Abstract
Heat and mass transfer for a Forchheimer model of electrically conducting fluid with Soret and Dufour effects over a vertical heated plate is studied. The governing equations for the physical problem in consideration are highly coupled and nonlinear in nature. A shooting technique is applied to the first-order ODEs’ which are obtained by using similarity transformations to PDEs’ and then to higher-order ordinary differential equations. The effects of various non-dimensional significant parameters such as Richardson number, Prandtl number, magnetic parameter, Soret and Dufour parameters and so on are interpreted. Attenuation with the velocity of fluid flow occurs due to the cause of magnetic force. The diffusion effects which are crossed in the energy and solutal equation enhance the thermal effects. Skin friction, rate of heat, and mass transfer are also computed. Results obtained are compared with the existing work and found to be in good agreement.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- \(U_{o}\) :
-
Free stream velocity
- G :
-
Gravitational field
- \(T_{w}\) :
-
Uniform constant temperature
- \(C_{w}\) :
-
Uniform constant concentration
- \(T_{\infty }\) :
-
Ambient temperature
- \(C_{\infty }\) :
-
Ambient concentration
- u :
-
Velocity component along x-direction
- \(v\) :
-
Velocity component along y-direction
- \(\varepsilon\) :
-
Porosity
- \(\mathop g\limits^{ \to }\) :
-
Acceleration due to gravity
- p :
-
Pressure
- T :
-
Temperature of the fluid
- C :
-
Concentration of the fluid
- \(\overline{\mu }\) :
-
Effective viscosity of the fluid
- \(\mu\) :
-
Fluid viscosity
- \(k(y)\) :
-
Variable permeability of the porous medium
- \(\varepsilon (y)\) :
-
Variable porosity of the saturated porous medium
- \(\alpha (y)\) :
-
Variable effective thermal diffusivity of the medium
- \(\gamma (y)\) :
-
Variable effective solutal diffusivity
- \(\sigma^{*}\) :
-
Ratio of the thermal conductivity of solid to the conductivity of the fluid
- \(\gamma^{*}\) :
-
Ratio of the thermal diffusivity of solid to the diffusivity of the fluid
- Df:
-
Dufour number
- \(\Pr\) :
-
Prandtl number
- \(\sigma\) :
-
Local permeability parameter
- \(\beta^{*}\) :
-
Local inertial parameter
- \({\text{Gr}}_{\text{C}}\) :
-
Solutal Grashof number
- \(\tau\) :
-
Skin friction
- Nu:
-
Nusselt number
- \(\vec{q} = \left( {u,v} \right),\) :
-
u and v are the velocity components along the x and y planes
- C :
-
Specific heat at constant pressure
- \(C_{b}\) :
-
Empirical constant of the second-order resistance term due to inertia effect
- \(\kappa\) :
-
Variable thermal conductivity
- \(\kappa_{c}\) :
-
Variable solutal diffusivity,
- \(\beta_{T}\) :
-
Coefficient of volume expansion volumetric
- \(\beta_{\text{C}}\) :
-
Coefficient of expansion with species concentration
- \(\phi\) :
-
Viscous dissipation term
- \(D_{12}\) :
-
Concentration gradient (i.e. Dufour coefficient)
- \(D_{21}\) :
-
Temperature gradient (i.e. Soret coefficient)
- ρ :
-
Density of the fluid
- \(\psi ({\text{x}},{\text{y}})\) :
-
Stream function
- N :
-
Buoyancy ratio
- \(\alpha^{*}\) :
-
Ratio of viscosities
- Ec:
-
Eckert number
- \(\eta\) :
-
Similarity variable
- \(k_{o}\) :
-
Permeability at the edge
- \(\varepsilon_{o}\) :
-
Porosity at the edge
- \(\alpha_{o}\) :
-
Thermal conductivity at the edge
- \(\gamma_{o}\) :
-
Solutal diffusivity at the edge of the boundary layer
- \(d\;{\text{and}}\;d^{ * }\) :
-
3.0 and 1.5 resp. for variable permeability and \(d = d^{ * } = 0\) for uniform permeability
- \({\text{Sr}}\) :
-
Soret number
- Sc:
-
Schmidt number
- Re:
-
Reynolds number
- \({\text{Gr}}_{T}\) :
-
Thermal Grashof number
- Ri:
-
Richardson number
- Sh:
-
Sherwood number
References
Eckert ERG, Drake RM (1972) Analysis of heat and mass transfer. McGraw-Hill Book, New York
Nield DA, Bejan A (1991) Convection in porous media. Springer-Verlag, Berlin
Anghel M, Takhar HS, Pop I (2000) Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. J Heat Mass Transfer 43:1265–1274
Postelnicu A (2004) Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Int J Heat Mass Transfer 47:1467–1472
Reddy G, Reddy B (2011) Finite element analysis of Soret and Dufour effects on unsteady MHD free convection flow past an impulsively started vertical porous plate with viscous dissipation. J Naval Archit Mar Eng 8:1–12
Alam MS, Rahmam MM (2006) Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction. Nonlinear Anal Modell Control 11(1):3–12
Balasubrahmanyam M, Sudarshan Reddy P, Siva Prasad R (2011) Soret effect on mixed convective heat and mass transfer through a porous medium confined in a cylindrical annulus under a radial magnetic field in the presence of a constant heat source/sink. Int J Appl Math Mech 7(8):1–17
Motsa S (2008) On the onset of convection in a porous layer in the presence of Dufour and Soret effects. SJPAM 3:58–65
Awad FG, Sibanda P, Motsa S (2010) On the linear stability analysis of a Maxwell fluid with double-diffusive convection. Appl Math Modell 34:3509–3517
Zili-Ghedira L, Slimi K, Ben Nasrallah S (2003) Double diffusive natural convection in a cylinder filled with moist porous grains and exposed to a constant wall heat flux. J Porous Media 6(2):123–136
Mohammadein AA, El-Shaer NA (2004) Influence of variable permeability on combined free and forced convection flow past a semi-infinite vertical plate in a saturated porous medium. Heat Mass Transfer 40:341–346
Nalinakshi N, Dinesh PA, Chandrashekhar DV (2013) Soret and Dufour effects on mixed convection heat and mass transfer with variable fluid properties. Int J Math Arch 4(11):203–215
Reddy G, Dinesh PA, Sandeep N (2017) Effects of variable viscosity and porosity of fluid, Soret and Dufour mixed double diffusive convective flow over an accelerating surface. IOP Conf Ser Mater Sci Eng 263: 062012, 1–13
Veera Krishna M, Swarnalathamma BV, Chamkha AJ Investigations of Soret, Joule and hall effects on MHD rotating mixed convective ßow past an inÞnite vertical porous plate. J Ocean Eng Sci 4: 263–275. https://doi.org/10.1016/j.joes.2019.05.002
Brinkman HC (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl Sci Res A1:27–34
Hady FM, Bakier AK, Gorla RSR (1996) Mixed convection boundary layer flow on a continuous flat plate with variable viscosity. Heat Mass Transfer 31:169–172
Acknowledgements
The authors are grateful to the Research Centre Atria Institute of Technology, Ramaiah Institute of Technology, for all the support and also the financial support from VTU research scheme project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Nalinakshi, N., Dinesh, P.A. (2021). Thermo-Diffusion and Diffusion-Thermo Effects for a Forchheimer Model with MHD Over a Vertical Heated Plate. In: Rushi Kumar, B., Sivaraj, R., Prakash, J. (eds) Advances in Fluid Dynamics. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-4308-1_27
Download citation
DOI: https://doi.org/10.1007/978-981-15-4308-1_27
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-4307-4
Online ISBN: 978-981-15-4308-1
eBook Packages: EngineeringEngineering (R0)