 X

X (d) 1.5

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(c) 1.5

R

1

1

R

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5

–1.5 –1

–0.5

0

0.5

1

–1

–0.5

0

0.5 X

X (e)

1

1.5 1

R

0.5 0 –0.5 –1 –1.5 –1

–0.5

0

0.5

1 X

 ¼ 0:6, f ¼ 0:5; (b) Q  ¼ 0:6, f ¼ 0:6; (c) Q  ¼ 0:6, f ¼ 0:7; (d) Q  ¼ 0:6, f ¼ 0:8; Figure 14. Streamlines in the wave frame at (a) Q  ¼ 0:6, f ¼ 0:9. (e) Q

maximum wave amplitude scenario, a secondary bolus is observed to appear and more strongly sinusoidal streamlines are generated along the horizontal symmetry axis. Evidently therefore, the transport phenomena in

bolus generation are strongly influenced by wave amplitude, and this has also been observed in particular for inclined tube models of embryological flows by Fauci and Dillon (2006).

D. Tripathi and O. Anwar Be´g

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5. Conclusions In the present study, a mathematical model for the peristaltic flow of a viscoelastic physiological liquid, utilizing a simplification of the fractional second-grade constitutive model, in an inclined wavy (sinusoidal) tube has been developed, under the assumptions of long wave length and low Reynolds number. Closed-form solutions to the normalized momentum differential equations have been obtained. Numerical Mathematica-based computations have revealed some interesting features of the flow with regard to the effects of the emerging geometric/ hydromechanical parameters, i.e. fractional parameter, material constant, inclination angle, Reynolds number, Froude number and wave amplitude. These computations have been found to concur with earlier studies using alternative rheological formulations and have shown the following: . Pressure decreases by increasing the magnitude of

fractional parameter in the interval ð0 , b , 1Þ.

. With an increase in the value of material constant,

pressure increases. . In the interval ðp=5 # a # p=2Þ, i.e. from small . . . .

. .

tube inclination to the vertical tube case, pressure increases with increasing the inclination of the tube. The pressure increases with an increase in Reynolds number. The pressure is suppressed with increasing Froude number. Pressure is accentuated with increasing peristaltic wave amplitude. Maximum averaged flow rate is the same for various values of fractional parameter and material constant, whereas it exhibits significant variation for various values of inclination angle, Reynolds number, Froude number and amplitude. The effect of each parameter on the frictional force is observed to be the opposite to that for pressure. For a fixed average flow rate, with increasing wave amplitude, the bolus size is observed to grow considerably with a secondary bolus emerging at the highest wave amplitude.

The current study is presently being extended to consider the effects of transverse magnetic field and magnetic induction (Rashidi et al. 2011; Anwar Be´g et al. 2012) on electrically conducting peristaltic rheological transport, a topic of some importance in magnetically assisted gastric motility control. Furthermore, a more generalized fractional viscoelastic model is being considered as an extension to the present study and the results of such studies will be communicated imminently. The present solutions also provide a good benchmark for more general investigations, wherein other researchers may be able to compare the validity of their numerical

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