Numerical study of self-similar natural
convection mass transfer from a rotating
cone in anisotropic porous media with
Stefan blowing and Navier slip
Beg, OA, Uddin, MJ, Beg, TA, Kadir, A, Shamshuddin, M and Babaie, M
http://dx.doi.org/10.1007/s12648-019-01520-9
Title Numerical study of self-similar natural convection mass transfer from a
rotating cone in anisotropic porous media with Stefan blowing and Navier
slip
Authors Beg, OA, Uddin, MJ, Beg, TA, Kadir, A, Shamshuddin, M and Babaie, M
Publication title Indian Journal Of Physics
Publisher Springer
Type Article
USIR URL This version is available at: http://usir.salford.ac.uk/id/eprint/51046/
Published Date 2020
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1
INDIAN JOURNAL OF PHYSICS
ISSN: 0973-1458 (print version); ISSN: 0974-9845 (electronic version); PUBLISHER: SPRINGER
Impact factor = 0.967
Accepted April 11
th
2019
NUMERICAL STUDY OF SELF-SIMILAR NATURAL CONVECTION MASS
TRANSFER FROM A ROTATING CONE IN ANISOTROPIC POROUS
MEDIA WITH STEFAN BLOWING AND NAVIER SLIP
O. Anwar Bég
1
, M.J. Uddin
2
, T. A. Bég
3
, A. Kadir
4
, MD. Shamshuddin
5*
and Meisam Babaie
6
1,4
Aeronautical & Mechanical Engineering Department, School of Computing, Science and Engineering, Newton Building,
University of Salford, Manchester, M54WT, UK.
2
Department of Mathematics, University Sains-Malaysia, Malaysia.
3
Computational Mechanics and Renewable Energy, Dickenson Road, Manchester, M13, UK.
5*
Department of Mathematics, Vaagdevi College of Engineering, Warangal, Telangana, India.
6
Energy Sciences, Petroleum and Gas Engineering, University of Salford, Manchester, M54WT, UK.
*Corresponding author: shammaths@gmail.com; shamshuddin_md@vaagdevi.edu.in
Abstract: A mathematical model is presented for laminar, steady natural convection mass transfer in boundary layer
flow from a rotating porous vertical cone in anisotropic high permeability porous media. The transformed boundary
value problem is solved subject to prescribed surface and free stream boundary conditions with a MAPLE 17
shooting method. Validation with a Chebyshev spectral collocation method is included. The influence of tangential
Darcy number, swirl Darcy number, Schmidt number, rotational parameter, momentum (velocity slip), mass slip and
wall mass flux (transpiration) on the velocity and concentration distributions is evaluated in detail. The computations
show that tangential and swirl velocities are enhanced generally with increasing permeability functions (i.e. Darcy
parameters). Increasing spin velocity of the cone accelerates the tangential flow whereas it retards the swirl flow. An
elevation in wall suction depresses both tangential and swirl flow. However, increasing injection generates
acceleration in the tangential and swirl flow. With greater momentum (hydrodynamic) slip, both tangential and swirl
flows are accelerated. Concentration values and Sherwood number function values are also enhanced with
momentum slip, although this is only achieved for the case of wall injection. A substantial suppression in tangential
velocity is induced with higher mass (solutal) slip effect for any value of injection parameter. Concentration is also
depressed at the wall (cone surface) with an increase in mass slip parameter, irrespective of whether injection or
suction is present. The model is relevant to spin coating operations in filtration media (in which swirling boundary
layers can be controlled with porous media to deposit thin films on industrial components), flow control of mixing
devices in distillation processes and also chromatographical analysis systems.
Keywords : Mass transfer, rotating cone, slip, anisotropic porous media, wall suction/injection.
PACS No. : 46.15.-X, 47.11. +j, 47.10. ab, 44.30. +v, 07.05. Tp.
1. Introduction
Mass transfer (species diffusion) with or without buoyancy forces is fundamental to many
diverse procedures in modern chemical, environmental and biomedical engineering. It arises
in catalytic packed-bed reactors [1], geological contaminant dispersion [2], oil spill penetration
into stratified soils [3] and oxygen diffusion in neurological tissue [4]. The established
methodology for simulating mass transfer is the Fickian law of mass diffusion. To simulate
transport in porous media, a variety of methods have been employed to solve the Fickian
diffusion equation in permeable systems ranging from drag-force formulations to percolation
2
theory, in order to recreate the permeability (hydraulic conductivity) properties of such media.
Ulson De Souza and Whitaker [5] used the volume-averaging method to simulate mass
transfer in a packed-bed reactor, examining in detail the dispersion in the main fluid phase,
internal diffusion of the reactant in the pores of the catalyst and surface reaction within the
catalyst. Helmer et al. [6] studied unsteady water diffusion in tumors (diseased tissue) with a
tortuous porous medium model. Cotta et al. [7] applied integral transforms to study a range of
convective boundary layer mass transfer problems in permeable systems. Piquemal et al. [8]
simulated species mobility in porous media with a dispersion-convection equation for the
mobile fluid and a diffusion equation for the stagnant fluid, considering a cylindrical tube with
stagnant pockets in its wall and also studied a stratified permeable medium. Vafai and Tien [9]
conducted experiments and numerical studies of convective buoyancy-driven mass transfer in
porous media with Brinkman friction and Forchheimer inertial drag effects, evaluating the time
and space-averaged mass flux of a species in the porous medium. In these studies, the
permeability of the material was considered to be isotropic i.e. permeability was assumed to
be the same in any direction. However, in certain industrial filtration materials and foams and
also invariably in geological systems, porous media are anisotropic. The variation in
permeability in different directions can have a dramatic effect on transport phenomena and
can influence, for example, the fate of contaminants, the rates of mass transfer on embedded
body surfaces etc. Several articles have addressed anisotropic porous media hydrodynamics.
Marcus [10] was among the first researchers to investigate anisotropic flows in porous media.
He defined a directional permeability and conducted laboratory tests on carefully designed
samples. Wang et al. [11] considered a variety of different structural models to analyze non-
Darcy flow in anisotropic porous media surrounding the near-wellbore region of high-capacity
gas and condensate reservoirs. They showed that as pore-scale anisotropic parameter is
increased, there is a reduction in permeability components. Adams et al. [12] used a finite
element method to simulate radial encroachment of a viscous liquid into a homogeneous,
anisotropic porous medium, deriving effective permeabilities as functions of the principal
permeabilities. Other studies [13-14] examined the topology of flow in three-dimensional non-
stationary anisotropic heterogeneous porous media with a Monte Carlo simulator. Nakayama
and Kuwahara [15] determined the permeability tensor for isothermal anisotropic porous
media.
The flow from a rotating curved body is also of great interest in chemical engineering
operations. The Coriolis forces which are generated by centrifugal fields encourage fluid to be
drawn along the curved surface and significantly alter momentum diffusion rates. If species
diffusion also occurs, mass transfer rates at the curved surface are also generally enhanced.
Although many investigations have been reported on heat transfer from spinning bodies,
relatively few investigations have considered mass transfer. Salzberg and Kezios [16]
presented an early experimental study of mass-transfer by sublimation from the surface of a
rotating naphthalene cone in an airstream. Newman [17] employed laminar boundary layer
theory to determine the limiting rates of mass transfer to a rotating sphere at high Schmidt
3
numbers. Smith and Colton [18] developed approximate solutions for mass transfer from a
disk to a rotating fluid, observing that mass transfer is dominated by the outer-most zone of
the disk owing to high transfer rate associated with the leading edge of the boundary layer,
and furthermore that the species concentration field close to the axis of rotation extends to
significant axial distances as a long slender plume. Ellison and Cornet [19] determined
experimentally the mass transfer rates for oxygen diffusion to a disk rotating in aqueous
sodium chloride solution and calculated average Sherwood numbers over an extensive range
of Reynolds and Schmidt numbers. Laminar-turbulent transition in mass transfer from a
rotating disk was investigated by Mohr and Newman [20]. Toren et al. [21] studied
centrifugally-driven flow (due to a density gradient between the surface of an infinite disk and
the ambient fluid) in rotating Von Karman mass transfer at high Schmidt number. They
identified a linear Ekman layer driven by a buoyancy sublayer. Rashaida et al. [22] evaluated
the laminar boundary layer mass transfer from a rotating disk to a Bingham non-Newtonian
fluid, deriving a Sherwood number as a function of Reynolds number and dimensionless yield
stress (Bingham number), and observing that higher Bingham number depresses wall mass
transfer rates.
In recent years slip flows have also attracted the attention of engineers. These flows arise
when the Navier no-slip boundary condition, a common approximation in viscous fluid
dynamics, is not applicable. Slip may occur in the velocity field, thermal field (“thermal jump”)
or mass distribution (species concentration distribution), at a boundary. It arises in certain
polymeric manufacturing processes as well as in rarefied gas dynamics in high speed flight.
Theoretical models often use the Navier condition for momentum slip. Taamneh and Omari
[23] examined computationally the slip-flow and heat transfer in non-Newtonian inelastic fluids
in a porous medium micro-channel, using a Knudsen number to evaluate slip effects and
showing that with greater Knudsen numbers, the wall shear stress is enhanced i.e. the flow is
accelerated. Miguel [24] established several discrete regimes for slip flow in porous media
including free molecular (ballistic) flows at very high Knudsen numbers. He also observed that
for a slip-flow regime, the dimensionless permeability of the porous medium is dependent on
the structure of the medium and exhibits a power-law increase with the Knudsen number for
high porosities. Other slip flow transport phenomena have been studied by Khan et al. [25] for
nanofluid heat and mass transfer, Bég et al. [26] for magnetohydrodynamic radiative
convective flow, Wang [27] for Newtonian flow from an extending wall with partial slip and
Prasad et al. [28] for Casson non-Newtonian boundary layer heat transfer from a cylinder. The
mass (solutal) slip phenomenon (in addition to velocity slip) was recently addressed by
Bhattacharyya [29] for reactive mass transfer in a porous medium, in which it was shown that
mass slip decreases mass transfer rate from the wall and, in addition, also depresses the
concentration magnitude.
In the present study, we examine, for the first time, momentum and mass slip in the rotating
mass transfer (species diffusion) and boundary layer flow from a spinning cone in an
4
anisotropic porous medium. An axisymmetric laminar steady-state formulation with the
generalized anisotropic Darcy law is used. The governing transport equations are rendered
dimensionless and solved subject to modified boundary conditions with Maple17 software.
The solutions are found to correlate well with Chebyshev spectral collocation computations
for the general model and with earlier non-porous, no-slip solutions from the literature.
2. Mathematical Model
The geometry of the problem is depicted in Figure 1, with respect to an (x, y,
) coordinate
system. Steady-state, laminar, incompressible, axisymmetric, natural convective mass
transfer in boundary layer flow from a rotating porous cone embedded in an anisotropic
saturated high-permeability medium is examined. The diffusing species is non-reactive.
Rotation is sufficiently weak to neglect compressibility effects. An anisotropic Darcy model is
employed to simulate different permeabilities in the medium, which is homogenous and fully-
saturated. Large permeabilities are considered which simulate foam-like materials.
Forchheimer drag (inertial) and Brinkman boundary vorticity effects are ignored. The X
direction is orientated along the cone slant surface, the Y direction normal to this and
designates the angle in a plane perpendicular to the vertical symmetry axis:
Fig. 1: Physical model for convective mass transfer from a rotating cone in an anisotropic
porous medium
The conservation equations for the problem may be presented by amalgamating the models in
Bég et al. [26] (for heat and mass diffusion), Wang [27] (for anisotropic slip effects), Prasad et
al. [28] (for anisotropic porous media drag forces and thermal slip) and Ece [31] (for the
convective mass transfer analogy and rotational body forces in spinning flow) and take the
form:
Mass:
( ) ( )
0
RU RV
XY
+=
(1)
