Application of Rabinowitsch Fluid Model for the Mathematical Analysis of Peristaltic Flow in a Curved Channel

1 Introduction

The transportation of fluid due to the symmetric contraction and expansion of flexible walls is termed “peristaltic transport.” The significance of this phenomenon is noticeable not only in physiological processes but also in engineering and industry. Peristalsis appears to be a vital tool for transportation of chyme in the gastrointestinal track, movement of ovum in the fallopian tube, and blood flow in cardiac chambers. Further biomedical instruments such as the heart lung machine work under this code. Application in industry covers the pumping of toxic and acidic fluids with the help of breaker and handle pumps, which display the peristaltic phenomena. Numerous experimental, as well as theoretical attempts, have been made to analyse peristaltic motion of Newtonian and non-Newtonian fluids in different flow geometries [1–7]. Rabinowitsch fluid model is one of the fluid models where a nonlinear relationship between the shear stress and strain rate exists. This model has its significance as the three major categories for fluid are depicted for different values of nonlinear factor β, i.e., for β=0 this model represents Newtonian fluids, for β<0 it represents shear thickening fluids, and for β>0 it exhibits behaviour of shear thinning fluids. Many researchers had considered this model for the experimental and theoretical analysis of squeeze film, annular disk, externally pressurized bearing, journal bearing, and pivoted curved slider bearings [8, 9]. However, no one has attempted to examine Rabinowitsch fluid model for peristaltic flows in a curved channel. Recently, S. Nadeem and E. N. Maraj [10–13] had carried out theoretical analysis for peristaltic flow of various non-Newtonian fluids in a curved channel. The aim of the present work is to discuss the peristaltic flow of Rabinowitsch fluid in a curved channel. The study of this fluid model for peristaltic flow problems in curved channel is not explored so far. In the present article, mathematical analysis is carried out, and exact solutions are obtained for fluid velocity. The effects of various significant parameters are discussed graphically for the velocity profile, pressure rise, pressure gradient, and shear stresses.

2 Mathematical Formulation

Consider a two-dimensional flow of an incompressible Rabinowitsch fluid in a curved channel of uniform thickness 2a. The sinusoidal waves of small amplitude are imposed on the flexible channel walls. Curvilinear coordinates (N̅, S̅, Z̅) are considered where N̅ is the coordinates in the cross stream, S̅ is the coordinate in the downstream and Z̅ is the coordinate in the vertical directions respectively and R^* as the radius of curvature (see Figure 1).

Figure 1:

Geometry of the problem.

The flow in the channel is induced by sinusoidal waves of small amplitude b traveling along the flexible walls of the channel. The walls of the channel are considered as follows:

(1)N¯ = H(S¯, t¯) = a + bsin[2πλ(S¯ − ct¯)],  upper wall (1)

(2)N¯ = −H(S¯, t¯) = −a − bsin[2πλ(S¯ − ct¯)].  lower wall (2)

In these equations c is the wave speed and λ denotes the wave length. Let V̅ and U̅ denote the velocity components in the cross stream and downstream directions, respectively. The governing equations of motion for curved channel are described as

(3)R∗N¯ + R∗∂∂N¯((N¯ + R*)R∗V¯) + R∗N¯ + R∗∂U¯∂S¯ = 0, (3)

(4)ρf(∂V¯∂t¯ + V¯∂V¯∂N¯ + R∗U¯N¯ + R∗∂V¯∂S¯ − U¯2N¯ + R∗) =      −∂P¯∂N¯ + ∂∂N¯(τ¯N¯ N¯) + R∗N¯ + R∗∂∂S¯(τ¯N¯S¯) − τ¯S¯ S¯N¯ + R∗, (4)

(5)ρf¯(∂U¯∂t¯ + V¯∂U¯∂N¯ + R∗U¯N¯ + R∗∂U¯∂S¯ + U¯V¯N¯ + R∗) =     −R∗N¯ + R∗∂P¯∂S¯ + ∂∂N¯(τ¯N¯ S¯) + R∗N¯ + R∗∂∂S¯(τ¯S¯ S¯). (5)

In these equations, P̅ is the pressure, V̅ and U̅ are the velocity components in cross stream N̅ and downstream S̅ directions respectively and τ̅’s represent the stresses. For Rabinowitsch fluid model following empirical stress-strain relation in component form holds [7, 8]

(6)τ¯N¯N¯ = 2μ∂V¯∂N¯, (6)

(7)τ¯S¯ N¯ + ατS¯ N¯3 = μ(∂U¯∂N¯ + R∗N¯ + R∗∂V¯∂S¯ − U¯N¯ + R∗), (7)

(8)τ¯S¯ S¯ = 2μ(R∗N¯ + R∗∂U¯∂S¯ + V¯N¯ + R∗), (8)

where μ is the zero shear rate viscosity and α is the coefficient of pseudo plasticity. The flow phenomena is unsteady in the fixed frame. To carry out a steady-state analysis. we switch from fixed frame to wave frame (n̅, s̅) moving with the wave speed c. The transformation between the two frames is given by

(9)s¯ = S¯ − ct¯, n¯ = N¯, u¯ = U¯ − c, v¯ = V¯, (9)

where v̅ and u̅ are the velocity components along n̅ and s̅– directions in the wave frame.

With the help of these transformations (3)–(5) take the form

(10)∂∂n¯[(n¯ + R*)v¯] + R∗n¯ + R∗∂u¯∂s¯ = 0, (10)

(11)ρ(−c∂v¯∂s¯ + v¯∂v¯∂n¯ + R∗(u¯ + c)n¯ + R∗∂v¯∂s¯ − (u¯ + c)2n¯ + R∗) =    −∂p¯∂n¯ + 1n¯ + R∗∂∂n¯[(n¯ + R*)τn¯ n¯] + R∗n¯ + R∗∂∂s¯[τn¯ s¯] − τs¯ s¯n¯ + R∗, (11)

(12)ρ(−c∂u¯∂s¯ + v¯∂u¯∂n¯ + R∗(u¯ + c)n¯ + R∗∂u¯∂s¯ + (u¯ + c)v¯n¯ + R∗) = −R∗n¯ + R∗∂p¯∂s¯ + 1n¯ + R∗∂∂n¯[(n¯ + R*)τn¯ s¯] + R∗n¯ + R∗∂∂s¯[τs¯ s¯]. (12)

The following non-dimensional variables and velocity stream function relation are introduced:

s = S¯λ, n = N¯a, u = U¯c, v = V¯c, ψ = ψ¯ca, k = R∗a, Re = ρfcaμ,ε = ba, δ = aλ, p = a2P¯cμλ, β = αμ2c2a, u = −∂ψ∂n, v = δkn + k∂ψ∂s,

where Re is the Reynolds number, δ is the wave number, k is the curvature parameter, and β is the dimensionless nonlinear factor.

Equation (10) is identically satisfied, and (11)–(12) under long wavelength and low Reynolds number approximations in dimensionless form becomes

(13)∂p∂n = 0, (13)

(14)∂τns∂n + 1n+kτns = kn+k∂p∂s. (14)

Equation (14) subject to the boundary conditions

(15)τns = 0  at  n = 0  (center line of the channel) (15)

yields

(16)τns = knn + k∂p∂s. (16)

The extra stress tensor for Rabinowitsch model in non-dimensional form becomes

(17)τns + βτns3 = (∂u∂n − u + 1n + k). (17)

By substituting value of τ_ns into (17) gives

(18)knn + k∂p∂s + β(knn + k∂p∂s)3 = (∂u∂n − u + 1n + k). (18)

The appropriate boundary conditions in the wave frame are defined as

(19)u = −1,  at  n = h = 1 + εsins,∂u∂n = 0,  at  n = 0, (19)

where ε=b / a is the amplitude ratio.

3 Solution of the Problem

The exact solution of (18) subject to the boundary conditions (19) is as follows:

(20)u(n, s) = −12(h + k)2(k + n)2∂p∂s(2k(h + k)2(h − n)(k + n)2 − 2k2(h + k)2(k + n)2(log(h + k) − log(k + n)))−12(h + k)2(k + n)2(2(h + k)2(k + n)2 + k3(h − n)(2h2(k + n)2+ k2(6k2 + 9kn + 2n2) + hk(9k2 + 14kn + 4n2))β − 6k4(h + k)2(k + n)2β(log(h + k) − log(k + n))). (20)

Defining the volume flow rate as

(21)Θ = ∫−hhu(n, s)dn (21)

Substituting (20) into (21), we get

(22)A + B∂p∂s + C(∂p∂s)3 = Θ (22)

where

A = −h5(h + k)4 − 4h4k(h + k)4 − 6h3k2(h + k)4 − 4h2k3(h + k)4 − hk4(h + k)4 + h5(−h + k)(h + k)3+ 2h4k(−h + k)(h + k)3 − 2h2k3(−h + k)(h + k)3 − hk4(−h + k)(h + k)3,

B = −h6k4(h + k)4 − h5k2(h + k)4 − 2h4k3(h + k)4 − 5h3k42(h + k)4 − 7h2k54(h + k)4 − hk62(h + k)4 − h6k4(−h + k)(h + k)3 − 3h5k22(−h + k)(h + k)3 − h4k32(−h + k)(h + k)3 + 2h3k4(−h + k)(h + k)3 + 3h2k54(−h + k)(h + k)3 − hk62(−h + k)(h + k)3 − 12h2klog(−h + k) + hk2log(−h + k) −12k3log(−h + k) + 12h2klog(h + k) + hk2log(h + k) + 12k3log(h + k) −hk(h2 − 3hk + 2k2) log(h + k)2(−h + k)−hk(h2 + 3hk + 2k2) log(h + k)2(h + k),

C = −h6k3β4(h + k)4 + 7h4k5β4(h + k)4 + 3h3k6β(h + k)4 + 3h2k7β2(h + k)4− hk8β3(h + k)4 − k9β3(h + k)4 − h6k3β4(−h + k)(h + k)3 − 9h5k4β2(−h + k)(h + k)3 − 11h4k5β4(−h + k)(h + k)3 + 11h3k6β3(−h + k)(h + k)3 + 9h2k7β2(−h + k)(h + k)3 + 5hk8β3(−h + k)(h + k)3 + k9β3(−h + k)(h + k)3 − 12h2k3βlog(−h + k) + hk4βlog(−h + k) + k5βlog(−h + k) + 12h2k3βlog(h + k) + hk4βlog(h + k)−k5βlog(h + k)−hk3(h2 −3hk + 2k2) βlog(h + k)2(−h + k) − hk3(h2 + 3hk + 2k2) βlog(h + k)2(h + k).

Using mathematics software Mathematica, the real solution of (22) is computed, and the result for pressure rise is obtained by substituting the value of ∂p/∂s in the following formula:

(23)Δp = ∫−hh∂p∂sds (23)

4 Results and Discussion

This section is devoted to the discussion of graphical results for pressure rise, fluid velocity, and stream function. In this article, we are interested in studying the effect of curvature parameter k and the coefficient of pseudo plasticity β. Figure 2 is the graph of pressure rise Δp against the flow rate Θ for different values of curvature parameter. Figure 2a and b show the effects of curviness of the channel for dilatant and pseudo plastic fluids, respectively. It is observed that the effect of curvature parameter contribution for dilatant and pseudo plastic fluids are quite opposite. In case of shear thickening fluid (dilatant, i.e., Silica and polyethylene glycol), increasing in curviness of the channel enhances pressure rise in the peristaltic pumping region and lowers the pressure rise in the augmented pumping region. This leads to the fact that the increase in curvature parameter upsurge the pressure rise which contributes in moving the bolus in the forward direction. However, in case of shear thinning fluids, pressure rise decreases in peristaltic pumping region and increases in augmented pumping region, respectively, with the increase in curvature parameter. Figure 3a and b illustrate the effect of coefficient of pseudo plasticity for shear thickening and shear thinning fluids, respectively. From Figure 3a, it is concluded that for shear thickening fluids pressure rise increases in peristaltic pumping region and decreases in augmented pumping region with the increase in β. However, in case of shear thinning fluids increasing values of β, lessen the pressure rise in peristaltic pumping region and enhances the pressure rise in augmented pumping region, respectively. This trend is displayed in Figure 3b. Moreover, free pumping (i.e., ΔP=0) occurs for both types of non-Newtonian fluids. This result is quite compatible with the flow of dilatant and pseudo plastic fluids. In case of dilatant fluids, flow motion is relatively slow compared to shear thinning fluids, which contributes in increasing the pressure rise for dilatant and a decrease in pressure rise for shear thinning fluids in peristaltic pumping region. Figures 4 and 5 demonstrate the effect of pertinent parameters on fluid velocity. Figure 4a and b are the graphical presentation for curvature parameter effects. It is observed that curvature parameter disturbs the shear thickening and shear thinning fluids significantly. In case of shear thickening fluids, fluid velocity decreases near the inner wall of the channel and increases in the center, as well as out half of the channel. Figure 4b depicts that curvature parameter affects the fluid velocity of shear thinning fluids oppositely. The contribution of coefficient of pseudo plasticity for shear thickening and shear thinning fluids are displayed in Figure 5a and b. It is noticed that in general for shear thickening fluids, velocity profile increases with the increase in β and for shear thinning fluids β plays role in decreasing fluid velocity. Graphical results for shear stress are displayed in Figures 6 and 7. The effect of curviness of the channel on stress profile are shown in Figure 6a and b for shear thickening and shear thinning fluids, respectively. It is observed that for the case of shear thickening fluids, shear stress improves near the lower wall of the channel and decays afterwards with the increase in the curviness of the channel, whereas for the case of shear thinning fluids reverse effect is observed (see Fig. 6b). Figure 7a and b demonstrate the effect of coefficient of pseudo plasticity on stress profile. It is notable that in the case of shear thickening, as well as shear thinning fluids, β contributes in enhancing shear stress in the inner half of the channel and lessening shear stress in the outer half of the channel, respectively. Figures 8 and 9 are the plots of streamlines for different values of curvature parameter k and β, respectively. Figure 8 shows that with the increase in curviness of the channel, the number of trapping streamlines increases, and size of the trapped bolus reduces. However, the coefficient of pseudo plasticity effects in an opposite way. Figure 9 illustrates that with the increase in β, the trapping bolus grows and the number of closed streamlines reduces. Tables 1 and 2 shows the numerical results of shear stress for different values of significant parameters at the lower and upper wall of the channel, respectively.

Figure 2:

Display of the effect of curvature parameter k, where s=0.0005, ε=0.03, and β=0.05.

Figure 3:

Plots are shown for different values of β, where s=0.0005, ε=0.03, and k=2.

Figure 4:

Plot is shown for velocity component u(n, s) against n; the plot shows effect of k, where s=0.01, ε=0.001, and Θ=0.5.

Figure 5:

Plot is shown for velocity component u(n, s) against n; the plot shows the effect of β, where k=2, s=0.01, ε=0.001, and Θ=0.5.

Figure 6:

Plot is shown for velocity component u(n, s) against n; the plot shows the effect of β, where s=0.01, ε=0.001, and Θ=0.5.

Figure 7:

Plot is shown for velocity component u(n, s) against n; the plot shows the effect of β, where k=2, s=0.01, ε=0.001, and Θ=0.5.

Figure 8:

Here, the streamlines are plotted for, against different values of k, i.e., (a) k=1.8, (b) k=1.9, (c) k=2.0, whereas ε=0.03, β=0.5, and Θ=0.5.

Figure 9:

Here, the streamlines are plotted for, against different values of β, i.e., (a) β=0.1, (b) β=0.5, (c) β=1.0, whereas k=2, ε=0.03, and Θ=0.5.

5 Conclusions

Current mathematical study for peristaltic flow of Rabinowitsch fluid in a curved channel leads to the following concluding remarks:

1. The curvature parameter affect significantly on dilatant as well as pseudo plastic fluids. i.e., for shear thickening fluid pressure rise enhances in the peristaltic pumping region and decays in the augmented pumping region while for shear thinning fluids pressure rise decreases in peristaltic pumping region and increases in augmented pumping region with the increase in curvature parameter.

2. Pressure rise increases in the peristaltic pumping region and decreases in the augmented pumping region for dilatant fluids. However, this trend is opposite in case of pseudo plastic fluids.

3. The curviness of the channel affects the dilatant quite opposite to the pseudo plastic fluid.

4. For both shear thickening, as well as shear thinning fluids, fluid velocity increases in the central region of the curved channel.

5. Stresses at the lower wall of the channel decreases with the increase in curvature parameter for dilatant fluids and for pseudo plastic fluid opposite trend is observed.