The peristaltic flow for Carreau fluid through an elastic channel

Dheia G. Salih Al-Khafajy; Radhwan R. Majeed Mashhadi

doi:10.1515/jmbm-2022-0257

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The peristaltic flow for Carreau fluid through an elastic channel

Dheia G. Salih Al-Khafajy und Radhwan R. Majeed Mashhadi

Veröffentlicht/Copyright: 14. März 2023

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Aus der Zeitschrift Journal of the Mechanical Behavior of Materials Band 32 Heft 1

Abstract

The purpose of this study is to investigate the effect of an elastic wall on the peristaltic flow of Carreau fluid between two concentric cylinders, where the inner tube is cylindrical with an inelastic wall and the outer wall is a regular elastic sine wave. For this problem, cylindrical coordinates were used, and a short wavelength “relative to channel width for its length,” as well as the governing equations of Carreau fluid in the Navier–Stokes equations. Then, the analytical solution has been investigated by using the regular perturbation technique. The solutions obtained by this perturbation are up to the fourth-order in dimensionless Weissenberg number ( W e ). The performed computations of various parameter values such as velocity, shear stress, and wave frame streamlines are discussed in detail for different values of the Weissenberg number ( W e ). The obtained results demonstrate that the fluid velocity increases with the increase in the value of W e and some features of the wall, while the opposite behavior is observed with the increase in other features of the wall. Hence, the presented numerical analysis reveals many aspects of the flow by considering a non-Newtonian Carreau fluid model, and the presented model can be equally applicable to other bio-mathematical studies. The results were evaluated using the Mathematica software program. The Mathematica program was used by entering various data for the parameters, where the program showed the graphs, then the effect of these parameters became clear and the results were mentioned in the conclusion.

Keywords: Carreau fluid; peristaltic flow; wall properties; cylindrical coordinates

Nomenclature

a: sinusoidal wave
G: flexural rigidity
H: longitudinal tension per unit width
I ̅: identity tensor
I: coefficient of viscous damping
J L: spring stiffness
m: mass per unit area
n: dimensionless power law index
p ̅: pressure
Q 1: flexural stiffness of the wall
Q 2: longitudinal tension per unit width
Q 3: mass per unit area
Q 4: coefficient of viscid damping
Q 5: spring stiffness
R: perpendicular to the pipe’s axis
Re: Reynolds number
S ̅: extra stress tensor
V 1 ̅: axial velocity
V 3 ̅: transverse velocity
W e: Weissenberg number
Z: parallel to the channel’s radius
ρ: density
μ 0: zero shear viscosity
Γ: time constant
J L: spring stiffness
∅: amplitude ratio
δ: dimensionless wave number

1 Introduction

“Clutching and compressing” is what the Greek word peristalsis means (peristalikos). To put it in another way, peristaltic transport involves the continuous stretching and contracting of fluid-filled distensible tube-like waves [1]. Examples of its broad application include the nuclear industry and digestive system functions such as swallowing, chyme motion, capillary, and arteriole movement, in addition to sanitary fluid transportation and hazardous liquid transfer, the kidneys transfer urine to the bladder [2].

A mathematical model of Carreau fluid flow with sinusoidal wall variations in asymmetric channels has an analytic solution [3]. Many researchers in engineering and biology have dedicated their lives to understanding how peristaltic fluxes operate. Since this is the case, a large range of commonly utilized fluids exhibit non-Newtonian behavior. A variety of non-Newtonian models’ peristaltic flows and the relationships between them have been investigated by several researchers. Findings from numerous experiments on peristaltic flows in various flow geometries [4,5,6,7,8] were quite varied. Peristaltic flow in cylindrical tubes is easily explained in terms of circular geometry [9].

The fluid movement was examined endoscopically and the rubber wall in the human body, and it is clear that there is no solid wall in the human body, rather, all those walls are rubber. The manuscripts that have dealt with this field of study are rare, especially in cylindrical coordinates. This study resulted in the fact of the effect of the mentioned parameters such as the flexural stiffness of the wall, the longitudinal tension per unit width, the mass per unit area and the effect of other parameters on the rubber wall.

Carreau fluid peristaltic flow through an elastic conduit is the subject of this investigation. For some significant elements, such as velocity distribution, stress, wave frame streamlines, and graphic findings are supplied at the end of the article.

2 Mathematical formulations

A peristaltic flow of Carreau fluid between the eccentric cylinders is depicted in this illustration. The inner tube is rigid, and a sinusoidal wave propagates down its length as a result of the flow geometry design. Although the inner tube radius is same, we focus on the fluid flow in the outer tube’s circumference. Figure 1 shows that the cylindrical coordinates are obtained when R is perpendicular to the pipe’s axis and Z is parallel to the channel’s radius, as demonstrated.

r ̅ = r 1 ̅ = a 1 .

Figure 1

The problem geometry.

Outer wall,

r ̅ = r 2 ̅ ( z ̅ . t ̅ ) = a 2 + b sin 2 π l ( z ̅ − c t ̅ ) .

The basic governing equations of the continuity and the Navier–Stokes equations are

(1) ∂ V 1 ̅ ∂ R ̅ + V 1 ̅ R ̅ + ∂ V 3 ̅ ∂ Z ̅ = 0 ( continuity equation ) ,

(2) ρ ∂ V ̅ 1 ∂ t ̅ + V ̅ 1 ∂ V ̅ 1 ∂ R ̅ + V ̅ 3 ∂ V ̅ 1 ∂ Z ̅ = − ∂ p ̅ ∂ R ̅ + 1 R ̅ ∂ ∂ R ̅ ( R ̅ S ̅ R ̅ R ̅ ) + ∂ ∂ Z ̅ ( S ̅ R ̅ Z ̅ ) ρ ∂ V ̅ 3 ∂ t ̅ + V ̅ 1 ∂ V ̅ 3 ∂ R ̅ + V ̅ 3 ∂ V ̅ 3 ∂ Z ̅ = − ∂ p ̅ ∂ Z ̅ + 1 R ̅ ∂ ∂ R ̅ ( R ̅ S ̅ R ̅ Z ̅ ) + ∂ ∂ Z ̅ ( S ̅ Z ̅ Z ̅ ) ( momentum equation ) ,

where V 1 ̅ is the axial velocity, V 3 ̅ is the transverse velocity, ρ is the density, p ̅ is the pressure, and S ̅ is the extra stress tensor of the Carreau fluid, which is defined as follows:

(3) S ̅ = − μ 0 1 + n − 1 2 Γ 2 α ̇ ̅ 2 α ̇ ̅ ,

where P ̅ is the pressure, I ̅ is the identity tensor, μ 0 is the zero shear viscosity, Γ is the time constant n is the dimensionless power law index, and α ̇ is given by

α ̇ = 1 2 ∑ i = 1 3 ∑ j = 1 3 α ̇ ij α ̇ ji ,

In this work, a case is studied where Γ < 1 in the constitutive equation. A Newtonian n = 1 or Γ = 0 model must be used for the above model to work.

To understand how an elastic wall behaves, the equation L ° ° = P ̅ − p ̅ 0 can be used, where the membrane motion with viscous damping forces is represented by the operator L ° ° as follows:

L ° ° = G ∂ 4 ∂ Z ̅ 4 − H ∂ 2 ∂ Z ̅ 2 + m ∂ 2 ∂ t ̅ 2 + I ∂ ∂ t ̅ + J L ,

where G is the flexural rigidity of the wall, H is the longitudinal tension per unit width, m is the mass per unit area, I is the coefficient of viscous damping, and J L is the spring stiffness.

The governing equation for flexible wall canal properties at r ̅ = q ̅ using the momentum equation is obtained as follows:

(4) ∂ ∂ Z ̅ L ° ° ( q ̅ ) = ∂ P ̅ ∂ z ̅ = 1 R ̅ ∂ ∂ R ̅ ( R ̅ S ̅ R ̅ Z ̅ ) + ∂ ∂ Z ̅ ( S ̅ Z ̅ Z ̅ ) − ρ ∂ V 3 ̅ ∂ t ̅ + V 1 ̅ ∂ V 3 ̅ ∂ R ̅ + V 3 ̅ ∂ V 3 ̅ ∂ Z ̅ .

The velocity along the channel’s rims is V 3 ̅ ( r 1 ̅ ) = V 3 ̅ ( r 2 ̅ ) = 0 .

3 Solution method

As long as you keep ( R ̅ , Z ̅ ) constant, the flow between the two tubes is inherently unstable. It changes into a wave structure ( r ̅ , z ̅ ) that operates at the same speed as a z-direction z-wave. Given the two structures, the transformations between them are as follows:

r ̅ = R ̅ , z ̅ = Z ̅ − c t ̅ , v 1 ̅ = V 1 ̅ , v 3 ̅ = V 3 ̅ − c ,

where ( v 1 ̅ , v 3 ̅ ) and ( V 1 ̅ , V 3 ̅ ) are velocity elements of the moving and stationary structures. With these transformations, the equations of continuity and momentum, respectively, are as follows:

(5) ∂ v ̅ 1 ∂ r ̅ + v ̅ 1 r ̅ + ∂ ( v ̅ 3 + c ) ∂ ( z ̅ + c t ) ̅ = 0 ,

(6) ρ ∂ v 1 ̅ ∂ t ̅ + v 1 ̅ ∂ v 1 ̅ ∂ r ̅ + ( v 3 ̅ + c ) ∂ v 1 ̅ ∂ ( z ̅ + c t ̅ ) = − ∂ p ̅ ∂ r ̅ + 1 r ̅ ∂ ∂ r ̅ ( r ̅ S ̅ r ̅ r ̅ ) + ∂ ∂ ( z ̅ + c t ̅ ) ( S ̅ r ̅ z ̅ ) ,

(7) ρ ∂ ( v 3 ̅ + c ) ∂ t ̅ + v 1 ̅ ∂ ( v 3 ̅ + c ) ∂ r ̅ + ( v 3 ̅ + c ) ∂ ( v 3 ̅ + c ) ∂ ( z ̅ + c t ̅ ) = − ∂ p ̅ ∂ ( z ̅ + c t ̅ ) + 1 r ̅ ∂ ∂ r ̅ ( r ̅ S ̅ r ̅ z ̅ ) + ∂ ∂ z ̅ ( S ̅ z ̅ z ̅ ) .

The relative boundary conditions are v 3 ̅ ( r 1 ̅ ) = v 3 ̅ ( r 2 ̅ ) = − c .

The governing equation for flow channel wall elasticity (4) would be

(8) ∂ ∂ Z ̅ G ∂ 4 ∂ Z ̅ 4 − H ∂ 2 ∂ Z ̅ 2 + m ∂ 2 ∂ t ̅ 2 + I ∂ ∂ t ̅ + J L ( q ̅ ) = ∂ P ̅ ∂ z ̅ = − ρ ∂ ( v ̅ 3 + c ) ∂ t ̅ + v ̅ 1 ∂ ( v ̅ 3 + c ) ∂ r ̅ + ( v ̅ 3 + c ) ∂ ( v ̅ 3 + c ) ∂ z ̅ + 1 r ̅ ∂ ∂ r ̅ ( r ̅ S ̅ r ̅ z ̅ ) + ∂ ∂ z ̅ ( S ̅ z ̅ z ̅ ) .

To solve the equations controlling motion, we present non-dimensional equations to simplify these equations.

(9) v 1 = v 1 ̅ l a 2 c , v 3 = v 3 ̅ c , r = r ̅ a 2 , z = z ̅ l , t = c t ̅ l , τ = a 2 τ ̅ c μ 0 , δ = a 2 l , W e = Γ c a 2 , α ̇ = a 2 α ̇ ̅ c p = a 2 2 p ̅ c l μ 0 , Re = ρ c a 2 μ 0 , r 1 = r ̅ 1 a 2 = ε < 1 , ∅ = b a 2 , r 2 = r ̅ 2 a 2 = 1 + ∅ sin ( 2 π z ̅ ) ,

where ∅ is the amplitude ratio, Re is the Reynolds number, δ is the dimensionless wave number, and W e is the Weissenberg number.

Substituting Eq. (9) into Eqs. (5)–(8), we reformulate the governing equations and accompanying boundary conditions as follows:

(10) c l ∂ v 1 ∂ r + v 1 r + ∂ v 3 ∂ z = 0 ,

(11) Re δ 3 ∂ v 1 ∂ t + v 1 ∂ v 1 ∂ r + ( v 3 + 1 ) ∂ v 1 ∂ z = − ∂ p ∂ r + δ 1 r ∂ ∂ r ( r S rr ) + δ ∂ ∂ z ( S rz ) ,

(12) Re δ ∂ v 3 ∂ t + v 1 ∂ v 3 ∂ r + ( v 3 + 1 ) ∂ v 3 ∂ z = − ∂ p ∂ z + 1 r ∂ ∂ r ( r S rz ) + δ ∂ ∂ z S zz ,

(13) G a 2 3 μ 0 c l 5 ∂ 5 q ∂ z 5 − H a 2 3 μ 0 c l 3 ∂ 3 q ∂ z 3 + mc a 2 3 μ 0 l 3 ∂ 3 q ∂ z ∂ t 2 + I a 2 3 μ 0 l 2 ∂ 2 q ∂ z ∂ t + J L a 2 3 μ 0 c l ∂ q ∂ z = 1 r S rz + ∂ ∂ r S rz + δ ∂ ∂ z S zz − Re δ ∂ v 3 ∂ t + v 1 ∂ v 3 ∂ r + ( v 3 + 1 ) ∂ v 3 ∂ z .

It is very difficult to solve the problem in the latter form, and therefore we assume a very small wave-number ( δ ≪ 1 ), hence Eqs. (10)–(13) become

(14) ∂ v 1 ∂ r + v 1 r + ∂ v 3 ∂ z = 0 ,

(15) ∂ p ∂ r = 0 ,

(16) ∂ p ∂ z = 1 r S rz + ∂ ∂ r S rz ,

(17) Q 1 ∂ 5 q ∂ z 5 − Q 2 ∂ 3 q ∂ z 3 + Q 3 ∂ 3 q ∂ z ∂ t 2 + Q 4 ∂ 2 q ∂ z ∂ t + Q 5 ∂ q ∂ z = 1 r S rz + ∂ ∂ r S rz ,

where Q 1 = G a 2 3 μ 0 c l 5 is the flexural stiffness of the wall, Q 2 = H a 2 3 μ 0 c l 3 is the longitudinal tension per unit width, Q 3 = ms a 2 3 μ 0 l 3 is the mass per unit area, Q 4 = I a 2 3 μ 0 l 2 is the coefficient of viscid damping, and Q 5 = J L a 2 3 μ 0 g l is the spring stiffness.

Using the accepted assumption ( δ ≪ 1 ), the components of the additional shear stress will be dimensionless as follow:

(18) S rz = 1 + n − 1 2 W e 2 ∂ v 3 ∂ r 2 ∂ v 3 ∂ r .

Replacing S rz in Eq. (17), we have

(19) 1 r ∂ v 3 ∂ r + n − 1 2 W e 2 ∂ v 3 ∂ r 3 + r ∂ 2 v 3 ∂ r 2 + 3 r n − 1 2 W e 2 ∂ v 3 ∂ r 2 ∂ 2 v 3 ∂ r 2 = Q ,

with boundary conditions is v 3 ( ε ) = v 3 ( r 2 ) = − 1 , where Q = Q 1 ∂ 5 q ∂ z 5 − Q 2 ∂ 3 q ∂ z 3 + Q 3 ∂ 3 q ∂ z ∂ t 2 + Q 4 ∂ 2 q ∂ z ∂ t + Q 5 ∂ q ∂ z .

The corresponding stream function is v 1 = − 1 r ∂ ψ ∂ z and v 3 = 1 r ∂ ψ ∂ r .

4 Perturbation method solution

The momentum function, shear stress, and flow functions will be determined in this section. It is conceivable that the precise solution to Eq. (19) is impossible, thus we use the regular perturbation approach in terms of a “Weissenberg number” for a fourth-order to obtain the answer. It has been extended as follows to find a solution to the problem of perturbation:

(20) v 3 = v 30 + W e 2 v 32 + W e 4 v 34 + O ( W e 6 ) .

Eq. (20) is substituted in Eq. (19) with the boundary conditions, and the like powers of W e are then equated.

r ∂ 2 v 03 ∂ r 2 + ∂ v 03 ∂ r = rQ (Zero-order), with boundary conditions v 30 = − 1 at r = ε and r = 1 + ∅ · sin ( 2 π ( z − t ) ) .

r r ∂ 2 v 32 ∂ r 2 + ∂ v 32 ∂ r = − 3 n − 1 2 r ∂ v 30 ∂ r 2 ∂ 2 v 30 ∂ r 2 + n − 1 2 ∂ v 30 ∂ r 3 (Second-order),

with boundary conditions v 32 = 0 at r = ε and r = 1 + ∅ · sin ( 2 π ( z − t ) ) .

∂ 2 v 34 ∂ r 2 + ∂ v 34 ∂ r = − 3 ( n − 1 ) r ∂ 2 v 30 ∂ r 2 ∂ v 32 ∂ r ∂ v 30 ∂ r − 3 n − 1 2 r ∂ 2 v 32 ∂ r 2 ∂ v 30 ∂ r 2 − 3 n − 1 2 ∂ v 30 ∂ r 2 ∂ v 32 ∂ r

(Fourth-order),

with boundary conditions v 34 = 0 at r = ε and r = 1 + ∅ · sin ( 2 π ( z − t ) ) .

The formula for the solutions obtained (velocity, shear stress, and stream function) is too long. The accompanying constants can be determined using the accompanying boundary conditions. Thus, we will discuss these solutions through graphs in Section 5.

5 Graphical results and discussion

In this section, the results will be displayed that were received after utilizing the perturbation approach to solve the problem’s equations and then drawing the results with the Mathematica application. This section is divided into three sections: the first covers the impact of parameters on fluid movement through the flow channel, the second discusses the impact of parameters on shear stress, and the third analyzes the impact of parameters on fluid flow routes.

5.1 Velocity profile

Figures 2–6, illustrate the effect of parameters W e , ε , ∅ , Q 1 , Q 2 , Q 3 , Q 4 , n , t , and Q 5 on the distribution of velocity v 3 vs r, respectively. We notice in Figures 2 and 3 that the velocity is positively affected by the increase in parameters W e , ∅ , Q 1 , and Q 2 , respectively. Increasing the values of the parameters ε , Q 3 , n , and Q 4 causes the velocity distribution to narrow (Figures 4 and 5). The velocity distribution rises with the increase in parameter Q 5 and decreases with the increase in t , as seen in Figure 6.

$Figure 2 Distribution of velocity for different values of W e {W}_{e} and ∅ \varnothing with ϵ = 0 . 15 \epsilon =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 2

Distribution of velocity for different values of W e and ∅ with ϵ = 0 . 15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 3 Distribution of velocity for different values of Q 1 {Q}_{1} and Q 2 {Q}_{2} with W e = 0.03 {W}_{e}=0.03 , ϵ = 0 . 15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 3

Distribution of velocity for different values of Q 1 and Q 2 with W e = 0.03 , ϵ = 0 . 15 , ∅ = 0.15 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 4 Distribution of velocity for different values of ϵ \epsilon and Q 3 {Q}_{3} with W e = 0.03 {W}_{e}=0.03 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 4

Distribution of velocity for different values of ϵ and Q 3 with W e = 0.03 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 5 Distribution of velocity for different values of n n and Q 4 {Q}_{4} with W e = 0.03 {W}_{e}=0.03 , ϵ = 0 . 15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , and t = 0.1 t=0.1 .$

Figure 5

Distribution of velocity for different values of n and Q 4 with W e = 0.03 , ϵ = 0 . 15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 5 = 0.1 , z = 0.4 , and t = 0.1 .

$Figure 6 Distribution of velocity for different values of t t and Q 5 {Q}_{5} with W e = 0.03 {W}_{e}=0.03 , ϵ = 0 . 15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , z = 0.4 z=0.4 , and n = 0.3 {\rm{and}}{n}=0.3 .$

Figure 6

Distribution of velocity for different values of t and Q 5 with W e = 0.03 , ϵ = 0 . 15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , z = 0.4 , and n = 0.3 .

5.2 Shear stress

Here we discuss the effect of parameters W e , ε , ∅ , Q 1 , Q 2 , Q 5 , Q 3 , Q 4 , n , and t on the shear stress, as noted in Figures 7–12. The stress decreases in the start of the channel on the elastic and solid wall (the outer and inner walls of the channel) where its value is negative and positive and then the stress turns once, twice, or three times under the influence of the increase in the parameters W e , ε , ∅ , Q 1 , Q 2 , and Q 5 , respectively. As the stress increases on the channel’s elastic and solid walls (the outer and inner walls), we see from Figures 13–15 that parameters Q 3 , Q 4 , and n compared to the previous parameters have opposite effect when their values are negative and positive at the beginning of the channel, and then the stress turns once or twice during the channel. According to Figure 16, it is noted that parameter t has a direct impact on stress.

$Figure 7 Effect of different values of W e {W}_{{\rm{e}}} on stress with ϵ = 0 . 15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 \hspace{1em}{Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 7

Effect of different values of W e on stress with ϵ = 0 . 15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 8 Effect of different values of ϵ \epsilon on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 8

Effect of different values of ϵ on stress with W e = 0 . 03 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 9 Effect of different values of ∅ \varnothing on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 \epsilon =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 \hspace{1em}{Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 9

Effect of different values of ∅ on stress with W e = 0 . 03 , ϵ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 10 Effect of different values of Q 1 {Q}_{1} on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 , ∅ = 0.15 \epsilon =0.15,\varnothing =0.15 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 10

Effect of different values of Q 1 on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 11 Effect of different values of Q 2 {Q}_{2} on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 , ∅ = 0.15 \epsilon =0.15,\varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 {\rm{and}}t=0.1 .$

Figure 11

Effect of different values of Q 2 on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 12 Effect of different values of Q 5 {Q}_{5} on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 12

Effect of different values of Q 5 on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 13 Effect of different values of Q 3 {Q}_{3} on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 , ∅ = 0.15 \epsilon =0.15,\varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 13

Effect of different values of Q 3 on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 14 Effect of different values of stress for Q 4 {Q}_{4} on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , n = 0.3 n=0.3 , and t = 0.1 t=0.1 .$

Figure 14

Effect of different values of stress for Q 4 on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 5 = 0.1 , z = 0.4 , n = 0.3 , and t = 0.1 .

$Figure 15 Effect of different values of n n on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 , ∅ = 0.15 \epsilon =0.15,\varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , and t = 0.1 t=0.1 .$

Figure 15

Effect of different values of n on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , and t = 0.1 .

$Figure 16 Effect of different values of t t on stress with W e = 0 . 03 {W}_{{\rm{e}}}=0.03 , ϵ = 0.15 , ∅ = 0.15 \epsilon =0.15,\varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.1 {Q}_{4}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , z = 0.4 z=0.4 , and n = 0.3 {\rm{and}}n=0.3 .$

Figure 16

Effect of different values of t on stress with W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.1 , Q 5 = 0.1 , z = 0.4 , and n = 0.3 .

5.3 Phenomena trapping

Bolus is created due to the peristaltic motion of the wall of the flow channel influencing the fluid flow within the channel. In this section, we will discuss the effect of parameters W e , ε , ∅ , Q 1 , Q 2 , Q 3 , Q 4 , Q 5 , n , and t on this bolus.

Figures 17–21 show that as the parameters W e , ∅ , Q 1 , Q 2 , and Q 5 increase, the trapped bolus swells and stretches to the channel’s outer wall (the elastic wall), while Figures 22–25 show that as these parameters decrease, the trapped bolus contracts. This is the case for values ε , Q 3 , Q 4 , and n . In Figure 26 it is seen that the trapped bolus is not affected by the increase in the parameter t .

$Figure 17 Various values of the wave frame streamlines of We = {0.010, 0.015, 0.020} at ϵ = 0 . 15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 {\rm{and}}t=0 .$

Figure 17

Various values of the wave frame streamlines of W_e = {0.010, 0.015, 0.020} at ϵ = 0 . 15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 18 Various values of the wave frame streamlines of ∅ = \varnothing = {0.160, 0.165, 0.170} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 t=0 .$

Figure 18

Various values of the wave frame streamlines of ∅ = {0.160, 0.165, 0.170} at W e = 0 . 03 , ϵ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 19 Various values of the wave frame streamlines of Q 1 = {Q}_{1}= {0.10, 0.11, 0.12} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 t=0 .$

Figure 19

Various values of the wave frame streamlines of Q 1 = {0.10, 0.11, 0.12} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 20 Various values of the wave frame streamlines of Q 2 = {Q}_{2}= {0.5, 0.6, 0.7} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 t=0 .$

Figure 20

Various values of the wave frame streamlines of Q 2 = {0.5, 0.6, 0.7} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 21 Various values of the wave frame streamlines of Q 5 = {Q}_{5}= {0.2, 1.0, 1.8} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , n = 0.3 n=0.3 , and t = 0 t=0 .$

Figure 21

Various values of the wave frame streamlines of Q 5 = {0.2, 1.0, 1.8} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , n = 0.3 , and t = 0 .

$Figure 22 Various values of the wave frame streamlines of ϵ = \epsilon = {0.150, 0.155, 0.160} at W e = 0 . 03 {W}_{e}=0.03 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 t=0 .$

Figure 22

Various values of the wave frame streamlines of ϵ = {0.150, 0.155, 0.160} at W e = 0 . 03 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 23 Various values of the wave frame streamlines of Q 3 = {Q}_{3}= {0.1, 0.2, 0.3} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 {\rm{and}}t=0 .$

Figure 23

Various values of the wave frame streamlines of Q 3 = {0.1, 0.2, 0.3} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 4 = 0.4 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 24 Various values of the wave frame streamlines of Q 4 = {Q}_{4}= {0.4, 1.4, 2.4} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 5 = 0.1 {Q}_{5}=0.1 , n = 0.3 n=0.3 , and t = 0 {\rm{and}}t=0 .$

Figure 24

Various values of the wave frame streamlines of Q 4 = {0.4, 1.4, 2.4} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 5 = 0.1 , n = 0.3 , and t = 0 .

$Figure 25 Various values of the wave frame streamlines of n = n= {0.3, 0.5, 0.7} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , and t = 0 {\rm{and}}t=0 .$

Figure 25

Various values of the wave frame streamlines of n = {0.3, 0.5, 0.7} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , and t = 0 .

$Figure 26 Various values of the wave frame streamlines of t = {t}= {0.02, 0.04, 0.06} at W e = 0 . 03 {W}_{e}=0.03 , ϵ = 0.15 \epsilon =0.15 , ∅ = 0.15 \varnothing =0.15 , Q 1 = 0.1 {Q}_{1}=0.1 , Q 2 = 0.5 {Q}_{2}=0.5 , Q 3 = 0.1 {Q}_{3}=0.1 , Q 4 = 0.4 {Q}_{4}=0.4 , Q 5 = 0.1 {Q}_{5}=0.1 , and n = 0.3 n=0.3 .$

Figure 26

Various values of the wave frame streamlines of t = {0.02, 0.04, 0.06} at W e = 0 . 03 , ϵ = 0.15 , ∅ = 0.15 , Q 1 = 0.1 , Q 2 = 0.5 , Q 3 = 0.1 , Q 4 = 0.4 , Q 5 = 0.1 , and n = 0.3 .

5 Concluding remarks

Important results were obtained by studying the effect of peristaltic flow of Carreau fluid through a channel of two overlapping tubes having the same center “the inner tube is cylindrical and solid, while the outer tube is elastic rubber,” these results will be reviewed briefly. Through the cylindrical coordinates that were dealt with, the effect of the mentioned parameters on the rubber channels in the human body within became clear.

The effect of parameters W e , ∅ , Q 1 , Q 2 , and Q 5 on the fluid velocity is direct, while the effect of parameters Q 3 , Q 4 , ε , n , and t are indirect.
It is observed that the velocity curve resembles a parabola and that the greatest fluid velocity is at the beginning of the flow channel with respect to parameters W e , ∅ , Q 1 , Q 2 , Q 5 , and t and in the middle of the flow channel with respect to parameters ε , ∅ , Q 3 , Q 4 , and n , while the flow in both sides of the channel (channel wall) is the least.
Several fluctuations were observed along the channel concerning the shear stress, as it begins its path in the channel decreasing with the increase in the parameters W e , ε , ∅ , Q 1 , Q 2 , and Q 5 , and its journey begins to increase with the increase in the parameters Q 3 , Q 4 , n , and t .
Increases in the parameters W e , ∅ , Q 1 , Q 2 , and Q 5 result in the trapped bolus growing and expanding to the outside wall (elastic), while decreases in the parameters ε , Q 3 , and Q 4 result in the trapped bolus shrinking, and not affected by the increase in the parameter t .
An increase in the Weissenberg number, flexural stiffness of the wall, and longitudinal tension per unit width increases blood velocity and also increases the wave frame streamlines, while decreasing the shear stress.

The present work may be an improvement to increase slow blood flow through a slight longitudinal tension and a slight increase in wall flexural stiffness.

dr.dheia.g.salih@gmail.com

Funding information: This manuscript was self-funded by the authors.
Author contributions: The authors relied on themselves to write the manuscript and did not use a specific party.
Conflict of interest: There is no conflict of interest with any of the parties.

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Received: 2022-04-09

Revised: 2022-08-19

Accepted: 2022-09-19

Published Online: 2023-03-14

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

Artikel in diesem Heft

https://doi.org/10.1515/jmbm-2022-0257

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Carreau fluid; peristaltic flow; wall properties; cylindrical coordinates

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