# Hydrodynamical Study of Creeping Maxwell Fluid Flow through a Porous Slit with Uniform Reabsorption and Wall Slip

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## Abstract

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## 1. Introduction

## 2. Formulation of the Problem

## 3. Solution of the Problem

#### 3.1. System of Equations for the 1st Order

#### 3.2. System of Equations for the 2nd Order

#### 3.3. System of Equations for the 3rd Order

- (a)
- The axial velocity is maximum along the center line of the slit as$${u}_{max}=\frac{1}{35H}\left(\frac{{Q}_{0}}{2W}-x{V}_{0}\right)\left(\frac{105{b}_{1}{b}_{2}}{2}-6{{b}_{1}}^{4}{b}_{4}{\delta}^{2}\right).$$
- (b)
- The maximum radial velocity occurs at the slit walls, i.e.,$${v}_{max}=\frac{{V}_{0}{b}_{1}}{70}(-35+105{b}_{2}+6{{b}_{1}}^{2}(-1+3{b}_{1}{b}_{3}-2{b}_{1}{b}_{4}){\delta}^{2}).$$
- (c)
- The axial flow rate is obtained using the summarized solution as$$\begin{array}{ccc}Q\left(x\right)\hfill & =& 2W{\int}_{0}^{H}\left(u(x,y)\right)dy,\hfill \\ & =& \frac{{b}_{1}({Q}_{0}-2W{V}_{0}x)}{70}\left(-35+105{b}_{2}+6{{b}_{1}}^{2}(-1+3{b}_{1}{b}_{3}-2{b}_{1}{b}_{4}){\delta}^{2}\right).\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}$$Here, the dependence of $Q\left(x\right)$ on $\delta $ is only due to the presence of $\varphi $, if $\varphi =0$ then$$\begin{array}{c}Q\left(x\right)=\left({Q}_{0}-2W{V}_{0}x\right)\hfill \end{array}$$
- (d)
- The leakage flux $q\left(x\right)$ is obtained as$$\begin{array}{ccc}q\left(x\right)\hfill & =& -{\displaystyle \frac{dQ}{dx}},\hfill \\ & =& \frac{{b}_{1}W{V}_{0}}{35}(-35+105{b}_{2}+6{{b}_{1}}^{2}(-1+3{b}_{1}{b}_{3}-2{b}_{1}{b}_{4}){\delta}^{2})).\hfill \end{array}$$
- (e)
- The fractional reabsorption ${F}_{a}$ is obtained as$$\begin{array}{ccc}{F}_{a}\hfill & =& {\displaystyle \frac{1}{Q\left(0\right)}}\left(Q\left(0\right)-Q\left(L\right)\right),\hfill \\ & =& \frac{2LW{V}_{0}}{{Q}_{0}}.\hfill \end{array}$$The contribution of $\delta $ in leakage flux $q\left(x\right)$ is only due to $\varphi $ and ${F}_{a}$ is only influenced by absorption parameter.
- (f)
- Pressure DistributionHere, we will find out the pressure for each order by utilizing Equations (30) and (31) into Equations (15)–(17) to get the pressure for the first order$$\begin{array}{ccc}{\displaystyle \frac{\partial {p}^{\left(1\right)}}{\partial x}}\hfill & =& -\frac{3{b}_{1}\mu \left({Q}_{0}-2Wx{V}_{0}\right)}{2{H}^{3}W},\hfill \end{array}$$$$\begin{array}{ccc}{\displaystyle \frac{\partial {p}^{\left(1\right)}}{\partial y}}\hfill & =& -\frac{3{b}_{1}\mu y{V}_{0}}{{H}^{3}}.\hfill \end{array}$$$${p}^{\left(1\right)}=-\frac{3{b}_{1}\mu \left(x{Q}_{0}-W{x}^{2}{V}_{0}\right)}{2{H}^{3}W}+B\left(y\right),$$Differentiating Equation (84) with respect to y, and on comparing with Equation (83), gives$${B}^{\prime}\left(y\right)=-\frac{3{b}_{1}\mu y{V}_{0}}{{H}^{3}},$$$${p}^{\left(1\right)}-{p}_{0}^{\left(1\right)}=-\frac{3{b}_{1}\mu \left(x{Q}_{0}-W{x}^{2}{V}_{0}\right)}{2{H}^{3}W}-\frac{3{b}_{1}\mu {y}^{2}{V}_{0}}{2{H}^{3}},$$The mean pressure is obtained as$$\begin{array}{ccc}{\overline{p}}^{\left(1\right)}\left(x\right)\hfill & =& \frac{1}{H}{\int}_{0}^{H}({p}^{\left(1\right)}-{p}_{0}^{\left(1\right)})dy,\hfill \\ & =& \frac{{b}_{1}\mu}{2H}\left(\frac{3x\left(Wx{V}_{0}-{Q}_{0}\right)}{{H}^{2}W}-{V}_{0}\right),\hfill \end{array}$$$$\begin{array}{ccc}\Delta \phantom{\rule{0.222222em}{0ex}}{\overline{p}}^{1}\left(L\right)\hfill & =& \left({\overline{p}}^{1}\left(0\right)-{\overline{p}}^{1}\left(L\right)\right),\hfill \\ & =& \frac{3{\mathrm{b}}_{1}\mu L\left({Q}_{0}-LW{V}_{0}\right)}{2{H}^{3}W}.\hfill \end{array}$$$$\begin{array}{ccc}{p}^{\left(2\right)}-{p}_{0}^{\left(2\right)}\hfill & =& \frac{3{b}_{1}^{2}\delta \mu}{2H}\left(\frac{3{b}_{2}x\left(-{Q}_{0}+Wx{v}_{0}\right)}{{H}^{2}W}-3{b}_{2}{v}_{0}{\left(\frac{y}{H}\right)}^{2}+2{v}_{0}{\left(\frac{y}{H}\right)}^{4}\right),\hfill \end{array}$$$$\begin{array}{ccc}{\overline{p}}^{\left(2\right)}\left(x\right)\hfill & =& \frac{3{b}_{1}^{2}\delta \mu}{10H}\left(\frac{15{b}_{2}x\left(-{Q}_{0}+Wx{v}_{0}\right)}{{H}^{2}W}+(2-5{b}_{2}){v}_{0}\right),\hfill \end{array}$$$$\begin{array}{ccc}\Delta \phantom{\rule{0.222222em}{0ex}}{\overline{p}}^{2}\left(L\right)\hfill & =& \frac{9{\mathrm{b}1}^{2}\mathrm{b}2L\delta \mu \left({Q}_{0}-LW{v}_{0}\right)}{2{H}^{3}W}\hfill \end{array}$$$$\begin{array}{ccc}{p}^{\left(3\right)}-{p}_{0}^{\left(3\right)}\hfill & =& \frac{9{b}_{1}^{3}{\delta}^{2}\mu}{70H}(-28{V}_{0}{(\frac{y}{H})}^{6}+(3\left(-35{b}_{2}^{2}+2{b}_{1}{b}_{3}\right){V}_{0}){\left(\frac{y}{H}\right)}^{2}\hfill \\ & -& \frac{3x\left({Q}_{0}-Wx{V}_{0}\right)}{{H}^{2}W}(\left(35{b}_{2}^{2}-2{b}_{1}{b}_{3}\right))),\hfill \end{array}$$$$\begin{array}{ccc}{\overline{p}}^{\left(3\right)}\left(x\right)\hfill & =& \frac{9{b}_{1}^{3}{\delta}^{2}\mu}{70H}(\frac{3x\left({Q}_{0}-Wx{V}_{0}\right)}{{H}^{2}W}(-35{b}_{2}^{2}+2{b}_{1}{b}_{3})+(-4\hfill \\ & & -35{b}_{2}^{2}+2{b}_{1}{b}_{3}){V}_{0})\hfill \end{array}$$$$\begin{array}{ccc}\Delta \phantom{\rule{0.222222em}{0ex}}{\overline{p}}^{3}\left(L\right)\hfill & =& \frac{27{b}_{1}^{3}L{\delta}^{2}\mu \left({Q}_{0}-LW{v}_{0}\right)}{70{H}^{3}W}\left(35{b}_{2}^{2}-2{b}_{1}{b}_{3}\right)\hfill \end{array}$$The summarized forms of total pressure difference, mean pressure and pressure drop are given as$$\begin{array}{ccc}p(x,y)-p(0,0)\hfill & =& \frac{3{b}_{1}\mu {v}_{0}}{70H}(-84{{b}_{1}}^{2}{\delta}^{2}{\left(\frac{y}{H}\right)}^{6}+70{b}_{1}\delta {\left(\frac{y}{H}\right)}^{4}+(-35-105{b}_{1}{b}_{2}\delta \hfill \\ & +& \left(-315{{b}_{1}}^{2}{{b}_{2}}^{2}+18{{b}_{1}}^{3}{b}_{3}\right){\delta}^{2})){\left(\frac{y}{H}\right)}^{2})+\frac{3{b}_{1}x\mu \left({Q}_{0}-Wx{v}_{0}\right)}{70{H}^{3}W}(\hfill \\ & -& 35-105{b}_{1}{b}_{2}\delta +\left(-315{{b}_{1}}^{2}{{b}_{2}}^{2}+18{{b}_{1}}^{3}{b}_{3}\right){\delta}^{2})\hfill \end{array}$$$$\begin{array}{ccc}\overline{p}\left(x\right)\hfill & =& {b}_{1}\mu (\frac{3x\left({Q}_{0}-Wx{v}_{0}\right)}{70{H}^{3}W}(-35-105{b}_{1}{b}_{2}\delta +(-315{{b}_{1}}^{2}{{b}_{2}}^{2}\hfill \\ & +& 18{{b}_{1}}^{3}{b}_{3}){\delta}^{2})+\frac{{v}_{0}}{70H}(-35+(42{b}_{1}-105{b}_{1}{b}_{2})\delta +9{{b}_{1}}^{2}(-4-35{{b}_{2}}^{2}\hfill \\ & +& 2{b}_{1}{b}_{3}){\delta}^{2}))\hfill \end{array}$$$$\begin{array}{ccc}\Delta \phantom{\rule{0.222222em}{0ex}}\overline{p}\left(L\right)\hfill & =& \frac{3{b}_{1}L\mu \left(-{Q}_{0}+LW{v}_{0}\right)}{70{H}^{3}W}(-35-105\mathrm{b}1\mathrm{b}2\delta +(-315{\mathrm{b}1}^{2}{\mathrm{b}2}^{2}\hfill \\ & +& 18{\mathrm{b}1}^{3}\mathrm{b}3){\delta}^{2})\hfill \end{array}$$
- (g)
- The wall shear stress is obtained as$$\begin{array}{ccc}{\tau}_{w}=-{\tau}_{xy}{|}_{y=H}\hfill & =& \frac{3{b}_{1}\mu \left({Q}_{0}-2Wx{V}_{0}\right)}{70{H}^{2}W}(35+35{b}_{1}(-4+3{b}_{2})\delta +9{{b}_{1}}^{2}(28+35{{b}_{2}}^{2}\hfill \\ & -& 2{b}_{1}{b}_{3}){\delta}^{2}).\hfill \end{array}$$
- (h)
- The expressions for normal stress differences are given as$$\begin{array}{ccc}{\tau}_{n}={\tau}_{xx}-{\tau}_{yy}\hfill & =& \frac{3{b}_{1}\mu {v}_{0}}{70H}(1596{{b}_{1}}^{2}{\delta}^{2}{(\frac{y}{H})}^{6}-35{b}_{1}(-4\delta +144{b}_{1}{b}_{2}{\delta}^{2}\hfill \\ & & +\frac{9{b}_{1}{\delta}^{2}\left({Q}_{0}-2Wx{v}_{0}\right){}^{2}}{{H}^{2}{W}^{2}{v}_{0}{}^{2}}){(\frac{y}{H})}^{4}+(140-420{b}_{1}{b}_{2}\delta -72{{b}_{1}}^{2}(-70{{b}_{2}}^{2}\hfill \\ & & +{b}_{1}{b}_{3}){\delta}^{2}+\frac{105{b}_{1}\delta \left({Q}_{0}-2Wx{v}_{0}\right){}^{2}}{{H}^{2}{W}^{2}{v}_{0}{}^{2}})\hfill \\ & & {(\frac{y}{H})}^{2}-140{b}_{2}-4{{b}_{1}}^{2}(315{{b}_{2}}^{3}-4{b}_{1}{b}_{4}){\delta}^{2})\hfill \end{array}$$$$\begin{array}{ccc}{\tau}_{n}{|}_{y=H}\hfill & =& \frac{3{b}_{1}\mu {v}_{0}}{70H}(-140(-1+{b}_{2})+140{b}_{1}(1-3{b}_{2})\delta +4{{b}_{1}}^{2}(399\hfill \\ & -& 315{(-2+{b}_{2})}^{2}{b}_{2}-18{b}_{1}{b}_{3}+4{b}_{1}{b}_{4}){\delta}^{2}\hfill \\ & & -\frac{105{b}_{1}\delta (-1+3{b}_{1}\delta )\left({Q}_{0}-2Wx{v}_{0}\right){}^{2}}{{H}^{2}{W}^{2}{v}_{0}^{2}}),\hfill \end{array}$$$$\begin{array}{ccc}{\tau}_{n}{|}_{y=0}\hfill & =& \frac{3{b}_{1}\mu {v}_{0}}{70H}(-140{b}_{2}+4{{b}_{1}}^{2}\left(-315{{b}_{2}}^{3}+4{b}_{1}{b}_{4}\right){\delta}^{2}).\hfill \end{array}$$

## 4. Discussion

## 5. Conclusions

- If the Maxwell fluid parameter $\delta =0$ and slip parameter $\varphi =0$, then results obtained by Haroon [15] are recovered.
- The axial velocity $u\left(\zeta \right)$ of creeping Maxwell fluid decreases downstream along the slit length on increasing porosity K and also for increasing values of K, backward flow can be seen near the exit region of the slit.
- The axial velocity profile has an increasing behavior near the slit walls and its decreasing trend is observed along the centerline of the slit with increasing $\delta $.
- The shear thickening and thinning behavior of the Maxwell fluid is observed along centerline and near the walls of the slit, respectively.
- Along the slit length, the magnitude of $u\left(\zeta \right)$ decreases as the fluid moves from the entrance to exit region of the slit.
- An increase in axial velocity of a creeping Maxwell fluid is observed due to the increasing value of $\varphi $.
- The slip parameter $\varphi $ significantly influenced the magnitude of axial and radial velocities in comparison to other parameters.
- A decreasing trend in pressure profile for increasing values of $\varphi $ and $\delta $, whereas pressure is increasing with increasing porosity parameter K.
- The wall shear stress ${\tau}_{w}$ is increasing significantly by increasing $\delta $ and $\varphi $, but with K and ${\tau}_{w}$ decreasing.
- The contribution of $\delta $ in axial flow rate and leakage flux is only due to the presence of a slip parameter $\varphi $.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

u, v | Components of velocity field |

x, y | Cartesian coordinates |

L | Length of the slit |

H | Width of slit |

W | Breadth of slit |

${V}_{0}$ | Uniform velocity |

Q | Axial flow rate at any point x |

$\mu $ | Coefficient of viscosity |

$\lambda $ | fluid relaxation time |

$\delta $ | Maxwell fluid parameter (Deborah number) |

K | Porosity parameter |

$\varphi $ | Slip parameter |

$\psi $ | Stream function |

$p(x,y)$ | Pressure in the slit |

${\tau}_{w}$ | Wall shear stress |

${\tau}_{n}$ | Normal stresses difference |

$q\left(x\right)$ | Leakage flux |

${F}_{a}$ | Fractional reabsorption |

## References

- Nikolay, V. Desalination Engineering: Planning and Design; McGraw-Hill Professional: New York, NY, USA, 2013. [Google Scholar]
- Espedal, M.S.; Mikelic, A. Filtration in Porous Media and Industrial Application: Lectures Given at the 4th Session of the Centro Internazionale Matematico Estivo (CIME) Held in Cetraro, Italy, 24–29 August 1998; Springer: New York, NY, USA, 2007. [Google Scholar]
- Macey, R.I. Pressure flow patterns in a cylinder with reabsorbing walls. Bull. Math. Biophys.
**1963**, 25, 1–9. [Google Scholar] [CrossRef] - Macey, R.I. Hydrodynamics in the renal tubule. Bull. Math. Biophys.
**1965**, 27, 117. [Google Scholar] [CrossRef] [PubMed] - Marshall, E.; Trowbridge, E. Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule. Bull. Math. Biol.
**1974**, 36, 457–476. [Google Scholar] [CrossRef] - Marshall, E.; Trowbridge, E.; Aplin, A. Flow of a Newtonian fluid between parallel flat permeable plates, The application to a flat plate hemodialyzer. Math. Biosci.
**1975**, 27, 119–139. [Google Scholar] [CrossRef] - Berman, A.S. Laminar flow in channels with porous walls. J. Appl. Phys.
**1953**, 24, 1232–1235. [Google Scholar] [CrossRef] - Sellars, J.R. Laminar flow in channels with porous walls at high suction Reynolds numbers. J. Appl. Phys.
**1955**, 26, 489–490. [Google Scholar] [CrossRef][Green Version] - Yuan, S. Further investigation of laminar flow in channels with porous walls. J. Appl. Phys.
**1956**, 27, 267–269. [Google Scholar] [CrossRef] - Wah, T. Laminar flow in a uniformly porous channel. Aeronaut. Q.
**1964**, 15, 299–310. [Google Scholar] [CrossRef] - Terrill, R. Laminar flow in a uniformly porous channel with large injection. Aeronaut. Q.
**1965**, 16, 323–332. [Google Scholar] [CrossRef] - Karode, S.K. Laminar flow in channels with porous walls revisited. J. Membr. Sci.
**2001**, 191, 237–241. [Google Scholar] [CrossRef] - Siddiqui, A.M.; Haroon, T.; Shahzad, A. Hydrodynamics of viscous fluid through porous slit with linear absorption. Appl. Math. Mech.
**2016**, 37, 361–378. [Google Scholar] [CrossRef] - Haroon, T.; Siddiqui, A.M.; Shahzad, A. Stokes flow through a slit with periodic reabsorption: An application to renal tubule. Alex. Eng. J.
**2016**, 55, 1799–1810. [Google Scholar] [CrossRef][Green Version] - Haroon, T.; Siddiqui, A.; Shahzad, A. Creeping flow of viscous fluid through a proximal tubule with uniform reabsorption: A mathematical study. Appl. Math. Sci.
**2016**, 10, 795–807. [Google Scholar] [CrossRef] - Haroon, T.; Siddiqui, A.; Shahzad, A.; Smeltzer, J. Steady creeping slip flow of viscous fluid through a permeable slit with exponential reabsorption1. Appl. Math. Sci.
**2017**, 11, 2477–2504. [Google Scholar] - Rajagopal, K. On the creeping flow of the second-order fluid. J. Non-Newton. Fluid Mech.
**1984**, 15, 239–246. [Google Scholar] [CrossRef] - Ullah, H.; Sun, H.; Siddiqui, A.M.; Haroon, T. Creeping flow analysis of slightly non-Newtonian fluid in a uniformly porous slit. J. Appl. Anal. Comput.
**2019**, 9, 140–158. [Google Scholar] [CrossRef] - Ullah, H.; Siddiqui, A.M.; Sun, H.; Haroon, T. Slip effects on creeping flow of slightly non-Newtonian fluid in a uniformly porous slit. J. Braz. Soc. Mech. Sci. Eng.
**2019**, 41, 412. [Google Scholar] [CrossRef] - Kahshan, M.; Siddiqui, A.; Haroon, T. A micropolar fluid model for hydrodynamics in the renal tubule. Eur. Phys. J. Plus
**2018**, 133, 546. [Google Scholar] [CrossRef] - Kahshan, M.; Lu, D.; Siddiqui, A. A Jeffrey fluid model for a porous-walled channel: Application to flat plate dialyzer. Sci. Rep.
**2019**, 9, 1–18. [Google Scholar] [CrossRef][Green Version] - Lu, D.; Kahshan, M.; Siddiqui, A. Hydrodynamical study of micropolar fluid in a porous-walled channel: Application to flat plate dialyzer. Symmetry
**2019**, 11, 541. [Google Scholar] [CrossRef][Green Version] - Langlois, W. A Recursive Approach to the Theory of Slow, Steady-State Viscoelastic Flow. Trans. Soc. Rheol.
**1963**, 7, 75–99. [Google Scholar] [CrossRef] - Langlois, W. The recursive theory of slow viscoelastic flow applied to three basic problems of hydrodynamics. Trans. Soc. Rheol.
**1964**, 8, 33–60. [Google Scholar] [CrossRef] - Léger, L.; Hervet, H.; Massey, G.; Durliat, E. Wall slip in polymer melts. J. Phys. Condens. Matter
**1997**, 9, 7719. [Google Scholar] [CrossRef] - Atwood, B.; Schowalter, W. Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit. Rheol. Acta
**1989**, 28, 134–146. [Google Scholar] [CrossRef] - Beavers, G.S.; Joseph, D.D. Boundary conditions at a naturally permeable wall. J. Fluid Mech.
**1967**, 30, 197–207. [Google Scholar] [CrossRef] - Saffman, P.G. On the boundary condition at the surface of a porous medium. Stud. Appl. Math.
**1971**, 50, 93–101. [Google Scholar] [CrossRef] - Beavers, G.S.; Sparrow, E.M.; Magnuson, R.A. Experiments on coupled parallel flows in a channel and a bounding porous medium. J. Basic Eng.
**1970**, 92, 843–848. [Google Scholar] [CrossRef] - Kohler, J.T. An Investigation of Laminar Flow through a Porous Walled Channel: I. A Perturbation Solution Assuming Slip at the Permeable Wall. II. An Experimental Measurement of the Velocity Distribution Utilizing a Dye Tracer Technique. Ph.D. Thesis, University of Massachusetts Amherst, Amherst, MA, USA, 1974. [Google Scholar]
- Mikelic, A.; Jäger, W. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math.
**2000**, 60, 1111–1127. [Google Scholar] [CrossRef] - Rao, I.; Rajagopal, K. The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mech.
**1999**, 135, 113–126. [Google Scholar] [CrossRef] - Elshahed, M. Blood flow in capillary under starling hypothesis. Appl. Math. Comput.
**2004**, 149, 431–439. [Google Scholar] [CrossRef] - Singh, R.; Laurence, R.L. Influence of slip velocity at a membrane surface on ultrafiltration performance I. Channel flow system. Int. J. Heat Mass Transf.
**1979**, 22, 721–729. [Google Scholar] [CrossRef] - Makinde, O.; Osalusi, E. MHD steady flow in a channel with slip at the permeable boundaries. Rom. J. Phys.
**2006**, 51, 319. [Google Scholar] - Eldesoky, I.M. Unsteady MHD pulsatile blood flow through porous medium in a stenotic channel with slip at the permeable walls subjected to time dependent velocity (injection/suction). In Proceedings of the International Conference on Mathematics and Engineering Physics, Kobry Elkobbah, Egypt, 29–31 May 2014; Volume 7, pp. 1–25. [Google Scholar]
- El-Shehawy, E.; El-Dabe, N.; El-Desoky, I. Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid. Acta Mech.
**2006**, 186, 141–159. [Google Scholar] [CrossRef] - Ellahi, R. Effects of the slip boundary condition on non-Newtonian flows in a channel. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 1377–1384. [Google Scholar] [CrossRef] - Hron, J.; Le Roux, C.; Málek, J.; Rajagopal, K. Flows of incompressible fluids subject to Naviers slip on the boundary. Comput. Math. Appl.
**2008**, 56, 2128–2143. [Google Scholar] [CrossRef][Green Version] - Hayat, T.; Khan, M.; Ayub, M. The effect of the slip condition on flows of an Oldroyd 6-constant fluid. J. Comput. Appl. Math.
**2007**, 202, 402–413. [Google Scholar] [CrossRef][Green Version] - Rajagopal, K.R.; Srinivasa, A.R. A thermodynamic frame work for rate type fluid models. J. Non-Newton. Fluid Mech.
**2000**, 88, 207–227. [Google Scholar] [CrossRef] - Choi, J.; Rusak, Z.; Tichy, J. Maxwell fluid suction flow in a channel. J. Non-Newton. Fluid Mech.
**1999**, 85, 165–187. [Google Scholar] [CrossRef] - Sadeghy, K.; Najafi, A.H.; Saffaripour, M. Sakiadis flow of an upper-convected Maxwell fluid. Int. J. Non-Linear Mech.
**2005**, 40, 1220–1228. [Google Scholar] [CrossRef] - Abbas, Z.; Sajid, M.; Hayat, T. MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel. Theor. Comput. Fluid Dyn.
**2006**, 20, 229–238. [Google Scholar] [CrossRef] - Bhatti, K.; Siddiqui, A.M.; Bano, Z. Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube. Mathematics
**2020**, 8, 1170. [Google Scholar] [CrossRef]

**Figure 2.**Behavior of $u\left(\zeta \right)$ with K at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $\varphi =\delta =0.0$.

**Figure 3.**Behavior of $u\left(\zeta \right)$ with K at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $\varphi =0.2,\delta =0.0$.

**Figure 4.**Behavior of $u\left(\zeta \right)$ with K at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $\varphi =\delta =0.2$.

**Figure 5.**Behavior of $u\left(\zeta \right)$ with $\delta $ at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $K=0.2,\varphi =0.0$.

**Figure 6.**Behavior of $u\left(\zeta \right)$ with $\delta $ at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $K=0.2,\varphi =0.1$.

**Figure 7.**Behavior of $u\left(\zeta \right)$ with $\varphi $ at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $K=0.2,\delta =0.0$.

**Figure 8.**Behavior of $u\left(\zeta \right)$ with $\varphi $ at (

**a**) $x=0.1$, (

**b**) $x=0.5$, (

**c**) $x=0.9$, when $K=0.2,\delta =0.2$.

**Figure 9.**Behavior of $v\left(\zeta \right)$ with (

**a**) K when $\varphi =\delta =0.2,$ (

**b**) $\delta $ when $K=\varphi =0.2,$ (

**c**) $\delta $ when $K=\varphi =0.2$.

**Figure 10.**Behavior of $p(x,0)-p(0,0)$ due to (

**a**) K when $\varphi =\delta =0.2,$ (

**b**) $\delta $ when $K=\delta =0.2,$ (

**c**) $\varphi $ when $K=\delta =0.2$.

**Figure 11.**Behavior of ${\tau}_{w}$ with (

**a**) K when $\varphi =\delta =0.2,$ (

**b**) $\delta $ when $K=\delta =0.2,$ (

**c**) $\varphi $ when $K=\delta =0.2$.

**Figure 12.**Behavior of ${\tau}_{n}$ due to (

**a**) K when $\varphi =\delta =0.2,$ (

**b**) $\delta $ when $K=\delta =0.2,$ (

**c**) $\varphi $ when $K=\delta =0.2$.

**Figure 13.**Behavior of $Q\left(x\right)$ due to (

**a**) K when $\varphi =\delta =0.2,$ (

**b**) $\delta $ when $K=\delta =0.2,$ (

**c**) $\varphi $ when $K=\delta =0.2$.

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**MDPI and ACS Style**

Ullah, H.; Lu, D.; Siddiqui, A.M.; Haroon, T.; Maqbool, K. Hydrodynamical Study of Creeping Maxwell Fluid Flow through a Porous Slit with Uniform Reabsorption and Wall Slip. *Mathematics* **2020**, *8*, 1852.
https://doi.org/10.3390/math8101852

**AMA Style**

Ullah H, Lu D, Siddiqui AM, Haroon T, Maqbool K. Hydrodynamical Study of Creeping Maxwell Fluid Flow through a Porous Slit with Uniform Reabsorption and Wall Slip. *Mathematics*. 2020; 8(10):1852.
https://doi.org/10.3390/math8101852

**Chicago/Turabian Style**

Ullah, Hameed, Dianchen Lu, Abdul Majeed Siddiqui, Tahira Haroon, and Khadija Maqbool. 2020. "Hydrodynamical Study of Creeping Maxwell Fluid Flow through a Porous Slit with Uniform Reabsorption and Wall Slip" *Mathematics* 8, no. 10: 1852.
https://doi.org/10.3390/math8101852