Ferrofluid flow in magnetic field above stretching sheet with suction and injection
Abstract
The aim of this paper is to investigate the boundary layer of ferrofluid flow induced by a permeable stretching sheet. Fluid is electrically non-conducting in the presence of non-uniform magnetic field. The governing non-linear partial differential equations are reduced to non-linear ordinary differential equations by applying a similarity transformation. Numerical solutions are obtained by using Maple. The effects of the magnetic field, the Reynolds number and the porosity on the velocity and thermal fields are investigated. The impact of the parameters on the skin friction and the local Nusselt number is numerically examined. The skin friction and heat transfer coefficients are decreasing with enhancing the stretching, the values of porosity and the ferromagnetic parameter.
Keyword : stretching sheet, suction and injection, magnetic field, ferrofluid, self-similar solution
How to Cite
Bognár, G., & Hriczó, K. (2020). Ferrofluid flow in magnetic field above stretching sheet with suction and injection. Mathematical Modelling and Analysis, 25(3), 461-472. https://doi.org/10.3846/mma.2020.10837
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G. Bognár. On similarity solutions of boundary layer problems with upstream moving wall in non-Newtonian power-law fluids. IMA Journal of Applied Mathematics, 77(4):546–562, 2011. https://doi.org/10.1093/imamat/hxr033
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G. Bognár and K. Hriczó. Similarity transformation approach for a heated ferrofluid flow in the presence of magnetic field. Electronic Journal of Qualitative Theory of Differential Equations, 2018(42):1–15, 2018. https://doi.org/10.14232/ejqtde.2018.1.42
G. Bognár and K. Hriczó. Ferrofluid flow in the presence of magnetic dipole. Technische Mechanik, 39(1):3–15, 2019.
R. Cortell. A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet. International Journal of Non-Linear Mechanics, 41(1):78–85, 2006. https://doi.org/10.1016/j.ijnonlinmec.2005.04.008
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P.S. Gupta and A.S. Gupta. Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian Journal of Chemical Engineering, 55(6):744– 746, 1977. https://doi.org/10.1002/cjce.5450550619
T. Hayat, Q. Hussain and T. Javed. The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet. Nonlinear Analysis: Real World Applications, 10(2):966–973, 2009. https://doi.org/10.1016/j.nonrwa.2007.11.020
A.U. Khan, S. Nadeem and S.T. Hussain. Phase flow study of MHD nanofluid with slip effects on oscillatory oblique stagnation point flow in view of inclined magnetic field. Journal of Molecular Liquids, 224:1210–1219, 2016. https://doi.org/10.1016/j.molliq.2016.10.102
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S.-J. Liao. On the analytic solution of magnetohydrodynamic flows of nonNewtonian fluids over a stretching sheet. Journal of Fluid Mechanics, 488:189– 212, 2003. https://doi.org/10.1017/S0022112003004865
A. Majeed, A. Zeeshan, S.Z. Alamri and R. Ellahi. Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction. Neural Computing and Applications, 30(6):1947–1955, 2018. https://doi.org/10.1007/s00521-016-2830-6
S. Nadeem, M.R. Khan and A.U. Khan. MHD oblique stagnation point flow of nanofluid over an oscillatory stretchingshrinking sheet: Existence of dual solutions. Physica Scripta, 94(7):075204, 2019. https://doi.org/10.1088/1402-4896/ab0973
S. Nadeem, M.R. Khan and A.U. Khan. MHD stagnation point flow of viscous nanofluid over a curved surface. Physica Scripta, 94(11):115207, 2019. https://doi.org/10.1088/1402-4896/ab1eb6
S. Nadeem, N. Ullah, A.U. Khan and T. Akbar. Effect of homogeneousheterogeneous reactions on ferrofluid in the presence of magnetic dipole along a stretching cylinder. Results in Physics, 7:3574–3582, 2017. https://doi.org/10.1016/j.rinp.2017.09.006
J.L. Neuringer. Some viscous flows of a saturated ferro-fluid under the combined influence of thermal and magnetic field gradients. International Journal of Non-Linear Mechanics, 1(2):123–137, 1966. https://doi.org/10.1016/0020-7462(66)90025-4
J.L. Neuringer and R.E. Rosensweig. Ferrohydrodynamics. The Physics of Fluids, 7(12):1927–1937, 1964. https://doi.org/10.1063/1.1711103
S.S. Papell. Low viscosity magnetic fluid obtained by the colloidal suspension of magnetic particles, 1965. US Patent 3,215,572
K.B. Pavlov. Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnitnaya Gidrodinamika, 10(4):146– 147, 1974. https://doi.org/10.22364/mhd (in Russian)
M.M. Rashidi. The modified differential transform method for solving MHD boundary-layer equations. Computer Physics Communications, 180(11):2210– 2217, 2009. https://doi.org/10.1016/j.cpc.2009.06.029
B.C. Sakiadis. Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal, 7(1):26–28, 1961. https://doi.org/10.1002/aic.690070108
B.C. Sakiadis. Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface. AIChE journal, 7(2):221–225, 1961. https://doi.org/10.1002/aic.690070211
H. Schlichting and K. Gersten. Boundary-layer theory. Springer, Berlin, 2016. https://doi.org/10.1007/978-3-662-52919-5
P.G. Siddheshwar and U.S. Mahabaleswar. Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. International Journal of Non-Linear Mechanics, 40(6):807–820, 2005. https://doi.org/10.1016/j.ijnonlinmec.2004.04.006
J. Singh, U.S. Mahabaleshwar and G. Bognár. Mass transpiration in nonlinear MHD flow due to porous stretching sheet. Scientific Reports, 9(1):18484, 2019. https://doi.org/10.1038/s41598-019-52597-5
K. Vajravelu. Flow and heat transfer in a saturated porous medium over a stretching surface. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fu¨r Angewandte Mathematik und Mechanik, 74(12):605–614, 1994. https://doi.org/10.1002/zamm.19940741209
K. Vajravelu and J.R. Cannon. Fluid flow over a nonlinearly stretching sheet. Applied Mathematics and Computation, 181(1):609–618, 2006. https://doi.org/10.1016/j.amc.2005.08.051
A. Zeeshan, A. Majeed and R. Ellahi. Effect of magnetic dipole on viscous ferrofluid past a stretching surface with thermal radiation. Journal of Molecular Liquids, 215:549–554, 2016. https://doi.org/10.1016/j.molliq.2015.12.110
H.I. Andersson. MHD flow of a viscoelastic fluid past a stretching surface. Acta Mechanica, 95(1):227–230, 1992. https://doi.org/10.1007/BF01170814
P.D. Ariel. MHD flow of a viscoelastic fluid past a stretching sheet with suction. Acta Mechanica, 105(1):49–56, 1994. https://doi.org/10.1007/BF01183941
G.I. Barenblatt. Scaling, Self-similarity, and Intermediate Asymptotic. Cambridge University Press, Cambridge, 1996. https://doi.org/10.1017/CBO9781107050242
H. Blasius. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Inaugural-Dissertation ... von H. Blasius, ... Druck von B.G. Teubner, 1907. Available from Internet: https://books.google.lt/books?id=VDV3QwAACAAJ
G. Bognár. Analytic solutions to the boundary layer problem over a stretching wall. Computers & Mathematics with Applications, 61(8):2256–2261, 2011. https://doi.org/10.1016/j.camwa.2010.09.039
G. Bognár. On similarity solutions of boundary layer problems with upstream moving wall in non-Newtonian power-law fluids. IMA Journal of Applied Mathematics, 77(4):546–562, 2011. https://doi.org/10.1093/imamat/hxr033
G. Bognár and K. Hriczó. Ferrofluid flow along stretched surface under the action of magnetic dipole. WSEAS Transactions on Heat and Mass Transfer, 13:103–108, 2018.
G. Bognár and K. Hriczó. Similarity transformation approach for a heated ferrofluid flow in the presence of magnetic field. Electronic Journal of Qualitative Theory of Differential Equations, 2018(42):1–15, 2018. https://doi.org/10.14232/ejqtde.2018.1.42
G. Bognár and K. Hriczó. Ferrofluid flow in the presence of magnetic dipole. Technische Mechanik, 39(1):3–15, 2019.
R. Cortell. A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet. International Journal of Non-Linear Mechanics, 41(1):78–85, 2006. https://doi.org/10.1016/j.ijnonlinmec.2005.04.008
L.J. Crane. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4):645–647, 1970. https://doi.org/10.1007/BF01587695
E.G. Fisher. Extrusion of plastics. John Wiley & Sons, 1976.
P.S. Gupta and A.S. Gupta. Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian Journal of Chemical Engineering, 55(6):744– 746, 1977. https://doi.org/10.1002/cjce.5450550619
T. Hayat, Q. Hussain and T. Javed. The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet. Nonlinear Analysis: Real World Applications, 10(2):966–973, 2009. https://doi.org/10.1016/j.nonrwa.2007.11.020
A.U. Khan, S. Nadeem and S.T. Hussain. Phase flow study of MHD nanofluid with slip effects on oscillatory oblique stagnation point flow in view of inclined magnetic field. Journal of Molecular Liquids, 224:1210–1219, 2016. https://doi.org/10.1016/j.molliq.2016.10.102
X. Li, A.U. Khan, M.R. Khan, S. Nadeem and S.U. Khan. Oblique stagnation point flow of nanofluids over stretching/shrinking sheet with CattaneoChristov heat flux model: Existence of dual solution. Symmetry, 11(9):1070, 2019. https://doi.org/10.3390/sym11091070
S.-J. Liao. On the analytic solution of magnetohydrodynamic flows of nonNewtonian fluids over a stretching sheet. Journal of Fluid Mechanics, 488:189– 212, 2003. https://doi.org/10.1017/S0022112003004865
A. Majeed, A. Zeeshan, S.Z. Alamri and R. Ellahi. Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction. Neural Computing and Applications, 30(6):1947–1955, 2018. https://doi.org/10.1007/s00521-016-2830-6
S. Nadeem, M.R. Khan and A.U. Khan. MHD oblique stagnation point flow of nanofluid over an oscillatory stretchingshrinking sheet: Existence of dual solutions. Physica Scripta, 94(7):075204, 2019. https://doi.org/10.1088/1402-4896/ab0973
S. Nadeem, M.R. Khan and A.U. Khan. MHD stagnation point flow of viscous nanofluid over a curved surface. Physica Scripta, 94(11):115207, 2019. https://doi.org/10.1088/1402-4896/ab1eb6
S. Nadeem, N. Ullah, A.U. Khan and T. Akbar. Effect of homogeneousheterogeneous reactions on ferrofluid in the presence of magnetic dipole along a stretching cylinder. Results in Physics, 7:3574–3582, 2017. https://doi.org/10.1016/j.rinp.2017.09.006
J.L. Neuringer. Some viscous flows of a saturated ferro-fluid under the combined influence of thermal and magnetic field gradients. International Journal of Non-Linear Mechanics, 1(2):123–137, 1966. https://doi.org/10.1016/0020-7462(66)90025-4
J.L. Neuringer and R.E. Rosensweig. Ferrohydrodynamics. The Physics of Fluids, 7(12):1927–1937, 1964. https://doi.org/10.1063/1.1711103
S.S. Papell. Low viscosity magnetic fluid obtained by the colloidal suspension of magnetic particles, 1965. US Patent 3,215,572
K.B. Pavlov. Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnitnaya Gidrodinamika, 10(4):146– 147, 1974. https://doi.org/10.22364/mhd (in Russian)
M.M. Rashidi. The modified differential transform method for solving MHD boundary-layer equations. Computer Physics Communications, 180(11):2210– 2217, 2009. https://doi.org/10.1016/j.cpc.2009.06.029
B.C. Sakiadis. Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal, 7(1):26–28, 1961. https://doi.org/10.1002/aic.690070108
B.C. Sakiadis. Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface. AIChE journal, 7(2):221–225, 1961. https://doi.org/10.1002/aic.690070211
H. Schlichting and K. Gersten. Boundary-layer theory. Springer, Berlin, 2016. https://doi.org/10.1007/978-3-662-52919-5
P.G. Siddheshwar and U.S. Mahabaleswar. Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. International Journal of Non-Linear Mechanics, 40(6):807–820, 2005. https://doi.org/10.1016/j.ijnonlinmec.2004.04.006
J. Singh, U.S. Mahabaleshwar and G. Bognár. Mass transpiration in nonlinear MHD flow due to porous stretching sheet. Scientific Reports, 9(1):18484, 2019. https://doi.org/10.1038/s41598-019-52597-5
K. Vajravelu. Flow and heat transfer in a saturated porous medium over a stretching surface. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fu¨r Angewandte Mathematik und Mechanik, 74(12):605–614, 1994. https://doi.org/10.1002/zamm.19940741209
K. Vajravelu and J.R. Cannon. Fluid flow over a nonlinearly stretching sheet. Applied Mathematics and Computation, 181(1):609–618, 2006. https://doi.org/10.1016/j.amc.2005.08.051
A. Zeeshan, A. Majeed and R. Ellahi. Effect of magnetic dipole on viscous ferrofluid past a stretching surface with thermal radiation. Journal of Molecular Liquids, 215:549–554, 2016. https://doi.org/10.1016/j.molliq.2015.12.110