# A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

## 3. The Analytical Solution

## 4. Results and Discussion

_{c}. The presence of a larger slip drags the separation curve towards the slit. The viscous fluid flow in a permeable medium with slip in a stretching boundary is quite different from that of a contracting boundary.

_{C}with fixed Brinkman ratio result in exactly the opposite.

_{1}results in increasing shear at the wall boundary. In the case of mass injection, the shear wall boundary decreases faster for a smaller value of second-order slip parameter. Interestingly, however, the increasing value of Brinkman ratio leads to decreasing shear wall boundary as seen in Figure 8a.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Fisher, B.G. Extrusion of Plastics, 3rd ed.; Newnes-Butterworld: London, UK, 1976. [Google Scholar]
- Brinkman, H.C. On the permeability of the media consisting of closely packed porousparticles. Appl. Sci. Res.
**1947**, 1, 81–86. [Google Scholar] [CrossRef] - Darcy, H. Les Fontaines Publiques De La Ville De Dijon; Victor Dalmont: Paris, France, 1856. [Google Scholar]
- Forchheimer, P. Wasserbewegung Durch Boden; Zeitschrift des Vereines Deutscher In-Geneieure: Düsseldorf, Germany, 1901; Volume 45, pp. 1782–1788. [Google Scholar]
- Biot, M.A. Mechanics of Deformation and Acoustic Propagation in Porous Media. J. Appl. Phys.
**1962**, 33, 1482–1498. [Google Scholar] [CrossRef] - Truesdell, C. Sulla basi della thermomechanical. Rend. Lincei
**1957**, 22a, 158–166. [Google Scholar] - Truesdell, C. Sulla basi della thermomechanical. Rend. Lincei
**1957**, 22b, 33–38. [Google Scholar] - Chakrabarti, A.; Gupta, A.S. Hydromagnetic flow and heat transfer over a stretching sheet. Q. Appl. Math.
**1979**, 37, 73–78. [Google Scholar] [CrossRef] [Green Version] - Gupta, P.S.; Gupta, A.S. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng.
**1977**, 55, 744–746. [Google Scholar] [CrossRef] - Fang, T. Boundary layer flow over a shrinking sheet with power-law velocity. Int. J. Heat Mass Transf.
**2008**, 51, 5838–5843. [Google Scholar] [CrossRef] - Milavcic, M.; Wang, C.Y. Viscous flow due to a shrinking sheet. Q. Appl. Math.
**2006**, 64, 283–290. [Google Scholar] [CrossRef] [Green Version] - Nakayama, A. PC-Aided Numerical Heat Transfer and Convective Flow; CRC Press: Boca Raton, FL, USA, 1995. [Google Scholar]
- Sakiadis, B.C. Boundary-layer behavior on continuous solid surface. AIChE J.
**1961**, 7, 26–28. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary-layer behavior on continuous solid surfaces. II. The boundary layer on a continuous flat surface. AIChE J.
**1961**, 7, 221–225. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Tao, L. Mechanics of Mixtures; World Scientific: Singapore, 1995. [Google Scholar]
- Givler, R.C.; Altobelli, S.A. A Determination of the Effective Viscosity for the Brinkman-Forchheimer Flow Model. J. Fluid Mech.
**1994**, 258, 355–370. [Google Scholar] [CrossRef] - Shao, Q.; Fahs, M.; Hoteit, H.; Carrera, J.; Ackerer, P.; Younes, A. A 3D semi-analytical solution for density-driven flow in porous media. Water Res. Res.
**2018**, 54, 10094–10116. [Google Scholar] [CrossRef] - Lesinigo, M.; D’Angelo, C.; Quarteroni, A. A multiscale Darcy-Brinkman model for fluid flow in fractured porous media. Numer. Math.
**2011**, 117, 717–752. [Google Scholar] [CrossRef] - Murali, K.; Naidu, V.K.; Venkatesh, B. Solution of Darcy-Brinkman-Forchheimer Equation for Irregular Flow Channel by Finite Elements Approach. J. Phys. Conf. Ser.
**2019**, 1172, 012033. [Google Scholar] [CrossRef] - Kumaran, V.; Tamizharasi, R. Pressure in MHD/Brinkman flow past a stretching sheet. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 4671–4681. [Google Scholar] - Nield, D.A.; Bejan, A. Convection in Porous Media; Springer Verlag Inc.: New York, NY, USA, 1998. [Google Scholar]
- Nield, D.A.; Kuznetsov, A.V. Forced convection in porous media: Transverse heterogeneity effects and thermal development. In Handbook of Porous Media, 2nd ed.; Vafai, K., Ed.; Taylor and Francis: New York, NY, USA, 2005; pp. 143–193. [Google Scholar]
- Nield, D.A. The modeling of viscous dissipation in a saturated porous medium. J. Heat Transf.
**2007**, 129, 1459–1463. [Google Scholar] [CrossRef] - Nield, D.A.; Bejan, A. Convection in Porous Media, 4th ed.; Springer: New York, NY, USA, 2013. [Google Scholar]
- Pantokratoras, A. Flow adjacent to a stretching permeable sheet in a Darcy-Brinkman porous medium. Transp. Porous Med.
**2009**, 80, 223–227. [Google Scholar] [CrossRef] - Pop, I.; Ingham, D.B. Flow past a sphere embedded in a porous medium based on the Brinkman model. Int. Commun. Heat Mass Transf.
**1996**, 23, 865–874. [Google Scholar] [CrossRef] - Pop, I.; Na, T.Y. A note on MHD flow over a stretching permeable surface. Mech. Res. Commun.
**1998**, 25, 263–269. [Google Scholar] [CrossRef] - Pop, I.; Cheng, P. Flow past a circular cylinder embedded in a porous medium based on the Brinkman model. Int. J. Eng. Sci.
**1992**, 30, 257–262. [Google Scholar] [CrossRef] - Wang, C.Y. Darcy-Brinkman Flow with Solid Inclusions. Chem. Eng. Commun.
**2010**, 197, 261–274. [Google Scholar] [CrossRef] - Srinivasan, S.; Rajagopal, K.R. A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations. Int. J. Non-Linear Mech.
**2014**, 58, 162–166. [Google Scholar] [CrossRef] - Ingham, D.B.; Pop, I. Transport in Porous Media; Pergamon: Oxford, UK, 2002. [Google Scholar]
- Magyari, E.; Keller, B. Exact solutions for self-similar boundary-layer flows induced by permeable stretching surfaces. Eur. J. Mech. B
**2000**, 19, 109–122. [Google Scholar] [CrossRef] - Mastroberardino, A.; Mahabaleshwar, U.S. Mixed convection in viscoelastic flow due to a stretching sheet in a porous medium. J. Porous Media
**2013**, 16, 483–500. [Google Scholar] [CrossRef] - Siddheshwar, P.G.; Mahabaleshwar, U.S. Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. Int. J. Non-Linear Mech.
**2005**, 40, 807–820. [Google Scholar] [CrossRef] [Green Version] - Siddheshwar, P.G.; Chan, A.; Mahabaleshwar, U.S. Suction-induced magnetohydrodynamics of a viscoelastic fluid over a stretching surface within a porous medium. IMA J. Appl. Math.
**2014**, 79, 445–458. [Google Scholar] [CrossRef] - Tamayol, A.; Hooman, K.; Bahrami, M. Thermal analysis of flow in a porous medium over a permeable stretching wall. Transp. Porous Media
**2010**, 85, 661–676. [Google Scholar] [CrossRef] - Shao, Q.; Fahs, M.; Younes, A.; Makradi, A. A High Accurate Solution for Darcy-Brinkman Double-Diffusive Convection in Saturated Porous Media. J. Numer. Heat Transf. Part B Fundam.
**2015**, 69, 26–47. [Google Scholar] [CrossRef] - Navier, C.L.M.H. Mémoire sur les lois du mouvement des fluids. Mém. Acad. R. Sci. Inst. Fr.
**1827**, 6, 389–440. [Google Scholar] - Brinkman, H.C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res.
**1949**, 1, 27–34. [Google Scholar] [CrossRef] - Mahabaleshwar, U.S.; Nagaraju, K.R.; Kumar, P.N.V.; Baleanu, D.; Lorenzini, G. An exact analytical solution of the unsteady magnetohydrodynamics nonlinear dynamics of laminar boundary layer due to an impulsively linear stretching sheet. Contin. Mech. Thermodyn.
**2017**, 29, 559–567. [Google Scholar] [CrossRef] - Rajagopal, K.R. On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Model. Methods Appl. Sci.
**2007**, 17, 215–252. [Google Scholar] [CrossRef] - Vafai, K.; Tien, C.L. Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf.
**1981**, 24, 195–203. [Google Scholar] [CrossRef] - Fang, T.; Aziz, A. Viscous flow with second-order slip velocity over a stretching sheet. Zeitschrift Für Naturforschung A
**2010**, 65, 1087–1092. [Google Scholar] [CrossRef] - Lin, W. Mass transfer induced slip effect on viscous gas flows above a shrinking/stretching sheet. Int. J. Heat Mass Transf.
**2016**, 93, 17–22. [Google Scholar] - Andersson, H.I. Slip flow past a stretching surface. Acta Mech.
**2002**, 158, 121–125. [Google Scholar] [CrossRef] - Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys.
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Pavlov, K.B. Magnetohydrodynamic flow of an incompressible viscous liquid caused by deformation of plane surface. Magn. Gidrodin.
**1974**, 4, 146–147. [Google Scholar] - Wang, C.Y. Flow due to a stretching boundary with partial slip—An exact solution of the Navier-Stokes equations. Chem. Eng. Sci.
**2002**, 57, 3745–3747. [Google Scholar] [CrossRef] - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed.; Dover: New York, NY, USA, 1972; pp. 17–18. [Google Scholar]
- Birkhoff, G.; MacLane, S. A Survey of Modern Algebra; Macmillan: New York, NY, USA, 1996; pp. 107–108. [Google Scholar]
- Fang, T.-G.; Zhang, J.; Yao, S.-S. Slip Magnetohydrodynamic Viscous Flow over a Permeable Shrinking Sheet. Chin. Phys. Lett.
**2010**, 27, 124702. [Google Scholar] [CrossRef] - Fang, T.; Yao, S.; Zhang, J.; Aziz, A. Viscous flow over a shrinking sheet with a second order slip flow model. Commun. Nonlinear Sci. Numer. Simul.
**2010**, 15, 1831–1842. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**–

**d**) The solution domain of D

_{1}for K

_{1}verses mass transpiration V

_{c}with Brinkman ratio $\Lambda =1$ for the case of a shrinking boundary with different choices of ${\Gamma}_{1}$ and ${\Gamma}_{2}$.

**Figure 3.**(

**a**–

**d**) The solution domain of β versus V

_{c}for different values of K

_{1}for the case of a shrinking boundary.

**Figure 4.**(

**a**) The solution domain for β versus V

_{c}for different values of ${\Gamma}_{1}$ for the case of a shrinking boundary in the absence of ${\Gamma}_{2}$. (

**b**) The solution domain for β versus V

_{c}for different values of K

_{1}for the case of a shrinking boundary with ${\Gamma}_{1}=0.5$. (

**c**) Solution domain for β versus V

_{c}for different values of K

_{1}for the case of a shrinking boundary with ${\Gamma}_{1}=5$.

**Figure 5.**(

**a**,

**b**) Upper and lower solution branches of axial velocity profile, ${f}_{\eta}\left(\eta \right)$ verses $\eta $, for different values of Navier slip parameter ${\Gamma}_{1}$ when K

_{1}= 0.5 and K

_{1}= 2 for the case of a shrinking boundary.

**Figure 6.**(

**a**) Axial velocity ${f}_{\eta}\left(\eta \right)$ verses $\eta $ for different values of ${\Gamma}_{1}$ for the case of a stretching boundary. (

**b**) Transverse velocity $f\left(\eta \right)$ verses $\eta $ for different values of ${\Gamma}_{1}$ with ${\Gamma}_{2}=-0.1$ in the presence of K

_{1}for the case of a stretching boundary. (

**c**) Axial velocity ${f}_{\eta}\left(\eta \right)$ verses $\eta $ for different values of ${\Gamma}_{2}$ with ${\Gamma}_{1}=0.5$ in the presence of K

_{1}for the case of a stretching boundary.

**Figure 7.**(

**a**) Axial velocity ${f}_{\eta}\left(\eta \right)$ verses $\eta $ for different values of Λ in the presence of K

_{1}for the case of a stretching boundary.(

**b**) Transverse velocity $f\left(\eta \right)$ verses $\eta $ for different values of Λ in the presence of K

_{1}for the case of a stretching boundary. (

**c**) Axial velocity ${f}_{\eta}\left(\eta \right)$ verses $\eta $ for different values of V

_{c}in the presence of K

_{1}for the case of a stretching boundary. (

**d**) Transverse velocity $f\left(\eta \right)$ verses $\eta $ for different values of V

_{C}in the presence of K

_{1}for the case of the stretching boundary.

**Figure 8.**(

**a**) Shear stress ${f}_{\eta \eta}\left(\eta \right)$ verses $\eta $ for different values of $\Lambda $ in the presence of K

_{1}for the case of the stretching boundary. (

**b**) Shear stress ${f}_{\eta \eta}\left(\eta \right)$ verses $\eta $ for different values V

_{c}in the presence of K

_{1}for the case of a stretching boundary.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mahabaleshwar, U.S.; Vinay Kumar, P.N.; Nagaraju, K.R.; Bognár, G.; Nayakar, S.N.R.
A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions. *Fluids* **2019**, *4*, 125.
https://doi.org/10.3390/fluids4030125

**AMA Style**

Mahabaleshwar US, Vinay Kumar PN, Nagaraju KR, Bognár G, Nayakar SNR.
A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions. *Fluids*. 2019; 4(3):125.
https://doi.org/10.3390/fluids4030125

**Chicago/Turabian Style**

Mahabaleshwar, U. S., P. N. Vinay Kumar, K. R. Nagaraju, Gabriella Bognár, and S. N. Ravichandra Nayakar.
2019. "A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions" *Fluids* 4, no. 3: 125.
https://doi.org/10.3390/fluids4030125