# Flow in a two dimensional channel with deforming and peristaltically moving walls

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## Abstract

In this paper, effects of deforming walls on peristaltic flow in a two dimensional channel have been investigated. The two dimensional form of the governing equations is simplified by using appropriate transformations and well established approximations, which are used extensively for solution of such models. The transformations are designed so that the complex problem is reduced into an ordinary differential equation (ODE). New and simple non-linear ODE is formed in view of adopted procedures and techniques. Its solutions are exactly matched with the solutions of classical problems. Solutions of the final problem are provided for small values of the surface expansion (contraction) ratio and Reynolds number with the help of the perturbation technique and non-linear shooting method. The velocity field, pressure and shear stress are evaluated analytically and numerically. Meanwhile, effects of all parameters are observed on the velocity field, pressure rise per wavelength and shear stress profiles with the help of tables and different figures. Excellent agreement between solutions is found. Current results are apparently matched with the classical problems of peristaltic flow in deforming and non-deforming walls.

## Keywords

Surface deformation Peristaltic pumping## 1 Introduction

Peristalsis is the mechanism of fluid transport through elastic channels, chambers and pipes by means of sinusoidal waves. There are many agents that are producing fluid motion in which some sources are very active and prominent. For example pressure gradients and surface (solid body, wall and plate) motion (deformation, peristaltic motion) may cause fluid motion. The phenomenon of peristalsis has many applications in biology and engineering. In many physiological systems, the peristaltic motion of different body parts are producing fluid transportation. The ureter’s muscles undergo repeated contraction (expansion) together with peristaltic movement and thoroughly pump urine from kidney into the bladder and similarly the spermatozoa moves in this way in ducts. On the other hand, the contraction and relaxation of muscles combined with peristaltic motion is also responsible for digestion of food. Besides these, it is widely used in manipulating the biomedical devices and machines, house pumps, finger and roller pumps that are used to force blood and other fluids. Due to its numerous applications, the phenomenon has attracted many scientists, engineers, mathematicians and tremendous research have been carried out on peristaltic pumping [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

The first attempt is made by Shapiro [1] and he solved the peristaltic flow problem for a two dimensional tube with assumptions of small Reynolds number and long wave length. He assumed a peristaltic wave of very long wave length and the aspect ratio of wave length to the diameter of tube is greater than four. The closed form solutions of Shapiro for the modeled equations are very famous and obviously reduced it to well-known Poiseuille solutions. The problem of peristaltic flow is further analyzed by assuming the sinusoidal waves [6] and arbitrary shapes waves [7]. The theoretical data is experimentally verified in [8] and latest techniques and methods are provided. These analysis are propitious to the fundamental concepts which can help us in analyzing the problems and provide solutions to a new hydrodynamic systems. The peristaltic flow problem for two dimensional channel is solved numerically see [9, 10] and found excellent agreement between the experimental and theoretical inquiries.

A lot of research articles are appeared in the literature witch presents the flow between deforming and porous walls [11, 12, 13, 14, 15]. Majdalani [12] extended the scope of deformable channels flow and used similarity variables for the solution of model problem. Matebese [16] studied the flow model of a deformable channel where the gap between walls in filled with porous medium and assumed a variable magnetic field. A more complex model is studied by Khan Marwat and Asghar [17] and they investigated the coupled effects of wall contraction (expansion) and peristalsis in a deformable channel and recovered the solutions of [1] for zero wall deformation ratio (α). A lot of research work are available for static and moving walls for both Newtonin, non-Newtonin and Jeffery fluids in nano or in micropolar fluid, see [18, 19, 20, 21, 22, 23]. Akram et al. [18] analyzed the Peristaltic pumping of a Jeffrey fluid with double-diffusive convection in nanofluids in the presence of inclined magnetic field. Sadaf et al. [19] and Shehzadi and Nadeem [20] discussed the fluid flow between walls for peristaltic transportation in a chanel. Sadaf et al. [19] discuss this phenominan for Nano fluid with MHD effects while Shehzadi and Nadeem [20] consider a porous medium with porous boundary conditions for this study with out MHD effects.

For the best of auther’s informations, we include some other slassical work in this section which are discussed by [24, 25]. Saleem et al. [24] modeled the problem associated with fow, heat, and mass transfer features of gyrotactic microorganisms containing MHD Jefrey fuid over a vertical cone with nanoparticles. Qasim et al. [25] examined the effects of nonlinear thermal radiation on the flow of Jeffrey fluid over a radially stretching sheet with variable thermal conductivity.

Here we present a model of fluid flow between peristaltically moving and deforming walls. More precisely, we study the consequences of deformable surfaces on peristaltic flow in a two dimensional channel. Note that the channel is contracting (expanding) with time. A set of transformations is introduced such that the generalized wall’s geometry is invoked in the new variables. In view of these new variables, the governing equations are converted into the simplest PDE’s. Later on, the wall geometry is specified and modified by the well-known peristaltic and deforming walls shapes. Besides that, the assumptions and approximation of long wavelength and small deformation ratio are employed so that the governing PDE’s are converted into an ODE. The ODE is new, simple and gives the results of [17] for fixed and special value of the parameters. Perturbation and numerical methods are used for the solution of modeled problem. The flow field properties have been evaluated by numerical means and new results are found for velocity components, pressure and shear stress. The profiles of velocity field, pressure rise per wavelength and shear stress are discussed in detail for different values of Reynolds number and wall deformation ratio. Note that the behavior of all these quantities is recorded in different tables. The two different solutions are compared in tables and graphs. The ranges of parameters are so chosen for which error between the solutions is very negligible. New results are also matched with [17] and further analysis of the transformation in hand, may give rise to new research problems. The advantage and usefulness of the transformation and approximation is very clear and obviously converts the complex problem into a simplest one. The last problem is easy to solve and provides accurate and authentic results.

## 2 Mathematical formulation

*H′*\(\to\) 0, \(H^{2} \to\) 0. Moreover, it is also assumed that \(\alpha\) is of small order. All these assertions are invoked into Eq. (8) and finally we get a most suitable candidate for the vorticity function:

## 3 Perturbation solution of the problem

Upon substituting the assumed solutions (15)–(16) into Eqs. (12), (13), we get the following differential systems:

### 3.1 Zeroth-order system

### 3.2 First-order system

## 4 Results

The governing equation are reduced into a simplest ODE with the help of useful transformations and established approximations. The final problem contains several dimensionless parameters and perturbation method is used for the solution of this problem for small value of the parameters involved in the problem. Moreover, a numerical solution of final problem is also formed. Effects of all the parameters i.e. surface deformation and peristaltic motion are seen on the flow characteristics and field quantities. Response of the main flow to the different parameters \(s\) and \(Re\) is also noted in different graphs. Both the velocity components, pressure, pressure rise (drop) per volume flow rate, and shear stress profiles are presented in different figures. A set of different results against different parameters is formed and the new observations are presented in graphs and tables.

Besides that the two different solutions for each of these quantities are compared in different tables and excellent results are found.

### 4.1 Axial velocity

*c*,

*h,*\(F_{00}\), \(F_{01}\), \(F_{10}\), and \(F_{11}\), \(Re\) has values in the interval 0 ≤ \(Re\) ≤ 0.5 and wall expansion (contraction) \(s\) has small values. Note that the solution strictly varies with \(F\) and \(s\) small variation in \(F\) gives rise to significant changes in error between the two solutions. Any small changes in \(s\) and \(Re\) creates variation in \(F\). For manipulated choices of \(s\), \(Re\) and different parts of \(F\), it is suggested that \(F\) must be unity or smaller in order to produce more accurate results. In view of these restrictions, the analytical solution provides good results for bit larger values of the perturbation parameters involved in the problem.

The two solutions are presented for \(\varvec{u}_{1}\) at \(\varvec{\eta}\) = 0.4949, when \(\varvec{s}\) = 0.1 and 0.5

| Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | |

0.0025 | 2.6578 | 3.2789 | 2.65759 | 3.27406 | 0.00790 | 0.147829 |

0.0625 | 2.7580 | 3.4166 | 2.75768 | 3.41110 | 0.01160 | 0.161238 |

0.2500 | 3.0724 | 3.8489 | 3.07045 | 3.83937 | 0.06351 | 0.248218 |

0.4750 | 3.4526 | 4.3730 | 3.44577 | 4.35329 | 0.19821 | 0.452761 |

0.5000 | 3.4951 | 4.4316 | 3.48748 | 4.41040 | 0.21855 | 0.480682 |

The two solutions are presented for \(u_{1}\) at \(\eta\) = 0.4949 when \(s\) = − 0.1 and − 0.5

| Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = − 0.1 | \(s\) = − 0.5 | \(s\) = − 0.1 | \(s\) = − 0.5 | \(s\) = − 0.1 | \(s\) = − 0.5 | |

0.0025 | 2.3496 | 1.7377 | 2.34936 | 1.73289 | 0.010216 | 0.277571 |

0.0625 | 2.4312 | 1.7828 | 2.43096 | 1.77754 | 0.009872 | 0.295915 |

0.2500 | 2.6874 | 1.9240 | 2.68599 | 1.91706 | 0.052495 | 0.362013 |

0.4750 | 2.9969 | 2.0942 | 2.99201 | 2.08449 | 0.163435 | 0.465821 |

0.5000 | 3.0314 | 2.1131 | 3.02602 | 2.10310 | 0.177791 | 0.475488 |

### 4.2 Normal velocity

The two solutions are shown for \(v_{1}\) at \(\eta\) = 0.2424 when \(s\) = 0.1 and 0.5

\(Re\) | Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | |

0.0025 | − 0.1695 | − 0.2258 | − 0.169502 | − 0.225945 | 0.00118 | 0.05867 |

0.0625 | − 0.1788 | − 0.2385 | − 0.178824 | − 0.238640 | 0.01342 | 0.11103 |

0.2500 | − 0.2079 | − 0.2780 | − 0.207956 | − 0.278309 | 0.02693 | 0.11103 |

0.4750 | − 0.2427 | − 0.3252 | − 0.242915 | − 0.325913 | 0.08851 | 0.21877 |

0.5000 | − 0.2465 | − 0.3304 | − 0.246799 | − 0.331202 | 0.12115 | 0.24215 |

The two solutions are shown for \(v_{1}\) at \(\eta\) = 0.2424 when \(s\) = − 0.1 and − 0.5

\(Re\) | Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = − 0.1 | \(s\) = − 0.5 | \(s\) = − 0.1 | \(s\) = − 0.5 | \(s\) = − 0.1 | \(s\) = − 0.5 | |

0.0025 | − 0.1413 | − 0.0847 | − 0.141280 | − 0.0848361 | 0.014156 | 0.016043 |

0.0625 | − 0.1489 | − 0.0889 | − 0.148916 | − 0.0891006 | 0.010744 | 0.225139 |

0.2500 | − 0.1727 | − 0.1022 | − 0.172780 | − 0.1024270 | 0.046302 | 0.221621 |

0.4750 | − 0.2012 | − 0.1181 | − 0.201416 | − 0.1184190 | 0.107241 | 0.269382 |

0.5000 | − 0.2044 | − 0.1199 | − 0.204598 | − 0.1201960 | 0.096775 | 0.246264 |

### 4.3 Pressure distributions is compared with classical model

The two solutions are shown for \(\left( {\frac{{\partial p_{n} }}{\partial \eta }} \right)^{*}=\frac{{h^{2} }}{{Re\rho \upsilon^{2} }}\left( {\frac{{\partial p_{n} }}{\partial \eta }} \right)\)

\(\eta\) | Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | \(s\) = 0.1 | \(s\) = 0.5 | |

0.0100 | − 0.8020 | − 0.8118 | − 0.779433 | − 0.802667 | 2.81385 | 1.12504 |

0.0240 | − 0.7584 | − 0.7673 | − 0.741064 | − 0.760378 | 2.2659 | 0.9022 |

0.4760 | − 0.0021 | − 0.0021 | − 0.002310 | − 0.002190 | 10 | 4.2858 |

0.5240 | − 0.0021 | − 0.0021 | − 0.002310 | − 0.002190 | 10 | 4.2858 |

0.9760 | − 0.7584 | − 0.7673 | − 0.741064 | − 0.760378 | 2.2859 | 0.9022 |

## 5 Wall shear stress

The two solutions are shown for shear stresses when \(\upeta\) = 0.2525, \({\text{s}}\) = 0.3, − 0.3

\(Re\) | Numerical solution | Perturbation solution | Percent error | |||
---|---|---|---|---|---|---|

\(s\) = 0.3 | \(s\) = − 0.3 | \(s\) = 0.3 | \(s\) = − 0.3 | \(s\) = 0.3 | \(s\) = − 0.3 | |

0.0025 | 3.4658 | 2.4934 | 3.46008 | 2.48758 | 0.165314 | 0.233962 |

0.0625 | 3.5879 | 2.5567 | 3.58118 | 2.55030 | 0.187648 | 0.250951 |

0.2500 | 3.9749 | 2.7567 | 3.95962 | 2.74629 | 0.385896 | 0.379057 |

0.4750 | 4.4528 | 3.0011 | 4.41374 | 2.98148 | 0.884960 | 0.658062 |

## 6 Conclusion

Viscous fluid flow in deformable and peristaltically moving walls in studied and different flows field properties are examined for small values of deformation rate and wave number. Set of variable is defined in such a way that the governing partial differential equation are transformed into a single ODE’s with boundary conditions. New variables are formed and used for the simplification of the equation of motion. Well-established approximations and assumption are used for further simplification of modeled equations. These simplification procedures helped us to simplify the governing PDE’s and converted them into a boundary value ODE, which is simplest one and easy to solve. The joint effects of physical parameters i.e. Reynolds number (\(Re\)) and deformation ratio (\(s\)) of the channel have been observed on the flow quantities. All results in the current analysis give rise to considerable influence on flow-field characters. Abrupt changes in pressure rise against volume flow rate (\(Q\)) are noted for large \(Re\) and profiles of [17] are also recovered. The solution of the last ODE is attempted by two methods. The two different solution are matched in different figures and tables and the two solution are same for small value of \(Re\) and \(s\). The problem of peristaltic flow in a channel of deformable walls is formulated in a convenient form with the help of established mathematical simulations and gives the bench mark solutions. Moreover, effects of peristalsis and deformation are analyzed independently and jointly.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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