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Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope
Physica A 387 (2008) 2403–2415 www.elsevier.com/locate/physa
Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope Kh.S. Mekheimer a,∗ , Y. Abd elmaboud b a Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, 11884 Cairo, Egypt b Mathematics Department, Faculty of Science, Al-Azhar University (Assiut Branch), Assiut, Egypt
Received 20 April 2007; received in revised form 21 November 2007 Available online 31 December 2007
Abstract This paper discusses the influence of an endoscope on the peristaltic flow of a couple stress fluid in an annulus under a zero Reynolds number and long wavelength approximation. The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. Analytical expressions for the axial velocity, stream function and axial pressure gradient are established. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Numerical calculations are carried out for the pressure rise, frictional forces and trapping. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail. c 2007 Elsevier B.V. All rights reserved.
Keywords: Peristaltic flow; Endoscope; Couple stress fluid; Pressure rise
1. Introduction The non-Newtonian fluids have received great attention during the recent years. The flow of non-Newtonian fluids is widely observed in industry and biology, e.g. enhanced oil recovery, chemical processes such as in distillation towers and fixed-bed reactors and, in the applications of pumping fluids such as synthetic lubricants, colloidal fluids, liquid crystals, and biofluids (e.g. animal and human blood). The couple stress fluid is a special case of non-Newtonian fluid which is intended to take into account the particle size effects. The micro-continuum theory of couple stress fluid proposed by Stokes [6], defines the rotational field in terms of the velocity field for setting up the constitutive relationship between the stress and the strain rate. Stokes’ micro-continuum theory is the simplest generalization of the classical theory of fluids, which allows for polar effects such as the presence of couple stresses, body couples and a non-symmetric stress tensor. The couple stress model plays an important role in understanding some of the non-Newtonian flow properties of blood. A number of studies for couple stress and non-Newtonian fluids have been reported [1–7]. Peristaltic pumping is a form of fluid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of distensible duct. Peristalsis is an inherent property of many biological systems having ∗ Corresponding author.
E-mail addresses: kh [email protected] (Kh.S. Mekheimer), yass [email protected] (Y. Abd elmaboud). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.12.017
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Fig. 1. Geometry of the problem.
smooth muscle tubes which transports biofluids by its propulsive movements and is found in the transport of urine from kidney to the bladder, the movement of chyme in the gastro-intestinal tract, intra-uterine fluid motion, vasomotion of the small blood vessels and in many other glandular ducts. The mechanism of peristaltic transport has been exploited for industrial applications like sanitary fluid transport, blood pumps in heart lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited. A vast amount of literature is available on the peristalsis involving viscous fluid under one or more assumption(s) of long wavelength, small Reynolds number, small amplitude ratio, small wave number etc. [8–28]. Most of the theoretical investigations just mentioned have been carried out by assuming that blood and most of the physiological fluids behave like non-Newtonian fluids, this approach provides a satisfactory understanding when the peristaltic mechanism is involved in small blood vessels, lymphatic vessels, intestine, ductus efferentes of the male reproductive tract and in transport of spermatozoa in the cervical canal [21,37,38]. The effect of an endoscope [29–32] on peristaltic motion of a fluid is very important for medical diagnosis and it has many clinical applications. Now the endoscope is a very important tool used for determining real reasons responsible for many problems in the human organs in which the fluid are transported by peristaltic pumping such as, stomach, small intestine, etc. Also from fluid dynamic point of view, there is no difference between an endoscope and a catheter. Furthermore, the insertion of a catheter in an artery will alter the flow field and modify the pressure distribution [33]. With the above discussion in mind, we propose to study the fluid mechanics effects of peristaltic transport in a gap between two coaxial tubes, filled with an incompressible couple stress fluid (as a blood model), the inner tube is rigid and the outer one has wave trains moving independently. The flow analysis is developed in a wave frame of reference moving with the velocity of the wave. The problem is solved analytically for the velocity, stream function, axial pressure gradient and frictional forces. In the present model we have accounted micro-continuum fluids proposed by Stokes [6]. These fluids have been referred as couple stress fluids wherein the parameters α, η accounts for the size effects in the flow field. Higher value of α implies that the flow is tending towards Newtonian whereas, the lower value of α implies that the flow has a dominance of particle size effects. η is the parameter which accounts for the effect of local viscosity due to particles in addition to bulk viscosity of the fluid (µ). A motivation of the present analysis is the hope that such a problem will be applicable in many clinical applications such as the endoscope problem. 2. Formulation of the problem Consider the axisymmetric flow of an incompressible viscous couple stress fluid through the gap between coaxial tubes. The inner tube is rigid and the outer has a sinusoidal wave traveling down its walls. The geometry of the wall surface is described in (Fig. 1), the equations for the radii are r10 = a1 , r20 = a2 + b cos
(1) 2π 0 (Z − ct 0 ), λ
(2)
where a1 , a2 are the radii of the inner and outer tubes, b is the amplitude of the wave, λ is the wavelength, c is the propagation velocity and t 0 is the time.
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For a couple stress fluid the constitutive equations and equation of motion in the absence of body force and body couple are [6] T ji, j = ρ
∂vi , ∂t
(3)
ei jk T jkA + M ji, j = 0,
(4)
τi j = − p 0 δi j + 2µdi j ,
(5)
µi j = 4ηw j,i + η wi, j ,
(6)
0
where vi is the velocity vector, τi j and Ti Aj are the symmetric and antisymmetric parts of the stress tensor Ti j respectively, Mi j is the couple stress tensor, µi j is the deviatoric part of Mi j , wi is the vorticity vector, di j is the symmetric part of the velocity gradient, η and η0 are the constants associated with the couple stress, p 0 is the pressure, and the other terms have their usual meaning from tensor analysis. Introducing a wave frame (r 0 , z 0 ) moving with velocity c away from the fixed frame (R 0 , Z 0 ) by the transformation z 0 = Z 0 − ct, (u 0 , v 0 )
r 0 = R,
v 0 = V 0 − c,
u0 = U 0,
(7)
(U 0 , V 0 )
where and are the velocity components in the moving and fixed frame respectively. After using these transformation, the equations of motion are ∂u 0 ∂v 0 u0 + + = 0, ∂r 0 ∂z 0 r0 ∂ ∂ ∂ ∂ ∂u 0 ∂ p0 1 ∂2 1 ∂2 ∂u 0 µ − η u0, + + + + ρ u 0 0 + v0 0 = − 0 + ∂r ∂z ∂r ∂r 0 ∂r 0 r 0 ∂r 0 ∂r 0 r 0 ∂z 0 2 ∂z 0 2 0 0 1 ∂ ∂ p0 ∂2 1 ∂ ∂2 0 ∂v 0 ∂ 0 ∂ 0 ∂v µ−η 0 0 r 0 + v0, ρ u 0 +v 0 =− 0 + 0 0 r 0 + ∂r ∂z ∂z r ∂r ∂r r ∂r ∂r ∂z 0 2 ∂z 0 2
(8) (9) (10)
where u 0 and v 0 are the velocity components in the r 0 and z 0 directions, respectively, ρ is the density, p 0 is the pressure and µ is the viscosity. We introduce the following non-dimensional variables a2 r0 z0 v0 λu 0 b r0 , z= , v= , u= , p = 2 p0 , r1 = 1 = , φ= a2 λ c a2 c λµc a2 a2 r r20 ρca2 µ a2 a1 r2 = α = αa2 = = 1 + φ cos(2π z), Re = , δ= , = , a2 , a2 µ λ a2 η r=
(11)
where φ is the amplitude ratio, Re is the Reynolds number, δ is the dimensionless wave number and α√is the couple stress fluid parameter indicating the ratio of the tube radius (constant) to material characteristic length ( η/µ, has the dimension of length). To proceed, we non-dimensionalize Eqs. (8)–(10), this yields 1 ∂(r u) ∂v + = 0, r ∂r ∂z ∂u ∂u ∂p ∂ ∂ 1 ∂2 1 ∂ ∂ 1 ∂2 Reδ 3 u +v =− + δ2 + + δ2 2 1 − 2 + + δ 2 2 u, ∂r ∂z ∂r ∂r ∂r r r ∂z ∂z α ∂r ∂r ∂v ∂v ∂p 1 ∂ ∂ ∂2 ∂ ∂2 1 1 ∂ Reδ u +v =− + r + δ2 2 1 − 2 r + δ 2 2 v. ∂r ∂z ∂z r ∂r ∂r ∂r ∂z ∂z α r ∂r
(12) (13) (14)
Using the long wavelength approximation and dropping terms of order δ and higher, it follows from Eqs. (12)–(14) that the appropriate equations describing the flow in the wave frame are 1 ∂(r u) ∂v + = 0, r ∂r ∂z
(15)
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∂p = 0, ∂r 1 2 ∂p 2 = ∇ 1 − 2 ∇ v, ∂z α
(16) (17)
where ∇2 =
1 ∂ r ∂r
∂ r . ∂r
Assuming the components of the couple stress tensor at the wall to be zero [7], we have the following dimensionless boundary conditions: v = −1, v = −1,
∂ 2v η ∂v − =0 r ∂r ∂r 2 η ∂v ∂ 2v =0 − 2 r ∂r ∂r
at r = r1 = , (18) at r = r2 ,
where η = η0 /η is a couple stress fluid parameter (η and η0 are the constants associated with the couple stress, when η → 1 (i.e. η0 → η) no couple stress effect; Newtonian fluid [6–8]). Eq. (16) shows that p is not a function of r then the expression for the velocity profile of the fluid, obtained as the solution of Eq. (17) subject to the boundary conditions (18) is given as v = −1 +
i 1 dp h 2 r + a36 I0 (αr ) − a35 K 0 (αr ) + a38 log r − a39 , 4 dz
(19)
where I0 , K 0 are the modified Bessel functions of the first and second kind respectively of order zero. 1 ∂ψ The corresponding stream function (u = − r1 ∂ψ ∂z and v = r ∂r ) is 1 1 dp ψ = − (r 2 − r12 ) + {α(r 2 − r12 )(r 2 + r12 − a38 − 2a39 ) + 2[2a36 (r I1 (αr ) − r1 I1 (αr1 )) 2 16α dz + 2a35 (r K 1 (αr ) − r1 K 1 (αr1 )) + a38 α(r log r − r1 log r1 )]}, where I1 and K 1 are the modified Bessel functions of first order, first and second kind, respectively. The instantaneous volume flow rate Q(z) is given by Z r2 i 1 dp h 2 Q(z) = 2 r wdr = −(r22 − r12 ) + α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44 , 8α dz r1
(20)
(21)
from Eq. (21) we get dp 8α 2 2 = 2 × Q(z) + (r − r ) , 2 1 dz α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44
(22)
where the constants in Eqs. (19)–(22) are stated in the Appendix. Following the analysis given by Shapiro et al. [12], the mean volume flow, Q over a period is obtained as φ2 − 2, Q = Q(z) + 1 + 2
(23)
which on using Eq. (22) yields dp 8α φ2 2 2 2 = 2 × Q + − 1 + + (r − r ) . 2 1 dz 2 α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44
(24)
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Fig. 2. The variation of 1p with Q for different values of α at φ = 0.4, η = 0.5, = 0.3.
The pressure rise 1p and the friction force (at the wall) on the outer and inner tubes are F (o) and F (i) respectively, in a tube of length L, in their non-dimensional forms, are given by Z 1 dp dz, (25) 1p = 0 dz Z 1 dp (o) 2 F = dz, (26) r2 − dz 0 Z 1 dp F (i) = r12 − dz. (27) dz 0 Substituting from Eq. (24) in Eqs. (25)–(27) with r1 = , r2 = 1 + φ cos(2π z), we get the pressure rise and the friction force (at the wall) on the outer and inner tubes. In the absence of the inner tube (i.e., r1 → 0), the results agree with the results obtained by Srivastava [8]. In the limit r1 → 0, η → 1 (i.e. η0 → η, no couple stress effect) and α > 0, Eqs. (25) and (26) give the corresponding expressions for a Newtonian fluid as −8 3 2 φ2 2 1p = Q 1+ φ + (φ − 16) , (28) 2 4 (1 − φ 2 )7/2 8 φ2 (o) 2 3/2 F = Q−1− + (1 − φ ) . (29) 2 (1 − φ 2 )3/2 The results obtained in Eqs. (28) and (29) are the same as those obtained by Shapiro et al. [12]. 3. Discussion of results The results of our analysis are presented as follows: 1. Pumping characteristics; 2. The streamlines and trapping regions for the parameters η, α, .
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Fig. 3. The variation of 1p with Q for different values of φ, at α = 10, η = 0.6.
Fig. 4. The variation of 1p with η for different values of α at Q = 0.6, φ = 0.4, = 0.32.
3.1. Pumping characteristics In order to have an estimate of the quantitative effects of the various parameters involved in the results of the present analysis we are using the MATHEMATICA programme. The numerical evaluations of the analytical results obtained for 1p, F (o) , F (i) , for different parameters values [29–36]: = 0.1 up to 0.5, L = λ = 8.01 cm, α > 0 and η = 0.0 up to 1.0. In Fig. √ 2 the variation of 1p versus Q is shown for different values of α (inverse effect of the material characteristic length η/µ, where the tube radius is constant) by fixing the other parameters = 0.3 (endoscope problem), η = 0.5 and φ = 0.4. As expected there is a linear relation between the pressure rise and the flow rate (as for a Newtonian fluid), also an increase in the flow rate reduces the pressure rise and thus maximum flow rate is achieved at zero pressure rise and maximum pressure occurs at zero flow rate. The length of interval of Q when 1p > 0 decreases
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Fig. 5. The variation of F (i) with Q for different values of α at φ = 0.4, η = 0.5, = 0.32.
Fig. 6. The variation of F (i) with Q for different values of φ, at α = 10, η = 0.6.
when we move from the couple stress fluid to Newtonian fluid. The pressure rise decreases as the couple stress fluid parameter α increases, and so the magnitude of the pressure rise under a given set of conditions is greater for the couple stress fluid than that for a Newtonian fluid. Fig. 3, depicts the variation of 1p with Q for different values of amplitude ratio φ and (the radius of the endoscope where the radius of the outer tube is constant) at η = 0.6, α = 10. It is evident that the pressure rise increases when φ and are increased. Figs. 2 and 3 are sectored so that the upper right-hand quadrant denotes the region of peristaltic pumping where Q > 0 (positive pumping) and 1p > 0 (adverse pressure gradient). The lower right-hand quadrant denotes the region of augmented pumping where Q > 0 (positive pumping) and 1p < 0 (favorable pressure gradient) and the upper left-hand quadrant denotes the region of retrograde pumping (or backward pumping) where Q < 0 and 1p > 0 (adverse pressure gradient) where, in this region the flow is opposite to the direction of the peristaltic motion. It is
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Fig. 7. The variation of F (i) with η for different values of α at Q = 0.5, φ = 0.4, = 0.32.
Fig. 8. The variation of F (o) with Q for different values of α at φ = 0.4, η = 0.5, = 0.32.
clear that the peristaltic pumping region becomes wider when the radius ratio and the amplitude ratio φ (the ratio between wave amplitude and the radius of the outer tube) are increased and narrower when the couple stress fluid parameter α is increased. The variation of 1p with the couple stress fluid parameter η for different values of α at = 0.32, φ = 0.4 and Q = 0.6 is presented in Fig. 4. It is observed that the relation between 1p and η is a non-linear relation, 1p increases with increasing η until η = 0.4 then 1p decreases in the range 0.4 < η ≤ 1 (Newtonian fluid) regardless of the values of α. Also it is shown that the pressure rise decreases when the couple stress fluid parameter α is increased. Figs. 5 and 6 describe the results obtained for the inner friction F (i) versus the flow rate Q and Fig. 7 depicts the variation of inner friction F (i) versus the couple stress fluid parameter η. Figs. 8 and 9 describe the results for the outer friction force F (o) versus the flow rate Q and Fig. 10 depicts the variation of outer friction F (o) versus the couple stress fluid parameter η.
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Fig. 9. The variation of F (o) with Q for different values of φ, at α = 10, η = 0.6.
Fig. 10. The variation of F (o) with η for different values of α at Q = 0.5, φ = 0.4, = 0.32.
Furthermore, the effect of important parameters as Q, η, α, and φ on the inner and outer friction force have been investigated. We notice from Figs. 5–10 that the inner and outer friction force have the opposite behavior compared to the pressure rise. It is clear that from Figs. 5 and 8 the inner and outer friction force increase as the couple stress fluid parameter α increase and they are decreased with increasing φ and as shown in Figs. 6 and 9. The variation of F (i) and F (o) with the couple stress fluid parameter η for different values of α at = 0.32, φ = 0.4 and Q = 0.5 is presented in Figs. 7 and 10 respectively. It is observed that the relation between the inner and outer friction force with η, is a non-linear relation. The inner and the outer friction force decrease by increasing η until η = 0.4 then the inner and the outer friction force increase in the range 0.4 < η ≤ 1 (Newtonian fluid) regardless of the values of α. The inner friction force behaves similar to the outer friction force for the same values of the parameters, moreover the outer friction force is greater than the inner friction force at the same values of the parameters.
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Fig. 11. Graph of the streamlines for three different values of η: (a) η = 0, (b) η = 0.5, (c) η = 1 at α = 0.5, Q = 0.4, = 0.3 and φ = 0.4, r ∈ [, r2 ].
3.2. Streamlines and fluid trapping The phenomenon of trapping, whereby a bolus (defined as a volume of fluid bounded by closed streamlines in the wave frame) is transported at the wave speed, and this trapped bolus is pushed ahead along with the peristaltic wave. The effects of η, α, and on trapping can be seen through Figs. 11–13. Fig. 11, illustrates the streamline graphs for different values of the couple stress fluid parameter η (the parameter which accounts for the effect of local viscosity due to particles in addition to bulk viscosity of the fluid (µ)) with a given fixed set of the other parameters. It is observed that the trapped bolus decrease in size when the couple stress fluid parameter η is increased and the trapped bolus disappear completely when η = 1 (i.e. η0 → η, no couple stress effect; Newtonian fluid). The effect of α on the trapping is illustrated in Fig. 12. It is evident that the trapped bolus appear, but the important observation is that when α is increased the trapped bolus disappear and the others streamline to create another smaller trapped bolus. Fig. 13, depicts the variation of on the trapping phenomena. It is observed that the bolus move towards the upper wall with elevating the values of and more trapped bolus appear. Also for more values of a new region of trapped bolus appears. 4. Concluding remarks We have presented a theoretical approach to study the annulus peristaltic flow of a couple stress fluid. The present analysis can serve as a model which may help in understanding the mechanism of physiological flows in an annulus
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Fig. 12. Graph of the streamlines for three different values of α: (a) α = 1, (b) α = 5, (c) α = 7 at η = 0.8, Q = 0.4, = 0.3 and φ = 0.4, r ∈ [, r2 ].
for fluids behaving like a couple stress fluid. From the point of view of mechanics, it is interesting to note how the peristaltic motion is influenced by the applied pressure gradient. Here we have analyzed the peristaltic flow through a gap between two coaxial tubes, the inner tube is rigid and the outer one has wave trains moving independently, the gap between them is filled with an incompressible viscous couple stress fluid (as a blood model). A long wavelength approximation is adopted. The exact expressions for axial velocity of the fluid, stream function and the axial pressure gradient are obtained analytically. Numerical integrations are used to analyze the novel features of pumping and trapping. The main findings can be summarized as follows: √ • The pressure rise decreases when the material characteristic length η/µ is decreased, also the magnitude of the pressure rise is greater for the couple stress fluid than that for a Newtonian fluid. • By elevating the values of the amplitude ratio more occlusion will be produced in the gap between the inner and outer tube and this leads to an increase in the pressure rise. • An increase in the radius of the inner tube will increase the pressure rise. • The inner and outer friction forces have an opposite behavior compared to the pressure rise. • The trapped bolus decrease in size as we move from couple stress fluid to a Newtonian fluid, and such trapped bolus disappear completely when η = 1 (i.e. η0 → η, Newtonian fluid). • A new region of the trapped bolus appear when the radius of the endoscope is increased. • In the absence of the inner tube (i.e., r1 → 0, without endoscope), and the limit η → 1 (i.e. η0 → η, no couple stress effect) where α > 0, the results coincide with the results given by Shapiro et al. [12].
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Fig. 13. Graph of the streamlines for three different values of : (a) = 0.1, (b) = 0.3, (c) = 0.5 at η = 0.6, Q = 0.6, α = 7 and φ = 0.4, r ∈ [, r2 ].
Appendix −(1 + η) −(1 + η) , a13 = , a14 = I0 (αr1 ) + I2 (αr1 ), a15 = I0 (αr2 ) − I0 (αr1 ), 2 r2 r12 = I0 (αr2 ) + I2 (αr2 ), a17 = K 0 (αr2 ) − K 0 (αr1 ), a18 = −(K 0 (αr1 ) + K 2 (αr1 )), 1 α ηI1 (αr1 ) 1 α ηI1 (αr2 ) = −(K 0 (αr2 ) + K 2 (αr2 )), a21 = α 2 a14 − , a22 = α 2 a16 − , 2 r1 2 r2 α ηK 1 (αr1 ) 1 2 α ηK 1 (αr2 ) 1 2 = − α a18 , a24 = − α a19 , a25 = a21 a11 − a13 a15 , r1 2 r2 2 (1 − η)a11 = a22 a11 − a12 a15 , a27 = a23 a11 − a13 a17 , a28 = a24 a11 − a12 a17 , a31 = , 2 (r 2 − r12 )a12 (r 2 − r12 )a13 = 2 , a33 = 2 , a34 = a26 a27 + a25 a28 , 4 4 (a32 − a31 )a25 + (a31 − a33 a13 )a26 (a31 − a32 )a27 − (a31 − a33 )a28 = , a36 = , a34 a a − a26 a27 25 28 1 1 2 (a31 − a32 )a27 a25 − (a31 − a33 )a17 a26 2 , a38 = a37 − (r2 − r1 ) − a15 a36 , = a25 a28 − a26 a27 a11 4
a11 = log a16 a19 a23 a26 a32 a35 a37
r2 , r1
a12 =
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1 a39 = r12 + a38 log r1 − a35 K 0 (αr1 ) + a36 I0 (αr1 ), a41 = r2 I1 (αr2 ) − r1 I1 (αr1 ), 4 a42 = r2 K 1 (αr2 ) − r1 K 1 (αr1 ), a43 = r2 log r2 − r1 log r1 , a44 = 2[2a36 a41 + 2a35 a42 + αa38 a43 ]. References [1] T. Hayat, A.H. Kara, E. Momoniat, Exact flow of a third grade fluid on a porous wall, Internat. J. Non-Linear Mech. 38 (2003) 1533. [2] T. Hayat, A.H. Kara, Couette flow of a third grade fluid with variable magnetic field, Math. Comput. Modelling 43 (2006) 132. [3] T. Hayat, S.B. Khan, M. Khan, The influence of Hall current on the rotating oscillating flows of an Oldroyd-B fluid in a porous medium, Non-Linear Dynam. 47 (4) (2007). [4] C. Fetecau, C. Fetecau, On some axial Couette flows of non-Newtonian fluids, Z. Angew. Math. Phys. 56 (2005) 1098. [5] Wang-Long Li, An average flow model for couple stress fluids, Tribology Lett. 15 (2003) 279. [6] V.K. Stokes, Couple stress fluid, Phys. Fluids 9 (1966) 1709. [7] K.C. Valanis, C.T. 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Numer. Simul. 12 (2007) 910. [28] T. Hayat, Masood Khan, A.M. Siddiqui, S. Asghar, Non-linear peristaltic flow of a non-Newtonian fluid under effect of a magnetic field in a planar channel, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 910. [29] Kh.S. Mekheimer, Peristaltic transport of a Newtonian fluid through a uniform and non-uniform annulus, Arab. J. Sci. Eng. 30 (1A) (2005) 69. [30] Abd El Hakeem Abd El Naby, A.E.M. El Misery, I.I. El Shamy, Effects of an endoscope and fluid with variable viscosity on peristaltic motion, Appl. Math. Comput. 158 (2004) 497. [31] Abd El Hakeem Abd El Naby, A.E.M. El Misiery, Effects of an endoscope and generalized Newtonian fluid on peristaltic motion, Appl. Math. Comput. 128 (2002) 19. [32] T. Hayat, N. Ali, S. Asghar, A.M. Siddiqui, Exact peristaltic flow in tubes with an endoscope, Appl. Math. Comput. 182 (2006) 359. [33] D.S. Sankar, K. Hemalatha, A non-Newtonian fluid flow model for blood flow through a catheterized artery-Steady flow, Appl. Math. Modelling 31 (2007) 1847. [34] T. Hayat, E. Momoniat, F.M. Mahomed, Endoscope effects on MHD peristaltic flow of a power-law fluid, Math. Probl. Eng. 1 (2006) 1. [35] N. Ali, T. Hayat, M. Sajid, Peristaltic flow of a couple stress fluid in an asymmetric channel, Biorheology 44 (2007) 125. [36] T. Hayat, E. Momoniat, F.M. Mahomed, Endoscope effects on MHD peristaltic flow of a power-law fluid, Math. Probl. Eng. doi:10.1155/MPE/2006/84276. [37] L.M. Srivastava, V.P. Srivastava, Peristaltic transport of blood: Casson model-II, J. Biomech. 17 (1984) 821. [38] L.M. Srivastava, V.P. Srivastava, S.N. Sinha, Peristaltic transport of a physiological fluid, part I: Flow in non-uniform geometry, Biorheology 20 (1983) 153.
Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope Kh.S. Mekheimer a,∗ , Y. Abd elmaboud b a Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, 11884 Cairo, Egypt b Mathematics Department, Faculty of Science, Al-Azhar University (Assiut Branch), Assiut, Egypt
Received 20 April 2007; received in revised form 21 November 2007 Available online 31 December 2007
Abstract This paper discusses the influence of an endoscope on the peristaltic flow of a couple stress fluid in an annulus under a zero Reynolds number and long wavelength approximation. The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. Analytical expressions for the axial velocity, stream function and axial pressure gradient are established. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Numerical calculations are carried out for the pressure rise, frictional forces and trapping. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail. c 2007 Elsevier B.V. All rights reserved.
Keywords: Peristaltic flow; Endoscope; Couple stress fluid; Pressure rise
1. Introduction The non-Newtonian fluids have received great attention during the recent years. The flow of non-Newtonian fluids is widely observed in industry and biology, e.g. enhanced oil recovery, chemical processes such as in distillation towers and fixed-bed reactors and, in the applications of pumping fluids such as synthetic lubricants, colloidal fluids, liquid crystals, and biofluids (e.g. animal and human blood). The couple stress fluid is a special case of non-Newtonian fluid which is intended to take into account the particle size effects. The micro-continuum theory of couple stress fluid proposed by Stokes [6], defines the rotational field in terms of the velocity field for setting up the constitutive relationship between the stress and the strain rate. Stokes’ micro-continuum theory is the simplest generalization of the classical theory of fluids, which allows for polar effects such as the presence of couple stresses, body couples and a non-symmetric stress tensor. The couple stress model plays an important role in understanding some of the non-Newtonian flow properties of blood. A number of studies for couple stress and non-Newtonian fluids have been reported [1–7]. Peristaltic pumping is a form of fluid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of distensible duct. Peristalsis is an inherent property of many biological systems having ∗ Corresponding author.
E-mail addresses: kh [email protected] (Kh.S. Mekheimer), yass [email protected] (Y. Abd elmaboud). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.12.017
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Fig. 1. Geometry of the problem.
smooth muscle tubes which transports biofluids by its propulsive movements and is found in the transport of urine from kidney to the bladder, the movement of chyme in the gastro-intestinal tract, intra-uterine fluid motion, vasomotion of the small blood vessels and in many other glandular ducts. The mechanism of peristaltic transport has been exploited for industrial applications like sanitary fluid transport, blood pumps in heart lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited. A vast amount of literature is available on the peristalsis involving viscous fluid under one or more assumption(s) of long wavelength, small Reynolds number, small amplitude ratio, small wave number etc. [8–28]. Most of the theoretical investigations just mentioned have been carried out by assuming that blood and most of the physiological fluids behave like non-Newtonian fluids, this approach provides a satisfactory understanding when the peristaltic mechanism is involved in small blood vessels, lymphatic vessels, intestine, ductus efferentes of the male reproductive tract and in transport of spermatozoa in the cervical canal [21,37,38]. The effect of an endoscope [29–32] on peristaltic motion of a fluid is very important for medical diagnosis and it has many clinical applications. Now the endoscope is a very important tool used for determining real reasons responsible for many problems in the human organs in which the fluid are transported by peristaltic pumping such as, stomach, small intestine, etc. Also from fluid dynamic point of view, there is no difference between an endoscope and a catheter. Furthermore, the insertion of a catheter in an artery will alter the flow field and modify the pressure distribution [33]. With the above discussion in mind, we propose to study the fluid mechanics effects of peristaltic transport in a gap between two coaxial tubes, filled with an incompressible couple stress fluid (as a blood model), the inner tube is rigid and the outer one has wave trains moving independently. The flow analysis is developed in a wave frame of reference moving with the velocity of the wave. The problem is solved analytically for the velocity, stream function, axial pressure gradient and frictional forces. In the present model we have accounted micro-continuum fluids proposed by Stokes [6]. These fluids have been referred as couple stress fluids wherein the parameters α, η accounts for the size effects in the flow field. Higher value of α implies that the flow is tending towards Newtonian whereas, the lower value of α implies that the flow has a dominance of particle size effects. η is the parameter which accounts for the effect of local viscosity due to particles in addition to bulk viscosity of the fluid (µ). A motivation of the present analysis is the hope that such a problem will be applicable in many clinical applications such as the endoscope problem. 2. Formulation of the problem Consider the axisymmetric flow of an incompressible viscous couple stress fluid through the gap between coaxial tubes. The inner tube is rigid and the outer has a sinusoidal wave traveling down its walls. The geometry of the wall surface is described in (Fig. 1), the equations for the radii are r10 = a1 , r20 = a2 + b cos
(1) 2π 0 (Z − ct 0 ), λ
(2)
where a1 , a2 are the radii of the inner and outer tubes, b is the amplitude of the wave, λ is the wavelength, c is the propagation velocity and t 0 is the time.
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For a couple stress fluid the constitutive equations and equation of motion in the absence of body force and body couple are [6] T ji, j = ρ
∂vi , ∂t
(3)
ei jk T jkA + M ji, j = 0,
(4)
τi j = − p 0 δi j + 2µdi j ,
(5)
µi j = 4ηw j,i + η wi, j ,
(6)
0
where vi is the velocity vector, τi j and Ti Aj are the symmetric and antisymmetric parts of the stress tensor Ti j respectively, Mi j is the couple stress tensor, µi j is the deviatoric part of Mi j , wi is the vorticity vector, di j is the symmetric part of the velocity gradient, η and η0 are the constants associated with the couple stress, p 0 is the pressure, and the other terms have their usual meaning from tensor analysis. Introducing a wave frame (r 0 , z 0 ) moving with velocity c away from the fixed frame (R 0 , Z 0 ) by the transformation z 0 = Z 0 − ct, (u 0 , v 0 )
r 0 = R,
v 0 = V 0 − c,
u0 = U 0,
(7)
(U 0 , V 0 )
where and are the velocity components in the moving and fixed frame respectively. After using these transformation, the equations of motion are ∂u 0 ∂v 0 u0 + + = 0, ∂r 0 ∂z 0 r0 ∂ ∂ ∂ ∂ ∂u 0 ∂ p0 1 ∂2 1 ∂2 ∂u 0 µ − η u0, + + + + ρ u 0 0 + v0 0 = − 0 + ∂r ∂z ∂r ∂r 0 ∂r 0 r 0 ∂r 0 ∂r 0 r 0 ∂z 0 2 ∂z 0 2 0 0 1 ∂ ∂ p0 ∂2 1 ∂ ∂2 0 ∂v 0 ∂ 0 ∂ 0 ∂v µ−η 0 0 r 0 + v0, ρ u 0 +v 0 =− 0 + 0 0 r 0 + ∂r ∂z ∂z r ∂r ∂r r ∂r ∂r ∂z 0 2 ∂z 0 2
(8) (9) (10)
where u 0 and v 0 are the velocity components in the r 0 and z 0 directions, respectively, ρ is the density, p 0 is the pressure and µ is the viscosity. We introduce the following non-dimensional variables a2 r0 z0 v0 λu 0 b r0 , z= , v= , u= , p = 2 p0 , r1 = 1 = , φ= a2 λ c a2 c λµc a2 a2 r r20 ρca2 µ a2 a1 r2 = α = αa2 = = 1 + φ cos(2π z), Re = , δ= , = , a2 , a2 µ λ a2 η r=
(11)
where φ is the amplitude ratio, Re is the Reynolds number, δ is the dimensionless wave number and α√is the couple stress fluid parameter indicating the ratio of the tube radius (constant) to material characteristic length ( η/µ, has the dimension of length). To proceed, we non-dimensionalize Eqs. (8)–(10), this yields 1 ∂(r u) ∂v + = 0, r ∂r ∂z ∂u ∂u ∂p ∂ ∂ 1 ∂2 1 ∂ ∂ 1 ∂2 Reδ 3 u +v =− + δ2 + + δ2 2 1 − 2 + + δ 2 2 u, ∂r ∂z ∂r ∂r ∂r r r ∂z ∂z α ∂r ∂r ∂v ∂v ∂p 1 ∂ ∂ ∂2 ∂ ∂2 1 1 ∂ Reδ u +v =− + r + δ2 2 1 − 2 r + δ 2 2 v. ∂r ∂z ∂z r ∂r ∂r ∂r ∂z ∂z α r ∂r
(12) (13) (14)
Using the long wavelength approximation and dropping terms of order δ and higher, it follows from Eqs. (12)–(14) that the appropriate equations describing the flow in the wave frame are 1 ∂(r u) ∂v + = 0, r ∂r ∂z
(15)
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∂p = 0, ∂r 1 2 ∂p 2 = ∇ 1 − 2 ∇ v, ∂z α
(16) (17)
where ∇2 =
1 ∂ r ∂r
∂ r . ∂r
Assuming the components of the couple stress tensor at the wall to be zero [7], we have the following dimensionless boundary conditions: v = −1, v = −1,
∂ 2v η ∂v − =0 r ∂r ∂r 2 η ∂v ∂ 2v =0 − 2 r ∂r ∂r
at r = r1 = , (18) at r = r2 ,
where η = η0 /η is a couple stress fluid parameter (η and η0 are the constants associated with the couple stress, when η → 1 (i.e. η0 → η) no couple stress effect; Newtonian fluid [6–8]). Eq. (16) shows that p is not a function of r then the expression for the velocity profile of the fluid, obtained as the solution of Eq. (17) subject to the boundary conditions (18) is given as v = −1 +
i 1 dp h 2 r + a36 I0 (αr ) − a35 K 0 (αr ) + a38 log r − a39 , 4 dz
(19)
where I0 , K 0 are the modified Bessel functions of the first and second kind respectively of order zero. 1 ∂ψ The corresponding stream function (u = − r1 ∂ψ ∂z and v = r ∂r ) is 1 1 dp ψ = − (r 2 − r12 ) + {α(r 2 − r12 )(r 2 + r12 − a38 − 2a39 ) + 2[2a36 (r I1 (αr ) − r1 I1 (αr1 )) 2 16α dz + 2a35 (r K 1 (αr ) − r1 K 1 (αr1 )) + a38 α(r log r − r1 log r1 )]}, where I1 and K 1 are the modified Bessel functions of first order, first and second kind, respectively. The instantaneous volume flow rate Q(z) is given by Z r2 i 1 dp h 2 Q(z) = 2 r wdr = −(r22 − r12 ) + α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44 , 8α dz r1
(20)
(21)
from Eq. (21) we get dp 8α 2 2 = 2 × Q(z) + (r − r ) , 2 1 dz α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44
(22)
where the constants in Eqs. (19)–(22) are stated in the Appendix. Following the analysis given by Shapiro et al. [12], the mean volume flow, Q over a period is obtained as φ2 − 2, Q = Q(z) + 1 + 2
(23)
which on using Eq. (22) yields dp 8α φ2 2 2 2 = 2 × Q + − 1 + + (r − r ) . 2 1 dz 2 α(r2 − r12 )(r12 + r22 − a38 − 2a39 ) + a44
(24)
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Fig. 2. The variation of 1p with Q for different values of α at φ = 0.4, η = 0.5, = 0.3.
The pressure rise 1p and the friction force (at the wall) on the outer and inner tubes are F (o) and F (i) respectively, in a tube of length L, in their non-dimensional forms, are given by Z 1 dp dz, (25) 1p = 0 dz Z 1 dp (o) 2 F = dz, (26) r2 − dz 0 Z 1 dp F (i) = r12 − dz. (27) dz 0 Substituting from Eq. (24) in Eqs. (25)–(27) with r1 = , r2 = 1 + φ cos(2π z), we get the pressure rise and the friction force (at the wall) on the outer and inner tubes. In the absence of the inner tube (i.e., r1 → 0), the results agree with the results obtained by Srivastava [8]. In the limit r1 → 0, η → 1 (i.e. η0 → η, no couple stress effect) and α > 0, Eqs. (25) and (26) give the corresponding expressions for a Newtonian fluid as −8 3 2 φ2 2 1p = Q 1+ φ + (φ − 16) , (28) 2 4 (1 − φ 2 )7/2 8 φ2 (o) 2 3/2 F = Q−1− + (1 − φ ) . (29) 2 (1 − φ 2 )3/2 The results obtained in Eqs. (28) and (29) are the same as those obtained by Shapiro et al. [12]. 3. Discussion of results The results of our analysis are presented as follows: 1. Pumping characteristics; 2. The streamlines and trapping regions for the parameters η, α, .
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Fig. 3. The variation of 1p with Q for different values of φ, at α = 10, η = 0.6.
Fig. 4. The variation of 1p with η for different values of α at Q = 0.6, φ = 0.4, = 0.32.
3.1. Pumping characteristics In order to have an estimate of the quantitative effects of the various parameters involved in the results of the present analysis we are using the MATHEMATICA programme. The numerical evaluations of the analytical results obtained for 1p, F (o) , F (i) , for different parameters values [29–36]: = 0.1 up to 0.5, L = λ = 8.01 cm, α > 0 and η = 0.0 up to 1.0. In Fig. √ 2 the variation of 1p versus Q is shown for different values of α (inverse effect of the material characteristic length η/µ, where the tube radius is constant) by fixing the other parameters = 0.3 (endoscope problem), η = 0.5 and φ = 0.4. As expected there is a linear relation between the pressure rise and the flow rate (as for a Newtonian fluid), also an increase in the flow rate reduces the pressure rise and thus maximum flow rate is achieved at zero pressure rise and maximum pressure occurs at zero flow rate. The length of interval of Q when 1p > 0 decreases
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Fig. 5. The variation of F (i) with Q for different values of α at φ = 0.4, η = 0.5, = 0.32.
Fig. 6. The variation of F (i) with Q for different values of φ, at α = 10, η = 0.6.
when we move from the couple stress fluid to Newtonian fluid. The pressure rise decreases as the couple stress fluid parameter α increases, and so the magnitude of the pressure rise under a given set of conditions is greater for the couple stress fluid than that for a Newtonian fluid. Fig. 3, depicts the variation of 1p with Q for different values of amplitude ratio φ and (the radius of the endoscope where the radius of the outer tube is constant) at η = 0.6, α = 10. It is evident that the pressure rise increases when φ and are increased. Figs. 2 and 3 are sectored so that the upper right-hand quadrant denotes the region of peristaltic pumping where Q > 0 (positive pumping) and 1p > 0 (adverse pressure gradient). The lower right-hand quadrant denotes the region of augmented pumping where Q > 0 (positive pumping) and 1p < 0 (favorable pressure gradient) and the upper left-hand quadrant denotes the region of retrograde pumping (or backward pumping) where Q < 0 and 1p > 0 (adverse pressure gradient) where, in this region the flow is opposite to the direction of the peristaltic motion. It is
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Fig. 7. The variation of F (i) with η for different values of α at Q = 0.5, φ = 0.4, = 0.32.
Fig. 8. The variation of F (o) with Q for different values of α at φ = 0.4, η = 0.5, = 0.32.
clear that the peristaltic pumping region becomes wider when the radius ratio and the amplitude ratio φ (the ratio between wave amplitude and the radius of the outer tube) are increased and narrower when the couple stress fluid parameter α is increased. The variation of 1p with the couple stress fluid parameter η for different values of α at = 0.32, φ = 0.4 and Q = 0.6 is presented in Fig. 4. It is observed that the relation between 1p and η is a non-linear relation, 1p increases with increasing η until η = 0.4 then 1p decreases in the range 0.4 < η ≤ 1 (Newtonian fluid) regardless of the values of α. Also it is shown that the pressure rise decreases when the couple stress fluid parameter α is increased. Figs. 5 and 6 describe the results obtained for the inner friction F (i) versus the flow rate Q and Fig. 7 depicts the variation of inner friction F (i) versus the couple stress fluid parameter η. Figs. 8 and 9 describe the results for the outer friction force F (o) versus the flow rate Q and Fig. 10 depicts the variation of outer friction F (o) versus the couple stress fluid parameter η.
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Fig. 9. The variation of F (o) with Q for different values of φ, at α = 10, η = 0.6.
Fig. 10. The variation of F (o) with η for different values of α at Q = 0.5, φ = 0.4, = 0.32.
Furthermore, the effect of important parameters as Q, η, α, and φ on the inner and outer friction force have been investigated. We notice from Figs. 5–10 that the inner and outer friction force have the opposite behavior compared to the pressure rise. It is clear that from Figs. 5 and 8 the inner and outer friction force increase as the couple stress fluid parameter α increase and they are decreased with increasing φ and as shown in Figs. 6 and 9. The variation of F (i) and F (o) with the couple stress fluid parameter η for different values of α at = 0.32, φ = 0.4 and Q = 0.5 is presented in Figs. 7 and 10 respectively. It is observed that the relation between the inner and outer friction force with η, is a non-linear relation. The inner and the outer friction force decrease by increasing η until η = 0.4 then the inner and the outer friction force increase in the range 0.4 < η ≤ 1 (Newtonian fluid) regardless of the values of α. The inner friction force behaves similar to the outer friction force for the same values of the parameters, moreover the outer friction force is greater than the inner friction force at the same values of the parameters.
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Fig. 11. Graph of the streamlines for three different values of η: (a) η = 0, (b) η = 0.5, (c) η = 1 at α = 0.5, Q = 0.4, = 0.3 and φ = 0.4, r ∈ [, r2 ].
3.2. Streamlines and fluid trapping The phenomenon of trapping, whereby a bolus (defined as a volume of fluid bounded by closed streamlines in the wave frame) is transported at the wave speed, and this trapped bolus is pushed ahead along with the peristaltic wave. The effects of η, α, and on trapping can be seen through Figs. 11–13. Fig. 11, illustrates the streamline graphs for different values of the couple stress fluid parameter η (the parameter which accounts for the effect of local viscosity due to particles in addition to bulk viscosity of the fluid (µ)) with a given fixed set of the other parameters. It is observed that the trapped bolus decrease in size when the couple stress fluid parameter η is increased and the trapped bolus disappear completely when η = 1 (i.e. η0 → η, no couple stress effect; Newtonian fluid). The effect of α on the trapping is illustrated in Fig. 12. It is evident that the trapped bolus appear, but the important observation is that when α is increased the trapped bolus disappear and the others streamline to create another smaller trapped bolus. Fig. 13, depicts the variation of on the trapping phenomena. It is observed that the bolus move towards the upper wall with elevating the values of and more trapped bolus appear. Also for more values of a new region of trapped bolus appears. 4. Concluding remarks We have presented a theoretical approach to study the annulus peristaltic flow of a couple stress fluid. The present analysis can serve as a model which may help in understanding the mechanism of physiological flows in an annulus
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Fig. 12. Graph of the streamlines for three different values of α: (a) α = 1, (b) α = 5, (c) α = 7 at η = 0.8, Q = 0.4, = 0.3 and φ = 0.4, r ∈ [, r2 ].
for fluids behaving like a couple stress fluid. From the point of view of mechanics, it is interesting to note how the peristaltic motion is influenced by the applied pressure gradient. Here we have analyzed the peristaltic flow through a gap between two coaxial tubes, the inner tube is rigid and the outer one has wave trains moving independently, the gap between them is filled with an incompressible viscous couple stress fluid (as a blood model). A long wavelength approximation is adopted. The exact expressions for axial velocity of the fluid, stream function and the axial pressure gradient are obtained analytically. Numerical integrations are used to analyze the novel features of pumping and trapping. The main findings can be summarized as follows: √ • The pressure rise decreases when the material characteristic length η/µ is decreased, also the magnitude of the pressure rise is greater for the couple stress fluid than that for a Newtonian fluid. • By elevating the values of the amplitude ratio more occlusion will be produced in the gap between the inner and outer tube and this leads to an increase in the pressure rise. • An increase in the radius of the inner tube will increase the pressure rise. • The inner and outer friction forces have an opposite behavior compared to the pressure rise. • The trapped bolus decrease in size as we move from couple stress fluid to a Newtonian fluid, and such trapped bolus disappear completely when η = 1 (i.e. η0 → η, Newtonian fluid). • A new region of the trapped bolus appear when the radius of the endoscope is increased. • In the absence of the inner tube (i.e., r1 → 0, without endoscope), and the limit η → 1 (i.e. η0 → η, no couple stress effect) where α > 0, the results coincide with the results given by Shapiro et al. [12].
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Fig. 13. Graph of the streamlines for three different values of : (a) = 0.1, (b) = 0.3, (c) = 0.5 at η = 0.6, Q = 0.6, α = 7 and φ = 0.4, r ∈ [, r2 ].
Appendix −(1 + η) −(1 + η) , a13 = , a14 = I0 (αr1 ) + I2 (αr1 ), a15 = I0 (αr2 ) − I0 (αr1 ), 2 r2 r12 = I0 (αr2 ) + I2 (αr2 ), a17 = K 0 (αr2 ) − K 0 (αr1 ), a18 = −(K 0 (αr1 ) + K 2 (αr1 )), 1 α ηI1 (αr1 ) 1 α ηI1 (αr2 ) = −(K 0 (αr2 ) + K 2 (αr2 )), a21 = α 2 a14 − , a22 = α 2 a16 − , 2 r1 2 r2 α ηK 1 (αr1 ) 1 2 α ηK 1 (αr2 ) 1 2 = − α a18 , a24 = − α a19 , a25 = a21 a11 − a13 a15 , r1 2 r2 2 (1 − η)a11 = a22 a11 − a12 a15 , a27 = a23 a11 − a13 a17 , a28 = a24 a11 − a12 a17 , a31 = , 2 (r 2 − r12 )a12 (r 2 − r12 )a13 = 2 , a33 = 2 , a34 = a26 a27 + a25 a28 , 4 4 (a32 − a31 )a25 + (a31 − a33 a13 )a26 (a31 − a32 )a27 − (a31 − a33 )a28 = , a36 = , a34 a a − a26 a27 25 28 1 1 2 (a31 − a32 )a27 a25 − (a31 − a33 )a17 a26 2 , a38 = a37 − (r2 − r1 ) − a15 a36 , = a25 a28 − a26 a27 a11 4
a11 = log a16 a19 a23 a26 a32 a35 a37
r2 , r1
a12 =
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