# Peristaltic Transport of Hyperbolic Tangent Fluid in a Tapered Asymmetric Channel

N Naduvinamani, Anita Siddayya Guttedar, Laxmi Devindrappa

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N Naduvinamani, Anita Siddayya Guttedar, Laxmi Devindrappa. Peristaltic Transport of Hyperbolic Tangent Fluid in a Tapered Asymmetric Channel. Indian Journal of Science and Technology, 2021, 14 (20), pp.1677-1688. 10.17485/IJST/v14i20.480. hal-04429378

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RESEARCH ARTICLE

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OPEN ACCESS

**Received:** 20.03.2021

**Accepted:** 20.05.2021

**Published:** 08.06.2021

**Citation:** Naduvinamani NB, Guttedar AS, Devindrappa L (2021) Peristaltic Transport of Hyperbolic Tangent Fluid in a Tapered Asymmetric Channel. Indian Journal of Science and Technology 14(20): 1677-1688. https://doi.org/10.17485/IJST/v14i20.480

* Corresponding author.
naduvinamaninb@yahoo.co.in

**Funding:** None

**Competing Interests:** None

**Copyright:** © 2021 Naduvinamani et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Published By Indian Society for Education and Environment (iSee)

**ISSN**

Print: 0974-6846

Electronic: 0974-5645

# Peristaltic Transport of Hyperbolic Tangent Fluid in a Tapered Asymmetric Channel

N B Naduvinamani¹*, Anita Siddayya Guttedar¹, Laxmi Devindrappa¹

¹ Department of Mathematics, Gulbarga University, Kalaburagi, 585106, Karnataka, India

## Abstract

**Objectives:** The study intends to investigate the problem of peristalsis transport of hyperbolic tangent fluid in a tapered asymmetric channel.

**Methods:** The two-dimensional equations of a hyperbolic tangent fluid have been simplified under the suspicions of low Reynolds number and long wavelength approximations. The reduced equations are solved by using standard perturbation technique. The numerical results obtained are presented in the graphical form for various values of physical parameters and are discussed. **Findings:** It is showing that for the larger values of non-uniform parameter, pressure rise diminishes and the axial velocity diminishes at the core part of the channel and increases at the right and left side of the channel for the increasing values of non-uniform parameter. **Applications:** Hyperbolic tangent fluid model anticipate the shear thinning phenomenon very accurately and are being used mostly in laboratory experiments and industries.

**Keywords:** Reynolds number; Hyperbolic tangent fluid; peristaltic transport; tapered asymmetric channel

## 1 Introduction

The system of peristaltic transport has been coined by Latham. Now a day's many researchers are fascinated towards tapered asymmetric channel of non-Newtonian fluid with peristalsis because of its widespread applications in medical, physiological and industrial processes. Peristalsis can be found in many physiological processes such as in the chyme movement in gastrointestinal tract, blood circulation in small blood vessels and also in food digestion. In biomedical field the dialysis and heart lung machines worked on the peristaltic principle.

Nadeem and Akram(1) have analyzed the hyperbolic tangent fluid under long wavelength and low Reynolds number approximations in an asymmetric channel with peristalsis via regular perturbation method. Also they have extended their work for the same fluid under the influence of MHD with heat transfer in a vertical asymmetric channel(2). Many experiments(3-10) have been made for hyperbolic tangent fluid for different types of channels.

Kothandapani et al. (11) have studied the peristaltic transport in the presence of tapered asymmetric channel for Johnson-Segalman fluid. From their study it is revealed that as the values of amplitude increases the pressure rise increases also in the
increment of non-dimensionless parameter and Weissenberg number the pressure rise decreases. Hayat et al. (12) have scrutinized the effect of magnetic field and nonlinear radiation of peristaltic motion of Sisko fluid in an inclined tapered asymmetric channel. Kothandapani and Prakash (13) have analyses the peristaltic transport of Williamson nanofluids under the influence of magnetic field and thermal radiation parameter in a tapered asymmetric channel. Prakash et al. (14) have studied the peristaltic motion in a tapered asymmetric channel for a third grade fluid. From their study it is noticed that as the non-uniform parameter (m) increases the pressure rise decreases and axial velocity enhances at the boundaries by increasing in non-uniform parameter (m). The peristaltic phenomena in the presence of tapered asymmetric channel has discussed in the literatures (15-19). Kothandapani et al. (20) have observed the peristaltic flow of an incompressible non-Newtonian fourth grade fluid with the influence of uniform magnetic field in a tapered asymmetric channel. Further similar studies can be found in relevant literature (21-30).

To the best of our knowledge, no attempt has been made to analyses the peristaltic transport of the hyperbolic tangent fluid in a tapered asymmetric channel. The governing equations are simplified under the assumptions of long wavelength and low Reynolds number approximations. The stream function and pressure gradient have calculated by adopting regular perturbation method. The effect of various physical parameters on pressure rise and axial velocity are discussed and explained graphically.

## 2 Mathematical Modelling and Methods

<figure><img src="image_4.png" /><figcaption>Fig 1. Schematic diagram of a tapered asymmetric</figcaption></figure>

Let us consider the peristalsis of two-dimensional hyperbolic tangent fluid through the channel of width 2d. Here the medium is considered to the tangent hyperbolic fluid flow is produced by the propagation of sinusoidal wave trains with constant speed c along the walls of the tapered asymmetric channel, such that

$$H_2(\bar{X}, \bar{t}) = d + m'\bar{X} + a_2\sin\left[\frac{2\pi}{\lambda}(\bar{X} - c\bar{t})\right] \quad (\text{Upper wall 1(a)})$$

$$H_1(\bar{X}, \bar{t}) = -d - m'\bar{X} - a_1\sin\left[\frac{2\pi}{\lambda}(\bar{X} - c\bar{t}) + \phi\right] \quad (\text{Lower wall 1(b)})$$
Where, $a_1$ and $a_2$ are the amplitudes of the wave, $\lambda$ is the wave length, width of the channel is 2d, phase speed of the wave is depicted with $c, \bar{t}$ be the time, and $\bar{X}$ is the sinusoidal wave propagation direction, the non uniform parameter is denoted by $m'(m' \ll 1)$. The phase difference $\phi \in [0, \pi]$ and the quantity $\phi = 0$ imply the symmetric channel with waves out of phase i.e. both walls move towards the outward or inward simultaneously. And the physical variables $a_1, a_2, d$ and $\phi$ obey the following required relation.

$$a_1^2 + a_2^2 + 2a_1 a_2 \cos\phi \le (2d)^2 \quad (2)$$

Thus, based on the above considered geometry and assumptions, in the fixed frame of reference the governing equations for unsteady two-dimensional tangent hyperbolic fluid (1,2,4) are as follows:

$$\frac{\partial \bar{U}}{\partial \bar{X}} + \frac{\partial \bar{V}}{\partial \bar{Y}} = 0 \quad (3)$$

$$\rho \left( \frac{\partial \bar{U}}{\partial \bar{t}} + \bar{U} \frac{\partial \bar{U}}{\partial \bar{X}} + \bar{V} \frac{\partial \bar{U}}{\partial \bar{Y}} \right) = - \frac{\partial \bar{P}}{\partial \bar{X}} - \frac{\partial \bar{\tau}_{\bar{X}\bar{X}}}{\partial \bar{X}} - \frac{\partial \bar{\tau}_{\bar{Y}\bar{Y}}}{\partial \bar{Y}} \quad (4)$$

$$\rho \left( \frac{\partial \bar{V}}{\partial t} + \bar{U} \frac{\partial \bar{V}}{\partial \bar{X}} + \bar{V} \frac{\partial \bar{V}}{\partial \bar{Y}} \right) = - \frac{\partial \bar{P}}{\partial \bar{Y}} - \frac{\partial \bar{\tau}_{\bar{X}\bar{Y}}}{\partial \bar{X}} - \frac{\partial \bar{\tau}_{\bar{Y}\bar{Y}}}{\partial \bar{Y}} \quad (5)$$

Following transformations indicate the moment of a considered wave frame $(\bar{x}, \bar{y})$ travelling with velocity $c$ away from the immovable frame $(\bar{X}, \bar{Y})$ as follows.

$$\bar{x} = \bar{X} - c\bar{t}, \quad \bar{y} = \bar{Y}, \quad \bar{u} = \bar{U} - c, \quad \bar{v} = \bar{V}, \quad \text{and } \bar{p}(x) = \bar{P}(X, t) \quad (6)$$

The dimensionless variables for the purpose of non-dimensionalization are

$$\left.
\begin{aligned}
x &= \frac{\bar{x}}{\lambda}, & y &= \frac{\bar{y}}{\bar{d}_1}, & u &= \frac{\bar{u}}{c}, & t &= \frac{c}{\lambda}\bar{t}, & v &= \frac{\bar{v}}{c\delta}, & h_1 &= \frac{\bar{h}_1}{\bar{d}_1}, & h_2 &= \frac{\bar{h}_2}{\bar{d}_1}, & \tau_{xx} &= \frac{\lambda}{\eta_0 c} \bar{\tau}_{\bar{x}\bar{x}} \\
\tau_{xy} &= \frac{\bar{d}_1}{\eta_0 c} \bar{\tau}_{\bar{x}\bar{y}}, & \tau_{yy} &= \frac{\bar{d}_1}{\eta_0 c} \bar{\tau}_{\bar{y}\bar{y}}, & \delta &= \frac{\bar{d}_1}{\lambda}, & \text{Re} &= \frac{\rho c \bar{d}_1}{\eta_0}, & \text{We} &= \frac{\Gamma c}{\bar{d}_1}, & P &= \frac{\bar{d}_1^2}{c \lambda \eta_0} \bar{p}, & \dot{\gamma} &= \frac{\dot{\gamma} \bar{d}_1}{c} \\
& m &= \frac{m' \lambda}{d}
\end{aligned}
\right\} \quad (7)$$

By utilizing the above non-dimensional parameters in the Eqs (1-7) and in view of stream function $\psi(u = \frac{\partial \psi}{\partial y}, v = -\delta \frac{\partial \psi}{\partial x})$, the resultant equations are as follows:

$$\delta \mathrm{Re} \left[ \left( \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y} \right) \frac{\partial \psi}{\partial y} \right] = - \frac{\partial P}{\partial x} - \delta^2 \frac{\partial \tau_{xx}}{\partial x} - \frac{\partial \tau_{xy}}{\partial y} \quad (8)$$

$$\delta^3 \mathrm{Re} \left[ \left( \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y} \right) \frac{\partial \psi}{\partial x} \right] = - \frac{\partial P}{\partial y} - \delta^2 \frac{\partial \tau_{xy}}{\partial x} - \delta \frac{\partial \tau_{yy}}{\partial y} \quad (9)$$

$$\tau_{xx} = -2[1 + n(We \dot{\gamma} - 1)] \frac{\partial^2 \psi}{\partial x \partial y}$$
$$ \tau_{xy} = -[1 + n(We \dot{\gamma} - 1)] \left( \frac{\partial^2 \psi}{\partial y^2} - \delta^2 \frac{\partial^2 \psi}{\partial x^2} \right) $$

$$ \tau_{yy} = 2\delta [1 + n(We \dot{\gamma} - 1)] \frac{\partial^2 \psi}{\partial x \partial y} $$

$$ \dot{\gamma} = \left[ 2\delta^2 \left( \frac{\partial^2 \psi}{\partial x \partial y} \right)^2 + \left( \frac{\partial^2 \psi}{\partial y^2} - \delta^2 \frac{\partial^2 \psi}{\partial x^2} \right)^2 + 2\delta^2 \left( \frac{\partial^2 \psi}{\partial x \partial y} \right)^2 \right]^{1/2} $$

In above equations of $τ_{xx}$, $τ_{xy}$ and $τ_{yy}$, $δ$ is wave number, $Re$ is Reynolds number and $We$ be the Weissenberg number respectively. According to the assumptions of low Reynolds number and long wave length approximation ($δ ≪ 1$), as well as ignoring the terms of higher order $δ$, we can reduce the Eqs. (8) and (9) to the following form.

$$ \frac{\partial P}{\partial x} = \frac{\partial}{\partial y} \left[ 1 + n \left( We \frac{\partial^2 \psi}{\partial y^2} - 1 \right) \right] \frac{\partial^2 \psi}{\partial y^2} \quad (10) $$

$$ \frac{\partial P}{\partial y} = 0 \quad (11) $$

After removal of pressure gradient from Eqs. (10-11)

$$ \frac{\partial^2}{\partial y^2} \left[ 1 + n \left( We \frac{\partial^2 \psi}{\partial y^2} - 1 \right) \right] \frac{\partial^2 \psi}{\partial y^2} = 0 \quad (12) $$

The boundary conditions are

$$ \left. \begin{array}{l} \psi = -\frac{F}{2}, \quad \frac{\partial \psi}{\partial y} = 0 \quad \text{for} \quad y = h_1(x) \\ \\ \psi = \frac{F}{2}, \quad \frac{\partial \psi}{\partial y} = 0 \quad \text{for} \quad y = h_2(x) \end{array} \right\} \quad (13) $$

Where

$$ h_2(x) = 1 + mx + b\sin(2\pi(x-t)) $$

$$ h_1(x) = -1 - mx - a\sin(2\pi(x-t) + \phi) $$

Also, the quantities $a, b, \phi$ and $d$ satisfy the following inequality:

$$ a^2 + b^2 + 2ab \cos \phi \le 4 $$

## 2.1 Method of Solution

Here the flow quantities $ψ, F$ and $P$ are expanded in terms of power series as follows to get the simplified form of Eqn (12) by utilising boundary conditions:

$$ \psi = \psi_0 + We \psi_1 + O(We^2) \quad (14) $$

$$ F = F_0 + We F_1 + O(We^2) \quad (15) $$

$$ P = P_0 + We P_1 + O(We^2) \quad (16) $$

Replacing Eqs. (14)-(16) in the above Eqs. (10), (12) and (13) and rearranging the coefficients of powers of $We$ gives the following system of equations.
## 2.2 System of order We⁰

$$ \frac{\partial^4 \psi_0}{\partial y^4} = 0 \qquad (17) $$

$$ \frac{\partial P_0}{\partial x} = (1 - n) \frac{\partial^3 \psi_0}{\partial y^3} \qquad (18) $$

$$ \psi_0 = \frac{F_0}{2}, \quad \frac{\partial \psi_0}{\partial y} = 0 \quad \text{for} \quad y = h_2(x) \qquad (19) $$

$$ \psi_0 = -\frac{F_0}{2}, \quad \frac{\partial \psi_0}{\partial y} = 0 \quad \text{for} \quad y = h_1(x) \qquad (20) $$

## 2.3 System of order We¹

$$ \frac{\partial^4 \psi_1}{\partial y^4} = \frac{n}{n-1} \frac{\partial^2}{\partial y^2} \left( \frac{\partial^2 \psi_0}{\partial y^2} \right)^2 \qquad (21) $$

$$ \frac{\partial P_1}{\partial x} = (1 - n) \frac{\partial^3 \psi_1}{\partial y^3} + n \frac{\partial}{\partial y} \left( \frac{\partial^2 \psi_0}{\partial y^2} \right)^2 \qquad (22) $$

$$ \psi_1 = \frac{F_1}{2}, \quad \frac{\partial \psi_1}{\partial y} = 0 \quad \text{for} \quad y = h_2(x) \qquad (23) $$

$$ \psi_1 = -\frac{F_1}{2}, \quad \frac{\partial \psi_1}{\partial y} = 0 \quad \text{for} \quad y = h_1(x) \qquad (24) $$

## 2.4 Solution for Zeroth order (We⁰)

Solution of the Eq. (17) satisfying the required boundary conditions defined in Eqs. (19-20) are written as

$$ \psi_0 = C_1 + C_2 y + C_3 y^2 + C_4 y^3 \qquad (25) $$

$$ \frac{\partial P_0}{\partial x} = -6 C_4 (-1 + n) \qquad (26) $$

## 2.5 Solution for First order (We¹)

Substituting Eq. (25) and satisfying the boundary conditions (23-24) into the Eq. (21), we get the solution of first order as follows

$$ \psi_1 = C_5 + C_6 y + C_7 y^2 + C_8 y^3 + 72 C_4^2 y^4 \qquad (27) $$

$$ \frac{\partial p_1}{\partial x} = -6 C_8 (-1 + n) + 24 C_4 (C_3 n + 3 C_4 (24 - 23 n)y) \qquad (28) $$
$$C_1 = \frac{1}{2} F_0 \left( 1 + \frac{2h_2^2(-3h_1 + h_2)}{(h_1 - h_2)^3} \right)$$

$$C_2 = \frac{6F_0h_1h_2}{(h_1 - h_2)^3}$$

$$C_3 = -\frac{3F_0(h_1 + h_2)}{(h_1 - h_2)^3}$$

$$C_4 = \frac{2F_0}{(h_1 - h_2)^3}$$

$$C_5 = 72C_4^2h_1^2h_2^2 + \frac{F_1(h_1 + h_2)(h_1^2 - 4h_1h_2 + h_2^2)}{2(h_1 - h_2)^3}$$

$$C_6 = \frac{6h_1h_2(F_1 - 24C_4^2(h_1 - h_2)^3(h_1 + h_2))}{(h_1 - h_2)^3}$$

$$C_7 = -\frac{3(F_1(h_1 + h_2) - 24C_4^2(h_1 - h_2)^3(h_1^2 + 4h_1h_2 + h_2^2))}{(h_1 - h_2)^3}$$

$$C_8 = 2 \left( \frac{F_1}{(h_1 - h_2)^3} - 72 C_4^2 (h_1 + h_2) \right)$$

By invoking

$$F = F_0 + We F_1 \qquad (29)$$

By gathering all perturbation solutions of the considered problem for small parameter We, up to $\psi$, $\frac{dP}{dx}$ and $\Delta p$ as

$$\psi = \psi_0 + We \psi_1 \qquad (30)$$

$$\frac{dP}{dx} = \frac{dP_0}{dx} + We \frac{dP_1}{dx} \qquad (31)$$

$$\Delta p = \Delta p_0 + We \Delta p_1 \qquad (32)$$

Utilizing $F_0 = F - We F_1$ and then ignoring the terms higher than $O(We)$ the results given by Eq. (30-32) can be explicitly calculated.

The axial velocity obtained from the equation $u = \frac{\partial \psi}{\partial y}$.

The non-dimensional term for the average rise in pressure $\Delta p$ is over one period of wavelength is given as follows

$$\Delta p = \int_{0}^{1} \int_{0}^{1} \left( \frac{\partial p}{\partial x} \right)_{y=0} dxdt \qquad (33)$$
**3 Results and discussion**

This category of the paper is to illustrate the outcomes of our problem graphically. MATHEMATICA software has been utilized
to calculate pressure rise. The influences of various physical parameters like wave amplitude parameters a and b, width of the
channel d, non uniform parameter m, Weissenberg number We, phase difference φ and power law index n for hyperbolic tangent
fluid in a tapered asymmetric channel are discussed. Figures 2, 3, 4, 5, 6, 7 and 8 shows the average rise in pressure ΔP against
time average flux Q. Figure 2. reveals that the pressure rise for the values of wave amplitude a. It is noticed that, pressure rise
increases in pumping section and co-pumping section. Figure 3. elucidate that behavior of wave amplitude b in pressure rise. It
is interesting to note down that an increase in b values results the increment in pressure rise. From Figure 4. it is observed that
pressure rise enhances as channel width increases. From Figure 5. it is interesting to note that as the non uniform parameter m,
increases the pressure rise decreases in the pumping section and opposite behavior occurred in co-pumping region. Figure 6.
shows that pressure rise decreases for the larger values of power law index in pumping region and increases in co-pumping
region. Similar behavior exposed in Figure 7. for the values of phase difference φ. It is noticed that from Figure 8. that as the
Weissenberg number increases (We) the pressure rise increases.

<figure><img src="image_5.png" /><figcaption>Fig 2. Variation of Δp versus Q for different values of a n = 0.5, d = 0.1, φ = 2π, we = 0.01, b = 0.4 and m = 0.3</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig 3. Variation of Δp versus Q for different values of b n = 0.3, d = 0.2, φ = 2π/3, we = 0.03, a = 0.3 and m = 0.3</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig 4. Variation of Δp versus Q for different values of d, n = 0.3, b = 0.4, φ = 2π/3, we = 0.03, a = 0.2 and m = 0.3</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig 5. Variation of Δp versus Q for different values of mn = 0.3, b = 0.4, φ = 2π/3, we = 0.03, a = 0.3 and d = 0.2</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig 6. Variation of Δp versus Q for different values of nm = 0.3, b = 0.4, φ = 2π/3, we = 0.03, a = 0.3 and d = 0.2</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig 7. Variation of $\Delta p$ versus $Q$ for different values of $\phi m = 0.3, b = 0.4, n = 0.3, we = 0.03, a = 0.3$ and $d = 0.2$</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig 8. Variation of $\Delta p$ versus $Q$ for different values of $we = 0.3, b = 0.4, n = 0.2, \phi = \frac{2\pi}{3}, a = 0.3$ and $d = 0.2$</figcaption></figure>

Figures 9, 10, 11 and 12 illustrate distribution of axial velocity ($u$) is plotted against $y$. Figure 9 shows that axial velocity decreases right and left side of the channel for the escalating values of wave amplitude $a$ and situation reversed in the core part of the channel. It is clear from the Figure 10.

<figure><img src="image_12.png" /><figcaption>Fig 9. Axial velocity for different values of $am = 0.5, b = 0.4, n = 0.2, \phi = \frac{2\pi}{3}, d = 0.02, t = 0.6, we = 0.01$</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig 10. Axial velocity for different values of bm = 0.5, n = 0.2, $\phi = \frac{2\pi}{3}$, a = 0.5, d = 0.02, t = 0.6, we = 0.01</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig 11. Axial velocity for different values of mb = 0.4, n = 0.2, $\phi = \frac{2\pi}{3}$, a = 0.5, d = 0.02, t = 0.6, we = 0.01</figcaption></figure>

That axial velocity increases in the region (-0.875, 0.125) for the increasing values of wave amplitude b and the opposite behavior is observed in the region (0.125, 1). It is interesting to note that from Figure 11. An increase in non-uniform parameter m the axial velocity increases at the right and left side of the channel and at the core part of the channel of u gets decreased. In Figure 12. The impact of Weissenberg number on axial velocity is observed. It increases in the region (-0.75, 0) with an enhance in Weissenberg number and converse in behaviour is noticed in the region (0, 0.75) of the channel.

<figure><img src="image_15.png" /><figcaption>Fig 12. Axial velocity for different values of web = 0.5, n = 0.6, $\phi = \frac{2\pi}{3}$, a = 0.6, d = 0.01, t = 0.6, m = 0.5</figcaption></figure>
## 4 Conclusion

In this study, we have considered the peristaltic motion of hyperbolic tangent fluid in the presence of tapered asymmetric channel. This representation is one of the most important fluid models in the group of non-Newtonian fluids. From laboratory experiments, it is found that this model anticipates the shear thinning phenomenon very accurately and are being used mostly in laboratory experiments and industries. The exact solutions of governing equations are difficult due to highly non-linear system of partial differential equations. A standard perturbation method is used to produce semi analytic results. The impact of different physical parameters like wave amplitude parameters a and b, width of the channel d, non-uniform parameter m, Weissenberg number We, phase difference φ and power law index n are discussed. The following observations have been made. The average rise in pressure increases with the increase in Weissenberg number and amplitude parameter a and b. For the larger values of non-uniform parameter (m) pressure rise diminishes. The axial velocity decreases at the central part of the channel and increases at the right and left side of the channel for the increasing values of non-uniform parameter (m).

## Nomenclature

$a_1$ $a_2$ Amplitudes of the wave

$\lambda$ Wave length

2d Width of the channel

$\tau_{xx}$, $\tau_{yy}$ and $\tau_{yy}$ Components of extra stress tensor

$\delta$ Wave number

Re Reynolds number

We Weissenberg number

m Non uniform parameter

$\phi$ Phase difference

n Power law index

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