Abstract
This research looks at the Hall and Ion-slip currents with time variation in an unstable, incompressible, viscous fluid electrically conducting free convection flow that passes over an electrically non-conducting inclined plate in the presence of an inclined magnetic field. The fluid flow is generated by the moving and oscillating plate, and it is also influenced by the magnetic force, gravitational force, and viscous force. The buoyancy forces are produced by temperature and concentration variance in the gravity field due to the oscillating plate in its own plane. The governing equations are derived from the Navier–Stokes equation, energy equation, and concentration equation and then applied to the boundary layer approximation. The magnetic Reynolds number of flows is kept relatively small, so this analysis has not used the magnetic induction equation. On the velocity distribution, the angle of inclination has a retarding impact. The results of this study have been shown to explain the drag on flow at inclined surfaces instantly. The impact of the relevant parameters on fluid velocity, temperature, and concentration distributions has been explained and visualized using graphs. Numerical data for skin friction, rate of heat transmission, and mass transfer in terms of shear stress, Nusselt, and Sherwood numbers are visualized graphically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \({\mathbf{q}}\) :
-
Fluid velocity vector
- \({\mathbf{J}}\) :
-
Current density vector
- \(u\) :
-
Velocity components along x-axis
- \(v\) :
-
Velocity components along y- axis
- \(w\) :
-
Velocity components along z- axis
- \(\beta\) :
-
Volumetric coefficient of thermal expansion
- \(\beta^{*}\) :
-
Volumetric coefficient of mass expansion
- \(B_{0}\) :
-
Constant magnetic induction along y-directions
- \(\omega\) :
-
Acceleration frequency
- \(\omega_{e}\) :
-
Cyclotron frequency
- \(\rho\) :
-
Density of the fluid
- \(\sigma\) :
-
Conductivity of the fluid
- \(\tau_{e}\) :
-
Electron collision time
- \(c_{p}\) :
-
Specific heat at the constant pressure
- \(k_{T}\) :
-
Thermal diffusion ratio
- \(\upsilon\) :
-
Kinematic viscosity
- \({\mathbf{B}}\) :
-
Magnetic field vector
- \({\mathbf{g}}\) :
-
Gravitational acceleration
- \(U_{0}\) :
-
Constant uniform velocity
- \(\alpha\) :
-
Inclined angle
- \(t\) :
-
Time
- \(\Delta t\) :
-
Time increment
- \(\tau\) :
-
Maximum time
- \(c_{s}\) :
-
Concentration susceptibility
- \(\sigma^{\prime}\) :
-
Electric conductivity
- \(\mu\) :
-
Fluid viscosity coefficient
- \(\mu_{e}\) :
-
Magnetic permeability
- \(\beta_{e}\) :
-
Hall parameter
- \(\beta_{i}\) :
-
Ion-slip parameter
- \(u^{*}\) :
-
Dimensionless velocity component in x-axis
- \(v^{*}\) :
-
Dimensionless velocity component in y-axis
- \(t^{*}\) :
-
Dimensionless time
- \(\omega^{*}\) :
-
Dimensionless acceleration parameter
- \(\theta\) :
-
Dimensionless temperature
- \(\varphi\) :
-
Dimensionless fluid concentration
- \(D_{m}\) :
-
Mass diffusion coefficient
- \(T\) :
-
Temperature of the fluid
- \(T_{w}\) :
-
Constant temperature near the plate
- \(T_{\infty }\) :
-
Outside boundary layer temperature
- \(C\) :
-
Concentration in the fluid
- \(C_{w}\) :
-
Constant concentration near the plate
- \(C_{\infty }\) :
-
Outside boundary layer concentration
- \(H_{a}\) :
-
Hartmann number
- \(G_{r}\) :
-
Thermal Grashof number
- \(G_{m}\) :
-
Solutal Grashof number
- \(P_{r}\) :
-
Prandtl number
- \(S_{c}\) :
-
Schmidt number
- \(D_{f}\) :
-
Dufour number
- \(\tau_{xL}\) :
-
Local primary shear stress
- \(\tau_{zL}\) :
-
Local secondary shear stress
- \(Nu_{L}\) :
-
Local Nusselt number
- \(Sh_{L}\) :
-
Local Sherwood number
- \(\tau_{xA}\) :
-
Average primary shear stress
- \(\tau_{zA}\) :
-
Average primary shear stress
- \(Nu_{A}\) :
-
Average Nusselt number
- \(Sh_{A}\) :
-
Average Sherwood number
References
Abel, S., Veena, P.H.: Visco-elastic fluid flow and heat transfer in porous medium over a stretching sheet. Int. J. Non-Linear Mech. 33, 531–540 (1998)
Abernathy, F.H.: Flow over an inclined plate. J. Fluids Eng. 84(3), 380–388 (1962)
Ajay, K.S.: MHD free convection and mass transfer flow with Hall current, viscous dissipation, joule heating and thermal diffusion. Indian J. Pure Appl. Phys. 41, 24–35 (2003)
Angirasa, D., Peterson, G.P.: Natural convection heat transform from an isothermal vertical surface to a fluid saturated thermally stratified porous medium. Int. J. Heat Mass Transfer 14(8), 4329–4335 (1997)
Basant, K.J., Peter, B.M.: Hall current and ion-slip effects on free convection flow in a vertical micro channel with an induced magnetic field”. Heat Transf. Asian Res. (2019). https://doi.org/10.1002/htj.21569
Beg, O.A., Sim, J.L., Zueco, R., Bhargava, R.: Numerical study of magneto-hydrodynamic viscous plasma flow in rotating porous media with Hall currents and inclined magnetic field influence. Commun. Nonlinear Sci. Numer. Simul. 15, 345–359 (2010)
Bhpendra, K.S., Abhay, K.J., Chaudhary, R.C.: Hall effect on MHD mixed convective flow of a viscous incompressible fluid past a vertical porous plate immersed in porous medium with heat source/sink. Rom. J. Phys. 52, 5–7 (2007)
Daniel, S., Daniel, Y.S.: Convective flow two immiscible fluids and heat transfer with porous along an inclined channel with pressure gradient. Int. J. Eng. Sci. 2(4), 12–18 (2013)
Eraslan, A.H.: Temperature distributions in MHD channels with Hall effect. AIAAJ 7, 186–188 (1969)
Gupta, P.S., Gupta, A.S.: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, 744–746 (1997)
Javeri, V.: Combined influence of Hall effect, ion slip, viscous dissipation and Joule heating on MHD heat transfer in a channel. Heat Mass Transf. 8, 193–202 (1975)
Jyotsna, R.P., Gouranga, C.D., Suprava, S.: Radiation and mass transfer effects on MHD flow through porous medium past an exponentially accelerated inclined plate with variable temperature. Ain Shams Eng. J. 8, 67–75 (2017)
Kumar, B.V.R., Singh, P.: Effect of thermal stratification on free convection in a fluid saturated porous enclosure. Numer. Heat Transf. 34, 343–356 (1999)
Singh, N.K., Kumar, V., Sharma, G.K.: The effect of inclined magnetic field on unsteady flow past on moving vertical plate with variable temperature. IJLTEMAS 5(2), 34–37 (2016)
Opanuga, A.A., Agboola, O.O., Okagbue, H.I., Olanrewaju, A.M.: Hall current and ion-slip effects on the entropy generation of couple stress fluid with velocity slip and temperature jump. Int. J. Mech. 12, 221–231 (2018)
Prasada, V., Sivaprasad, R., Rao, U.R.: Hall effects on hydromagnetic channel flow under an inclind magnetic field. Indian Natn. Sci. Acad. 52(3), 573–583 (1986)
Rajput, U.S., Gupt, N.K.: Dafour effect on unsteady free convection MHD flow past an exponentially accelerated plate through porous media with variable temperature and constant mass diffusion in an inclined magnetic field. IRJET 3(8), 2135–2140 (2016)
Ram, P.C.: The effect of Hall and ion slip current on free convection heat generating flow in a rotating fluid. Int. J. Energy Res. 19(5), 371–376 (1995)
Hanvey, R.R., Khare, R.K., Paul, A.: MHD flow of incompressible fluid through parallel plates in inclind magnetic field having porous medium with heat and mass transfer. IJSIMR 5(4), 18–22 (2017)
Sato, H.: The Hall effect in the viscous flow of ionized gas between two parallel plates under transverse magnetic field. J. Phys. Soc. Jpn. 16, 1427–1433 (1961)
Seth, G.S., Nandkeolyar, R., Mahto, N., Singh, S.K.: MHD couette flow in a rotating system in the presence of an inclined magnetic field. Appl. Math. Sci. 3(59), 2919–2932 (2009)
Seth, G.S., Nandkeolyer, R., Ansary, M.S.: Hartmann flow in a rotating system in the present of inclined magnetic field with Hall effects. Tamkang J. Sci. Eng. 13(3), 243–252 (2010)
Seth, G.S., Mandal, P.K., Chamkha, A.J.: MHD free convection flow past an impulsively moving vertical plate with ramped heat flux through porous medium in the presence of inclined magnetic field. FHMT 7, 23 (2016)
Agarwalla, S., Ahmed, N.: MHD mass transfer flow past an inclined plate with variable temperature and plate velocity embedded in a porous medium. Heat Transf. 47(1), 27–41 (2018)
Singh, J.K., Joshi, N., Begum, S.G., Srinivasa, C.T.: Unsteady hydromagnetic heat and mass transfer natural convection flow past an exponentially accelerated vertical plate with Hall current and rotation in the presence of thermal and mass diffusions. Front. Heat Mass Transf. (FHMT) 7, 24 (2016). https://doi.org/10.5098/hmt.7.24
Singh, J.K., Joshi, N., Rohidas, P.: Unsteady MHD natural convective flow of a rotating walters’-b fluid over an oscillating plate with fluctuating wall temperature and concentration. J. Mech. 34(4), 519–532 (2018). https://doi.org/10.1017/jmech.2017.25
Singh, J.K., Seth, G.S., Joshi, N., Srinivasa, C.T.: Mixed convection flow of a viscoelastic fluid through a vertical porous channel influenced by a moving magnetic field with Hall and ion-slip currents, rotation, heat radiation and chemical reaction. Bulg. Chem. Commun. 52(1), 147–158 (2020). https://doi.org/10.34049/bcc.52.1.468
Singh, J.K., Vishwanath, S.: Hall and ion-slip effects on MHD free convective flow of a viscoelastic fluid through porous regime in an inclined channel with moving magnetic field. Kragujevac J. Sci. 42, 5–18 (2020)
Krishna, M.V., Ahmad, N.A., Chamkha, A.J.: Hall and ion slip effects on unsteady MHD free convective rotating flow through a saturated porous medium over an exponential accelerated plate. Alex. Eng. J. 59(2), 565–577 (2020)
Author information
Authors and Affiliations
Contributions
M. R. Islam and M. M. Alam have written the literature review. M. R. Islam and S. Nasrin have developed the geometrical configuration, and illustrated the algorithm and written codes by using MATLAB R2015a tools. M. R. Islam and S. Nasrin has drawn the graphs and written the manuscript. All authors have checked the code and the simulated data.
Corresponding author
Ethics declarations
Conflict of interest
It is declaring that, on behalf of all authors, the corresponding author states that there is no conflict of interest; we have no affiliations or involvement with an organization with a financial or non-financial interest in this research.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Islam, M.R., Nasrin, S. & Alam, M.M. Unsteady MHD fluid flow over an inclined plate, inclined magnetic field and variable temperature with Hall and Ion-slip current. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00728-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-022-00728-y