Proceedings of the 7th World Congress on Mechanical, Chemical, and Material Engineering (MCM'21)
Prague, Czech Republic Virtual Conference – August, 2021
Paper No. HTFF 120
DOI: 10.11159/htff21.120

Hydromagnetic Mixed Convection along a Vertical Cylinder in a
Stratified Non-Darcian Porous Medium
Saddam Atteyia Mohammad1, Abdelkrim Younis Fathi2

University of Mosul-College of Engineering-Mechanical Engineering Department
Almajmuea 41002, Mosul, Iraq
saddamatteyia@uomosul.edu.iq; abd59you@gmail.com

1,2

Abstract - In the present problem, the simultaneous heat and mass transfer by steady, laminar, incompressible, viscous and conductive
to electricity combined convection flow straight a vertical cylinder placed inside a non-Darcian thermally stratified porous medium areconsidered. The condition of variable surface temperature and variable surface concentration are analyzed. All the regime of
convection (free, mixed, and forced) will be studied. The governing nonlinear partial differential equations will be converted to nonsimilar form by using suitable transformations. Then they are solved by a finite difference method. Hydrodynamic, thermal, andconcentration profiles as well as local Nusselt and Sherwood numbers that reflect the amount of heat and mass transfer will be analyzed
and discussed. The results show that increasing the value of the power law index of the surface temperature of the cylinder, curvatureparameter, and buoyancy ratio leads to enhance the rates of heat and mass transfer. On contrast the rates of heat and mass transfer are
reduced when the value of inertia effect parameter, square of the Hartmann number, and local thermal stratification parameter areraised. Enhancing the value of Lewis number causes lowering the rate of heat transfer and growing the rate of mass transfer.
Keywords: Vertical cylinder, Heat and Mass transfer, Porous medium, Mixed convection, Thermal stratification

1. Introduction

The subject of convective heat transfer from cylindrical geometry has received more attention from the researchers.
This is due to the fact that cylinders have been used in chemical catalytic reactors, geothermal systems, nuclear waste
disposal, heat exchangers and so on (Chen and Horng [1]). Stratification is a physical phenomenon may be arise fromtemperature difference, that causes a divergence in the density of the fluid and in this case this process is called as thermal
stratification. This is due to the input of thermal energy into the medium by thermal sources or heated bodies.
Concentration difference is another situation that cause a stratification like transmit process in the sea arising from salinity
distinction. The presence of different fluids may lead to arise of stratification. Stable state emerges when the fluid of lightdensity overlies the fluid of heavy density.
From the examples of convection operations that happen in ambience with stratification phenomenon areenvironmental chambers with heated walls and closed containers. Another example is the convection flow connected with
heat-removal systems for long-period profound ocean power modules. Furthermore, diverse buoyant flow systems
inclusive geological transport, geothermal systems, lake thermo hydraulics, condensation systems of power plant, andvolcanic flows and also in industrial thermal processing are of great interest (Mukhopadhyay and Ishak [2]).
The Darcy’s law is valid only for flows of low velocity through porous medium with low permeability. Including thenon-Darcy effects is necessary to model the real situation of flow in porous media that contain convection heat and masstransfer. The non-Darcian effects included boundary viscous resistance, porosity variation close the surface (due to packingof particles), inertia effects, and thermal dispersion (Chen [3]). The boundary viscous resistance may be regarded
considerable when heat transfer is treated in a zone very near to the surface. The Brinkman’s extension that is used to give
weight for boundary viscous resistance, join viscous shear stress term within the equation of momentum as well as the noslip boundary condition. The inertia effects is considered significant at a higher flow rate and it can be taken into
consideration through the addition of a velocity squared term (known as the Forchheimer’s extension) in the momentum
equation. While inertia effects are predominant, the thermal dispersion effect may become significant. Thermal dispersion
occur due to recirculation and mixing of local fluid streams as the fluid flows through tortuous paths formed by the solid
particles. Thus a good blending of the fluid happens inside the pores (Chen and Horng [1], Afify [4]).

HTFF 120-1

Researchers has gave good attention to the subject of magneto hydrodynamic MHD and heat and mass transfer indifferent geometries. The reason of this attention is the influence of magnetic field on the control of flow and execution ofmany systems that use fluids conductive to electricity. Many metallurgical processes in industry include cooling process ofcontinued strips or filaments via drawing them in fluid conducting to electricity in the existence of magnetic field. Fromthe examples of these processes is the drawing, annealing, and tinning of copper wires. In these processes the rate ofcooling can be controlled which can be affect the properties of the final product (Amanull et al. [5]; Ganapathirao et al.
[6]). Moreover, in different technological applications such as purification of molten metals, geothermal energy extraction,
nuclear reactor coolers, metal casting, and many others the effects of magnetic field are encountered. The fluids in
geothermal regions are electrically conducting and can be significantly influenced by the magnetic field. This fact ismotivated the study of the impact of magnetic field on the transport phenomenon over porous media (Ganapathirao et al.
[6]).
The problem of mixed convection flow past a vertical cylinder was investigated by Merkin and Pop [7]. Thermal
dispersion effect on the non-Darcy mixed convection flow over a vertical cylinder was investigated by Kumari et al. [8].
Using Darcy model, Hooper et al. [9] study the mixed convection along an isothermal vertical cylinder. The case of nonDarcy combined convection flow over vertical cylinder was investigated by Aldoss et al. [10]. With non-Darcy model,
Aldoss [11] in another study considered magneto hydrodynamic mixed convection flow adjacent to a vertical cylinder.
Mixed convection flow adjacent to a vertical heated or cooled permeable slender cylinder was investigated by Kumari et al.
[12] using the Darcy law in order to examine the impact of steady non-uniform suction or injection. The problem of mixed
convection flow over a vertical cylinder embedded in a nanofluid-saturated porous medium is studied by Gorla andHossain [13]. Steady mixed convection boundary layer flow on a vertical cylinder in a porous medium saturated with ananofluid for cooled and heated cylinder is studied by Rohni et al. [14].
The case of conjugate heat transfer of magneto hydrodynamic mixed convection flow of nanofluid past a vertical
slender hollow cylinder in a high porosity porous medium is analyzed numerically by Jafarian et al. [15]. Shu et al. [16]study the problem of steady mixed convection boundary layer flow on a cooled vertical permeable cylinder. Mohammad[17] investigated the problem of coupled heat and mass transfer by steady mixed convection flow of electricallyconducting fluid past a vertical circular cylinder embedded in non-Darcy porous medium with heat source. Mohammad[18] performed analysis to study the influences of variable viscosity on hydromagnetic coupled heat and mass transfer by
mixed convection non-Darcy flow adjacent to a vertical circular cylinder embedded in a porous medium.
In this paper, It will be study hydromagnetic simultaneous heat and mass transfer by mixed convection boundary layer
flow past a vertical circular cylinder embedded in a stratified non-Darcian porous medium. The entire regime of mixed
convection was considered. In other words calculations will be carried out from perspicuous natural convection edge toperspicuous forced convection edge. Numerical modeling is widely used in analyzing of such problems because of its
ability of providing specific conception about the physics of problems from this type, furthermore, the experimental workrequire high cost and time. This work finds applications in petroleum engineering, geothermal systems, insulation
technology, and nuclear engineering. To the best knowledge of the authors this problem does not appear in the previous
literature, therefore, the obtained numerical results are original and novel.

2. Problem Formulation

Consider laminar, incompressible, steady, and hydromagnetic coupled heat and mass transfer by non-Darcian
mixed convection flow along a vertical circular cylinder in a thermally stratified porous medium. Figure 1 depicts thephysical flow model and coordinate system. Moreover, the following assumptions are made in the formulation of thepresent problem:
1. The fluid in the medium is Newtonian and electrically conducting.
2. The fluid has constant properties exclude the density of the buoyancy term of the balance of momentum equation.
3. Density is approximated by the Boussinesq approximation.
4. Boundary-layer approximations are applicable.
Under the above assumptions the governing nonlinear equations are given as:

HTFF 120-2

Fig. 1: Physical flow model and coordinate system.
2.1. Continuity Equation𝜕𝜕𝜕𝜕𝜕𝜕(𝑟𝑟𝑟𝑟)+ 𝜕𝜕𝜕𝜕 = 0
𝜕𝜕𝜕𝜕

(1)

Where 𝑢𝑢 and 𝑣𝑣 represents the components of fluid velocity in the 𝑥𝑥 and 𝑟𝑟 axes respectively.

2.2. Momentum Equation (Mohammad [17]; Mohammad [18])−𝐾𝐾 𝜕𝜕𝜕𝜕
𝜎𝜎𝛽𝛽 2 𝑢𝑢
𝑐𝑐 √𝐾𝐾𝑢𝑢2
� + 𝜌𝜌𝜌𝜌 + 𝑜𝑜 � − 𝜈𝜈𝜇𝜇 𝜕𝜕𝜕𝜕
𝜙𝜙−𝐾𝐾 𝜕𝜕𝜕𝜕
𝑐𝑐√𝐾𝐾 2
𝑣𝑣 =
� �−
𝑣𝑣
𝜈𝜈
𝜇𝜇 𝜕𝜕𝜕𝜕
1.5
2 3 ⁄[150(1

(2)

𝑢𝑢 =
𝑐𝑐 = 1.75⁄�√150 𝜙𝜙�

𝐾𝐾 = 𝑑𝑑 𝜙𝜙(3)− 𝜙𝜙)2 ](4)

Where 𝐾𝐾 is the permeability of the porous medium, 𝜙𝜙 is the porosity of the porous medium, 𝑑𝑑 is the particle diameter,
𝑐𝑐 is constant, 𝜎𝜎 is the electrical conductivity of the fluid, 𝛽𝛽𝑜𝑜 is the magnetic induction, 𝑃𝑃 is the pressure, 𝜌𝜌 is the density of
the fluid, 𝜇𝜇 is the dynamic viscosity of the fluid, and 𝑔𝑔 is the gravitational acceleration. It can be eliminate pressure term bydifferentiating Eq. (2) with respect to 𝑟𝑟 and Eq. (3) with respect to 𝑥𝑥. Apply approximation of the Boussinesq
𝜌𝜌 = 𝜌𝜌∞ [1 − 𝛽𝛽𝑇𝑇 (𝑇𝑇 − 𝑇𝑇∞ ) − 𝛽𝛽𝐶𝐶 (𝐶𝐶 − 𝐶𝐶∞ )](5)

Where 𝑇𝑇 and 𝐶𝐶 are temperature and concentration respectively. 𝑇𝑇∞ and 𝐶𝐶∞ are free stream temperature andconcentration respectively. 𝜌𝜌∞ is the free stream density. 𝛽𝛽𝑇𝑇 and 𝛽𝛽𝐶𝐶 are the coefficients of thermal and concentration
expansion respectively. As stated earlier, the boundary layer approximations are applicable, thus, within the boundary layer(𝑣𝑣 ≪ 𝑢𝑢, 𝜕𝜕𝜕𝜕 ⁄𝜕𝜕𝜕𝜕 ≪ 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕). According to the above information the final format of the momentum equation can be givenby−𝜇𝜇 𝜕𝜕𝜕𝜕
2𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝜎𝜎𝛽𝛽 2 𝜕𝜕𝜕𝜕− 𝐾𝐾 𝑢𝑢 − 𝑔𝑔 − 𝜙𝜙𝑜𝑜 = 0
𝐾𝐾 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
√

HTFF 120-3(6)

2.3. Energy Equation

𝑢𝑢

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕2𝑇𝑇
1 𝜕𝜕𝜕𝜕
+ 𝑣𝑣 = 𝛼𝛼 � 2 +
�
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
𝑟𝑟 𝜕𝜕𝜕𝜕

(7)

𝑢𝑢

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝐷𝐷 𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝑣𝑣 =
�𝑟𝑟 �
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑟𝑟 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

(8)

In Eq. (7) 𝛼𝛼 represent the thermal diffusivity.

2.4. Mass Conservation Equation

In Eq. (8) 𝐷𝐷 represent the mass diffusivity.

2.5. Boundary Conditions

𝑟𝑟 = 𝑟𝑟𝑜𝑜 𝑣𝑣 = 0
𝑟𝑟 → ∞ 𝑢𝑢 = 𝑈𝑈∞

𝑇𝑇 = 𝑇𝑇𝑤𝑤 (𝑥𝑥) = 𝑇𝑇∞,0 + 𝐴𝐴𝑥𝑥 𝑚𝑚
𝑇𝑇 = 𝑇𝑇∞ (𝑥𝑥) = 𝑇𝑇∞,0 + 𝐵𝐵𝐵𝐵𝐶𝐶 = 𝐶𝐶𝑤𝑤 (𝑥𝑥) = 𝐶𝐶∞,0 + 𝐴𝐴𝑥𝑥 𝑚𝑚
𝐶𝐶 = 𝐶𝐶∞,0(9)

Where 𝑇𝑇∞,0 and 𝐶𝐶∞,0 are the free stream temperature and concentration at 𝑥𝑥 = 0 respectively. The subscript 𝑤𝑤 indicate
the condition at the cylinder wall.
2.6. Dimensionless Variables
The following dimensionless variables will be introduced to convert the system of governing differential
equations into a non-similar form. The non-similar system of equations is applicable from free convection limit
through mixed convection limit until reached the forced convection limit.
1
𝑥𝑥

𝑟𝑟 2

⁄

𝑟𝑟

𝜂𝜂 = 𝑃𝑃𝑃𝑃𝑥𝑥1 2 𝜁𝜁 −1 �2𝑟𝑟 − 2𝑜𝑜 �

𝑓𝑓 =𝜓𝜓(𝜁𝜁,𝜂𝜂)

1⁄2 −1𝜁𝜁

𝛼𝛼𝑟𝑟𝑜𝑜 𝑃𝑃𝑃𝑃𝑥𝑥

𝑜𝑜

𝑇𝑇−𝑇𝑇 (𝑥𝑥)∞𝜃𝜃(𝜁𝜁, 𝜂𝜂) = 𝑇𝑇 (𝑥𝑥)−𝑇𝑇
𝑤𝑤

∞,0

𝑅𝑅𝑅𝑅

1⁄2 −1𝜁𝜁 = �1 + � 𝑃𝑃𝑃𝑃𝑥𝑥 �𝑇𝑇 (𝑥𝑥)−𝑇𝑇𝑥𝑥𝑆𝑆 = 𝑇𝑇∞(𝑥𝑥)−𝑇𝑇∞,0
𝑤𝑤

∞,0

�

(10)
𝐶𝐶−𝐶𝐶

Φ(𝜁𝜁, 𝜂𝜂) = 𝐶𝐶 −𝐶𝐶∞,0
𝑤𝑤

∞,0

(11)

Where 𝜂𝜂 and 𝜁𝜁 are the pseudosimilarity variable and the non-similarity parameter respectively. Value of 𝜁𝜁 ranging
from 0 to 1. 𝜁𝜁 = 0 for pure free convection and 𝜁𝜁 = 1 for pure forced convection. 𝜓𝜓 and 𝑓𝑓 are the stream function anddimensionless stream function respectively. 𝜓𝜓 is given by 𝑢𝑢 = (1⁄𝑟𝑟)(𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕) and 𝑣𝑣 = −(1⁄𝑟𝑟)(𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕), thus, theequation of continuity is satisfied. 𝜃𝜃 and Φ are dimensionless temperature and dimensionless concentration respectively. 𝑆𝑆
is the local thermal stratification parameter. 𝑃𝑃𝑃𝑃𝑥𝑥 is the local Peclet number its equal to 𝑃𝑃𝑃𝑃𝑥𝑥 = 𝑈𝑈∞ 𝑥𝑥 ⁄𝛼𝛼. 𝑅𝑅𝑅𝑅𝑥𝑥 is the local
Rayleigh number its equal to 𝑅𝑅𝑅𝑅𝑥𝑥 = [𝑔𝑔𝛽𝛽𝑇𝑇 𝐾𝐾𝐾𝐾 (𝑇𝑇𝑤𝑤 − 𝑇𝑇∞ )⁄𝛼𝛼𝛼𝛼 ]. When we substituting the dimensionless variables into
momentum equation, energy equation, concentration equation, and boundary conditions the following non-similarboundary layer dimensionless equations can be obtained:
2.7. Dimensionless Momentum Equation

Where Γ =1⁄2

𝑐𝑐 √𝐾𝐾 �𝑃𝑃𝑃𝑃𝑥𝑥[1 + 2Γ𝑓𝑓 ′ + 𝑀𝑀]𝑓𝑓 ′′ + (1 − 𝜁𝜁)2 (−𝜃𝜃 ′ − 𝑁𝑁Φ′) = 0

1⁄2 2+𝑅𝑅𝑅𝑅𝑥𝑥 � 𝛼𝛼

𝜈𝜈𝜈𝜈
𝛽𝛽𝐶𝐶
𝑁𝑁 = is the buoyancy ratio.
𝛽𝛽𝑇𝑇

is the inertia effect parameter, 𝑀𝑀 =HTFF 120-4(12)𝜎𝜎𝛽𝛽𝑜𝑜2 𝐾𝐾
is the square of the Hartmann number, and𝜇𝜇𝜇𝜇2.8. Dimensionless Energy Equation
1

𝑚𝑚

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

−[2𝜂𝜂𝜂𝜂 + 1]𝜃𝜃 ′′ + �− �1 + 𝑚𝑚(1 − 𝜁𝜁)�𝑓𝑓 − 2𝜆𝜆� 𝜃𝜃 ′ + 𝑚𝑚𝑓𝑓 ′ 𝜃𝜃 = − 𝜁𝜁(1 − 𝜁𝜁) � 𝜃𝜃 ′ − 𝑓𝑓 ′ � − 𝑆𝑆𝑆𝑆𝑆𝑆′ (13)
2
2
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
Where 𝜆𝜆 =𝑥𝑥
1𝑟𝑟𝑜𝑜 �𝑃𝑃𝑃𝑃𝑥𝑥1⁄2 +𝑅𝑅𝑅𝑅𝑥𝑥1⁄2 �

corresponds to vertical flat plate.

is the curvature parameter. When the value of 𝜆𝜆 is equal to zero, this situation is

2.9. Dimensionless Concentration Equation

−

1
1
2𝜆𝜆
𝑚𝑚
𝜕𝜕𝜕𝜕
𝜕𝜕Φ
(2𝜂𝜂𝜂𝜂 + 1)Φ′′ + �− �1 + 𝑚𝑚(1 − 𝜁𝜁)�𝑓𝑓 − � Φ′ + 𝑚𝑚𝑓𝑓 ′ Φ = − 𝜁𝜁(1 − 𝜁𝜁) � Φ′ − 𝑓𝑓′�
𝐿𝐿𝐿𝐿
2
𝐿𝐿𝐿𝐿
2
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

(14)

Where 𝐿𝐿𝐿𝐿 = 𝛼𝛼 ⁄𝐷𝐷 is the Lewis number.

2.10. Dimensionless Boundary Conditions

𝑓𝑓(𝜁𝜁, 0) = 0
𝑓𝑓′(𝜁𝜁, ∞) = 𝜁𝜁 2

𝜃𝜃(𝜁𝜁, 0) = 1 − 𝑆𝑆𝜃𝜃(𝜁𝜁, ∞) = 0

(15)
(16)

Φ(𝜁𝜁, 0) = 1
Φ(𝜁𝜁, ∞) = 0

The formulas of velocity components, local Nusselt number and local Sherwood number in terms of dimensionlessvariables can be expressed by:
𝑈𝑈

𝑢𝑢 = 𝜁𝜁∞
2 𝑓𝑓′

𝑟𝑟 𝛼𝛼

(17)
⁄ 1 1

1

1

𝜕𝜕𝜕𝜕

𝑣𝑣 = − 𝑟𝑟𝑜𝑜 𝑃𝑃𝑃𝑃𝑥𝑥1 2 𝜁𝜁 � �1 + 𝑚𝑚(1 − 𝜁𝜁)�𝑓𝑓 − �1 − 𝑚𝑚(1 − 𝜁𝜁)�𝜂𝜂𝑓𝑓 ′ − 𝑚𝑚𝑚𝑚(1 − 𝜁𝜁) �
𝑥𝑥
2
2
2
𝜕𝜕𝜕𝜕
𝑁𝑁𝑁𝑁𝑥𝑥
1⁄2

𝑃𝑃𝑃𝑃𝑥𝑥 𝜁𝜁 −1𝑆𝑆ℎ𝑥𝑥
1⁄2 −1𝜁𝜁

𝑃𝑃𝑃𝑃𝑥𝑥= −𝜃𝜃′(𝜁𝜁, 0)

(18)
(19)
(20)

= −Φ′(𝜁𝜁, 0)

It is remind to state that, the presence of 𝜕𝜕⁄𝜕𝜕𝜕𝜕 in the above dimensionless equations makes them non-similar. Also, theprimes means partial differentiation with respect to 𝜂𝜂.

3. Numerical Solution procedure

The condition 𝜂𝜂 → ∞ in Eq. (16) is displaced by (𝜂𝜂 = 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 ) where 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 is adequately large value so that thecondition of velocity is satisfied. The domain of solution (𝜁𝜁, 𝜂𝜂) is divided into equal spaced mesh in the 𝜂𝜂 direction whereΔ𝜂𝜂 = 0.02 and another equal spaced mesh in the 𝜁𝜁 direction where Δ𝜁𝜁 = 0.1. To conserve space, the method used to solve
the non-similar system of Eqs. (12) to (14) are presented in Mohammad [18].

4. Results and Discussion

To assess the accuracy of the numerical results to be presented in the subsequent paragraphs, a compare on special
case of the problem with previously published work is conducted. The comparison of local Nusselt number for the whole
regime of combined convection with different values of power law index 𝑚𝑚 and for the case of Darcy flow when thecurvature parameter is equal to zero (𝜆𝜆 = 0 corresponds to vertical wall) is shown in Table 1.
HTFF 120-5

⁄Table 1: Comparison of 𝑁𝑁𝑁𝑁𝑥𝑥 ⁄�𝑃𝑃𝑃𝑃𝑥𝑥1 2 𝜁𝜁 −1 � values along vertical wall (𝜆𝜆 = 0) for previously publishedwork against present work at different values of 𝑚𝑚 and 𝜁𝜁 for Darcy flow.

𝜁𝜁
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

𝑚𝑚 = 00.44380.40350.37320.355
0.35060.36030.38320.41740.46030.50980.5642

Hsieh et al. [19]𝑚𝑚 = 0.50.77040.69910.64190.6026
0.58440.589
0.616
0.66290.72590.80140.8862𝑚𝑚 = 110.90710.83140.77830.75220.75550.78770.84570.925
1.0206
1.1284𝑚𝑚 = 00.44420.40390.3736
0.35530.35060.36030.38320.41740.46030.50990.5642Present work
𝑚𝑚 = 0.50.77080.69910.64180.60240.584
0.5886
0.61570.66270.72590.80150.8863𝑚𝑚 = 11.00040.90690.83110.77790.75160.75480.78720.84540.92481.02051.1284

From this table it is obvious that the agreement between the present work and the previous published work is good.
Again in order to conserve space the influence of the parameters considered in this study along with the parameter 𝜁𝜁 onvelocity, temperature, and concentration profiles will be described only without graphs. Furthermore, for some parameters
the local Nusselt and Sherwood numbers will be described only. At a given value of the exponent 𝑚𝑚 (in Eq. 9) and forlower values of the non-similarity parameter 𝜁𝜁 it was noticed that the velocity gradient is greater. At a fixed value of 𝜁𝜁, as
𝑚𝑚 increases the velocity, temperature, and concentration gradients are larger and the momentum, thermal, andconcentration boundary layer thicknesses are smaller. From the numerical results it was noticed for Γ = 𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 =2, 𝐿𝐿𝐿𝐿 = 5, and 𝑆𝑆 = 0.3 that the boost in the value of 𝑚𝑚 (𝑚𝑚 = 0, 0.5, 1) lead to increase in the amount of heat and masstransfer.
To a given value of 𝜁𝜁 the rise in the parameter of inertia lower the velocity of the fluid. Also, it was noticed that thevelocity profiles for natural convection thin near the cylinder surface and thick far from the cylinder surface. Furthermore,
at a given value of the mixed convection parameter an increase in the inertia parameter increases the temperature andconcentration of the fluid. It is observed that for pure forced convection, increasing of inertia effect parameter has noeffectiveness on the temperature and concentration profiles. When the inertia effect parameter enhances (Γ = 0.1, 1, 10)with 𝑚𝑚 = 0.5, 𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 = 2, 𝐿𝐿𝐿𝐿 = 5, and 𝑆𝑆 = 0.3 the local Nusselt and Sherwood numbers dwindling. This is
explicit because inertia effectiveness tends to delay the momentum transmit in the boundary layer and hence lessen the heatand mass transfer. The curves according to the different values of inertia effect parameter are converge to single point in
the forced convection frontier. We can infer from this behavior that inertia term has small weight in forced convection
when used the Ergun’s correlation.
Numerical solution presented that the increasing value of the square of the Hartmann number leads to increasing thevelocity gradient at a given value of 𝜁𝜁, while decreasing the temperature and concentration gradients. It can be interpreted
the reduction in the velocity of the fluid is due to the resistance of the magnetic force named Lorentz force. The action of
the magnetic force on the local Nusselt and Sherwood numbers are shown in Fig. 2 and Fig. 3 respectively.

HTFF 120-6

1.15
1.10

M= 0
M= 1
M= 2

ζ -1)
Nux /(Pe1/2
x

1.05
1.00
0.95
0.90
0.85
0.80
0.75
0.0

0.2

0.4

0.6

0.8

1.0

ζ
Fig. 2: Local Nusselt number variation for total regime of mixed convection withdifferent values of square of the Hartmann number. (𝑚𝑚 = 0.5, Γ = 𝜆𝜆 = 1, 𝑁𝑁 = 2,
𝐿𝐿𝐿𝐿 = 5, 𝑆𝑆 = 0.3)

It is clear that the increasing of magnetic force causes lowering in the amount of heat and mass transfer. Also, thiseffect decay as we transfer to the forced convection dominated regime. The results presented that as the curvatureparameter raises the velocity, temperature, and concentration gradient decreases. When (𝜆𝜆 = 0.5, 1, 2) with 𝑚𝑚 = 0.5, Γ =𝑀𝑀 = 1, 𝑁𝑁 = 2, 𝐿𝐿𝐿𝐿 = 5, and 𝑆𝑆 = 0.3 we can deduce that larger values of the curvature parameter give large amount of heatand mass transfer for whole region of mixed convection. This means that as the cylinder becomes slender, corresponding toa large value of (𝑥𝑥 ⁄𝑟𝑟𝑜𝑜 ) would generate a higher heat and mass transfer rates.
The impact of buoyancy ratio parameter on the boundary layer profiles can be explained as follows. As the buoyancy
ratio parameter excesses, the velocity gradient decreases and the temperature and
2.5
2.4

M= 0
M= 1
M= 2

2.3

Shx /(Pe1/2
ζ -1)
x

2.2
2.1
2.0
1.9
1.8
1.7
1.6
0.0

0.2

0.4

0.6

ζ

0.8

1.0

Fig. 3: Local Sherwood number variation for total regime of mixed convection
with different values of square of the Hartmann number. (𝑚𝑚 = 0.5, Γ = 𝜆𝜆 = 1,
𝑁𝑁 = 2, 𝐿𝐿𝐿𝐿 = 5, 𝑆𝑆 = 0.3)

HTFF 120-7

concentration gradients increases. Also it was observed that, for forced convection, the temperature and concentration are
not influenced by the increase in the value of buoyancy ratio parameter. According to the above reasons the amount of heatand mass transfer enhances as the value of buoyancy ratio raise as presented in Fig. 4 and Fig. 5 respectively. It is clear thatthe effect of buoyancy ratio on heat and mass transfer is weak in the forced convection dominated regime.

1.15
1.10

N=1
N=2
N=3

Nux /(Pe1/2
ζ -1)
x

1.05
1.00
0.95
0.90
0.85
0.80
0.75
0.0

0.2

0.4

0.6

0.8

1.0

ζ
Fig. 4: Local Nusselt number variation for total regime of mixed
convection with different values of buoyancy ratio. (𝑚𝑚 = 0.5, Γ = 𝑀𝑀 =𝜆𝜆 = 1, 𝐿𝐿𝐿𝐿 = 5, 𝑆𝑆 = 0.3)

2.4

N=1
N=2
N=3

1/2
x

-1

Shx /(Pe ζ )

2.2
2.0
1.8
1.6
0.0

0.2

0.4

0.6

ζ

0.8

1.0

Fig. 5: Local Sherwood number variation for total regime of mixed
convection with different values of buoyancy ratio. (𝑚𝑚 = 0.5, Γ =𝑀𝑀 = 𝜆𝜆 = 1, 𝐿𝐿𝐿𝐿 = 5, 𝑆𝑆 = 0.3)

HTFF 120-8

Increasing the value of the Lewis number has the effectiveness of increasing of velocity and concentration gradients
while decreasing of temperature gradients. This is due to the fact that the rise in the thermal diffusivity lead to rise in thethermal buoyancy forces and consequently the temperature of the fluid raise. On the other hand due to the reduction in themass diffusivity, the concentration buoyancy forces lowering and as a result the concentration of the fluid decrease. Fig. 6
indicates that the excess in the value of the Lewis number leads to reduction in the value of the local Nusselt number for
lower and moderate values of local non-similarity parameter. For higher value of local non-similarity parameter (i.e. forced
convection dominated region) this reduction can be neglected. On the contrary, Fig. 7 depicts

1.15
1.10

Le = 0.5
Le = 5
Le = 10

1.05

Nux /(Pe1/2
ζ -1)
x

1.00
0.95
0.90
0.85
0.80
0.0

0.2

0.4

0.6

0.8

1.0

ζ

Fig. 6: Local Nusselt number variation for total regime of mixed
convection with different values of the Lewis number. (𝑚𝑚 = 0.5, Γ =𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 = 2, 𝑆𝑆 = 0.3)

Shx /(Pe1/2
ζ -1)xthat the rise in the Lewis number leads to increase in the local Sherwood number for the total regime of mixed convection
and this increase is appreciable.
3.4
3.2
Le = 0.5
3.0
Le = 5
2.8
Le = 10
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 7: Local Sherwood number ζ
variation for total regime of mixed
convection with different values of the Lewis number. (𝑚𝑚 = 0.5, Γ =𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 = 2, 𝑆𝑆 = 0.3)

HTFF 120-9

Nux /(Pe1/2
ζ -1)x

From the numerical results it was observed that the enhance of the thermal stratification parameter will increase thevelocity, temperature, and concentration gradients and decrease the hydrodynamic, thermal, and concentration boundary
layer thicknesses. Furthermore, when the stratification parameter growing the local Nusselt and Sherwood numbers
decreases as shown in Fig. 8 and Fig. 9. Also from
1.35
1.30
S=0
1.25
S = 0.1
S = 0.2
1.20
S = 0.3
1.15
S = 0.4
1.10
S = 0.5
1.05
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.0
0.2
0.4
0.6
0.8
1.0
ζ
Fig. 8: Local Nusselt number variation for total regime of mixed convection
with different values of local thermal stratification parameter. (𝑚𝑚 = 0.5, Γ =𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 = 2, 𝐿𝐿𝐿𝐿 = 5)

Fig. 8 we can view that the entire regime of mixed convection is influenced significantly by the increasing of thestratification parameter. On the other hand, this action on the local Sherwood number is weak as compared with local
Nusselt number, and can be restricted to natural convection and natural convection dominated region (lower values of
mixed convection parameter). For forced convection dominated region this effect is very low and can be neglected aspresented in Fig. 9.

2.5
2.4

S=0
S = 0.1
S = 0.2
S = 0.3
S = 0.4
S = 0.5

Shx /(Pe1/2
ζ -1)
x

2.3
2.2
2.1
2.0
1.9
1.8
1.7
0.0

0.2

0.4

0.6

0.8

1.0

ζ
Fig. 9: Local Sherwood number variation for total regime of mixed
convection with different values of local thermal stratification parameter.
(𝑚𝑚 = 0.5, Γ = 𝑀𝑀 = 𝜆𝜆 = 1, 𝑁𝑁 = 2, 𝐿𝐿𝐿𝐿 = 5)
HTFF 120-10

5. Conclusions

The numerical results show that the raise in the value of 𝑚𝑚 lead to enhance the amounts of heat and mass transfer.
When the inertia effect parameter growing, the local Nusselt and Sherwood numbers lessening. Increasing of magnetic
force causes lowering amount of heat and mass transfer. Larger values of the curvature parameter would generate a higher
heat and mass transfer rates. The amount of heat and mass transfer enhances as the value of buoyancy ratio parameter raise.
The excess in the value of the Lewis number leads to reduction in the value of the local Nusselt number. On the contrary,
the rise in the Lewis number leads to increase in the local Sherwood number. Finally, when the stratification parameterrises the local Nusselt and Sherwood number decreases.

Acknowledgements

The author would like to thank the college of engineering-university of Mosul for the support.

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