Free Convective MHD Flow Past a Vertical Cone with Variable Heat and Mass Flux

Abstract

A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power law according to 𝑞_𝑤(𝑥)=𝑥^𝑚 and 𝑞^∗_𝑤(𝑥)=𝑥^𝑚, respectively, where 𝑥 denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then nondimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature, and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (𝑚), surface mass flux power-law exponent (𝑛), Schmidt number, buoyancy ratio parameter, and semivertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the published results and are found to be in excellent agreement. The local skin friction, Nusselt number, and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes, and hybrid solar energy systems.

1. Introduction

Combined heat and mass transfer in fluid-saturated porous media finds applications in a variety of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, and others. A thorough discussion of these and other applications is available in the monographs [1, 2]. Comprehensive reviews of the much of the work communicated in porous media transport phenomena have been presented in [3, 4]. Most studies dealing with porous media have employed the Darcy law. However, for high velocity flow situations, the Darcy law is inapplicable, since it does not account for inertial effects in the porous medium. Such flows can arise, for example, in the near-wellbore region of high capacity gas and condensate petroleum reservoirs and also in highly porous filtration systems under high blowing rates. The most popular approach for simulating high-velocity transport in porous media is the Darcy-Forchheimer drag force model. This adds a second-order (quadratic) drag force to the momentum transport equation. This term is related to the geometrical features of the porous medium and is independent of viscosity. A seminal study discussing the influence of Forchheimer inertial effects in porous media convection is presented by Vafai and Tien [5]. The mixed convective boundary layer flow from a vertical surface in a fluid-saturated non-Darcian porous medium including Forchheimer inertial effects is studied by C.-H. Chen and C.-K. Chen [6] and Chen et al. [7]. Thermal convection boundary layer flow with buoyancy and suction/blowing effects from a cone with nonuniform surface temperature is studied by Hossain and Paul [8]. The study is extended by Hossain and Paul [9] by considering nonuniform surface heat flux, both studies employing numerical methods. Chamkha et al. [10] studied the double-diffusive convection heat and mass transfer over a cone (and wedge) in Darcy-Forchheimer porous media. Magnetohydrodynamic (MHD) flowandheattransferisofconsiderableinterestbecauseitcanoccurin many geothermal, geophysical, technological, and engineering applications such as nuclear reactors and others. The geothermal gases are electrically conducting and are affected by the presence of a magnetic field. Vajravelu and Nayfeh [11] studied hydromagnetic convection from a cone anda wedge withvariablesurfacetemperatureandinternalheatgenerationorabsorption. Thus far the transient thermal convection flow over a cone in Darcy-Forchheimer porous media has not been studied in the literature despite important applications in geothermics, geophysics, and materials processing.

2. Mathematical Model

An axisymmetric unsteady natural convection boundary layer flow past a vertical cone with transverse magnetic field applied normal to the surface with variable heat and mass flux in a Darcy-Forchheimer fluid saturated porous medium in a cartesian (𝑥,𝑦) coordinate system is formulated mathematically in this section. Initially, it is assumed that the cone surface and the surrounding fluid which are at rest possess the same temperature 𝑇^′_∞ and concentration level 𝐶^′_∞ everywhere in the fluid. At time 𝑡^′>0, heat supplied from the cone surface to the fluid and concentration level near the cone surface are raised at a rate of 𝑞_𝑤(𝑥)=𝑥^𝑚 and𝑞^∗_𝑤(𝑥)=𝑥^𝑛, respectively, and they are maintained at the same level. It is assumed that the concentration 𝐶^′ of the diffusing species in the binary mixture is very less in comparison to the other chemical species, which are present and hence the Soret and Dufour effects are negligible. We consider viscous flow where pressure work, viscous dissipation, and thermal dispersion effects are neglected. The coordinate system chosen (as shown in Figure 1) is such that the 𝑥-direction is measured along the cone surface from the leading edge 𝑂, and the 𝑦-direction is normal to the cone generator. The cone apex is located at the origin (𝑥=𝑦=0).

Figure 1 

Physical Model.

Here 𝜙 designates the semivertical angle of the cone, and 𝑟 is the local radius of the cone.

Then under the previous assumptions, the governing boundary layer equations with Boussinesq’s approximation are 𝜕(𝑢𝑟)+𝜕()𝜕𝑥𝑣𝑟𝜕𝑦=0,𝜕𝑢𝜕𝑡^′+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝜕𝑦=𝜈²𝑢𝜕𝑦²−𝜎𝐵²₀𝜌𝑢+𝑔𝛽cos𝜙𝑇^′−𝑇^′_∞+𝑔𝛽^∗cos𝜙𝐶^′−𝐶^′_∞−𝜈𝐾𝑏𝑢−𝐾𝑢²,𝜕𝑇^′𝜕𝑡^′+𝑢𝜕𝑇^′𝜕𝑥+𝑣𝜕𝑇^′𝜕𝜕𝑦=𝛼²𝑇^′𝜕𝑦²,𝜕𝐶^′𝜕𝑡^′+𝑢𝜕𝐶^′𝜕𝑥+𝑣𝜕𝐶^′𝜕𝜕𝑦=𝐷²𝐶^′𝜕𝑦², (1) where all terms are defined in the nomenclature. Under the Boussinesq approximation, buoyancy effects are simulated only in the momentum equation, which is coupled to the energy equation, constituting a free convection regime. The corresponding spatial and temporal initial and boundary conditions at the surface and far from the cone take the following form: 𝑡^′≤0∶𝑢=0,𝑣=0,𝑇^′=𝑇^′_∞,𝐶^′=𝐶^′_∞𝑡∀𝑥,𝑦,_′>0∶𝑢=0,𝑣=0,𝜕𝑇^′𝑞𝜕𝑦=−^𝑤(𝑥)𝑘,𝜕𝐶^′𝑞𝜕𝑦=−^∗𝑤(𝑥)𝐷at𝑦=0,𝑢=0,𝑇^′=𝑇^′_∞,𝐶^′=𝐶^′_∞at𝑥=0,𝑢⟶0,𝑇^′⟶𝑇^′_∞,𝐶^′⟶𝐶^′_∞as𝑦⟶∞, (2) where all the parameters are defined in the nomenclature.

Equation (1) is highly coupled, parabolic, and nonlinear. An analytical solution is clearly intractable, and in order to facilitate a numerical solution we nondimensionalize the model. Proceeding with the analysis, we now introduce the following transformations: 𝑥𝑋=𝐿𝑦,𝑌=𝐿Gr_𝐿^1/4,𝑟𝑅=𝐿,where𝑟=𝑥sin𝜙,𝑉=𝑣𝐿𝜈Gr_𝐿^−1/4,𝑈=𝑢𝐿𝜈Gr_𝐿^−1/2,𝑡=𝜈𝑡^′𝐿²Gr_𝐿^1/2,𝑀=𝜎𝐵²₀𝐿²𝜇Gr_𝐿^−1/2,𝑇𝑇=^′−𝑇^′_∞𝑞_𝑤(𝐿)𝐿/𝑘,Gr_𝐿=𝑔𝛽cos𝜙𝑞_𝑤𝐿/𝑘𝐿⁴𝜈²,𝜈Pr=𝛼𝐾,Da=𝐿²𝐶,𝐶=^′−𝐶^′_∞𝑞^∗_𝑤(𝐿),𝛽𝐿/𝐷𝑁=^∗𝐶^′_𝑤−𝐶^′_∞𝛽𝑇^′_𝑤−𝑇^′_∞𝜈,Sc=𝐷𝑏,Fs=𝐿. (3)

The transport (1) is thereby reduced to the following dimensionless form: 𝜕(𝑈𝑅)+𝜕()𝜕𝑋𝑉𝑅𝜕𝑌=0,𝜕𝑈𝜕𝑡+𝑈𝜕𝑈𝜕𝑋+𝑉𝜕𝑈=𝜕𝜕𝑌₂𝑈𝜕𝑌²−𝑈−𝑀𝑈+𝑇cos𝜙+𝑁𝐶cos𝜙DaGr_𝐿−Fs𝑈Da²𝜕𝑇𝜕𝑡+𝑈𝜕𝑇𝜕𝑋+𝑉𝜕𝑇=1𝜕𝑌𝜕Pr²𝑇𝜕𝑌²,𝜕𝐶𝜕𝑡+𝑈𝜕𝐶𝜕𝑋+𝑉𝜕𝐶=1𝜕𝑌𝜕Sc²𝐶𝜕𝑌². (4)

The corresponding nondimensional initial and boundary conditions are given by𝑡≤0∶𝑈=0,𝑉=0,𝑇=0,𝐶=0∀𝑋,𝑌,𝑡>0∶𝑈=0,𝑉=0,𝜕𝑇𝜕𝑌=−𝑋^𝑚,𝜕𝐶𝜕𝑌=−𝑋^𝑛at𝑌=0,𝑈=0,𝑇=0,𝐶=0at𝑋=0,𝑈⟶0,𝑇⟶0,𝐶⟶0as𝑌⟶∞, (5) where again all the parameters are given in the nomenclature. The dimensionless local values of the skin friction, Nusselt number, and the Sherwood number are given by the following expressions: 𝜏_𝑥=−𝜕𝑈𝜕𝑌_𝑌=0Nu_𝑥=−𝑋𝜕𝑇𝜕𝑌_𝑌=0Sh_𝑥=−𝑋𝜕𝐶𝜕𝑌_𝑌=0. (6)

3. Numerical Solution

In order to solve the unsteady, nonlinear, coupled (4) under condition (5), an implicit finite difference scheme of Crank-Nicolson type has been employed which is discussed by many researchers [12–15]. The finite difference scheme of dimensionless governing equations is reduced to tridiagonal system of equations and is solved by Thomas algorithm as discussed elsewhere [16]. The region of integration is considered as a rectangle with 𝑋_max=1 and 𝑌_max=22 where 𝑌_max and corresponds to 𝑌=∞ which lies very well outside both the momentum and thermal boundary layers. The maximum of 𝑌 was chosen as 22, after some preliminary investigation so that the last two boundary conditions of (5) are satisfied within the tolerance limit 10⁻⁵. The mesh sizes have been fixed as Δ𝑋=0.05, Δ𝑌=0.05 with time step Δ𝑡=0.01. The computations are carried out first by reducing the spatial mesh sizes by 50% in one direction, and later in both directions by 50%. The results are compared. It is observed in all cases that the results differ only in the fifth decimal place. Hence, the choice of the mesh sizes seems to be appropriate. The scheme is unconditionally stable. The local truncation error is 𝑂(Δ𝑡²+Δ𝑌²+Δ𝑋), and it tends to zero as Δ𝑡, Δ𝑋 and Δ𝑌 tend to zero. Hence, the scheme is compatible. Stability and compatibility ensure the convergence. The derivatives involved in (6) are evaluated using five point approximation formula.

4. Results and Discussion

Only selective figures have been reproduced here for brevity. In the numerical computations, the following values for the dimensionless thermophysical parameters are prescribed: Grashof number (Gr_𝐿) = 1.0, Darcy number (Da) = 0.1 (high permeability), Forchheimer number (Fs) = 0.1 (weak quadratic drag), Prandtl number (Pr) = 7.0 (water), Schmidt number (Sc) = 0.6 (oxygen diffusing in air), surface heat flux power law exponent (𝑚)=0.5, surface mass flux power law exponent (𝑛)=0.5, buoyancy ratio parameter (𝑁)=1.0, and semivertical angle of the cone (𝜙)=20^∘. All graphs therefore correspond to these values unless otherwise indicated. To test the accuracy of the computations, the local shear stress and local Nusselt number computations for the nonporous case are compared with those of Hossain and Paul [9] for a heat flux gradient of 𝑚=0.5 and 𝑋=1.0 in the steady state, in Tables 1 and 2, respectively, and are found to be in good agreement.

Table 1 

Comparison of local skin friction values at 𝑋=1.0 and 𝑚=0.5 with those of Hossain and Paul [9] for steady state purely fluid (Da → ∞ in present model) case.

Table 2 

Comparison of local Nusselt number values at 𝑋=1.0 and 𝑚=0.5 with those of Hossain and Paul [9] for steady state purely fluid (Da → ∞ in present model) case.

In Figures 2(a) and 2(b), the influence of Grashof number (Gr_𝐿) on steady state velocity (𝑈) and temperature (𝑇) distributions with 𝑌-coordinate are shown. Free convection, that is, thermal buoyancy effects are analyzed via the Grashof number. For an increasing Gr_𝐿 from 0.1 through 1.0, 10.0, 50.0 to 100.0 cooling of the cone by free convection occurs; that is, heat is conducted away from the cone to the surrounding regime.

(a)

(b)

Figure 2 

(a) Steady state velocity profiles at 𝑋=1.0 for different Gr. (b) Steady state temperature profiles at 𝑋=1.0for different Gr.

Figures 3(a) and 3(b) show the effect of Darcy number (Da) on dimensionless velocity (𝑈) and temperature (𝑇) with transformed radial coordinate (𝑌) close to the leading edge (i.e., cone apex) at 𝑋=1.0. To study the influence of regime permeability from sparsely packed media to densely packed materials, the following values Da=1.0, 0.1, 0.01, 0.001 are considered. Da=𝐾/𝐿² for a fixed value of the reference length (𝐿) is directly proportional to permeability (𝐾) of the porous regime. Increasing Da increases the porous medium permeability and simultaneously decreases the Darcian impedance since progressively less solid fibers are present in the regime. The flow is therefore accelerated for higher Da values causing an increase in the velocity 𝑈 as shown in Figure 3(a). Maximum effect of rising Darcy number is observed at intermediate distance from the cone surface around 𝑌∼1. Conversely, temperature 𝑇 depicted in Figure 3(b) is opposed by increasing Darcy number. The presence of fewer solid fibers in the regime with increasing Da inhibits the thermal conduction in the medium which reduces distribution of thermal energy. The regime is therefore cooled when more fluid is present, and 𝑇 values in the thermal boundary layer are decreased. Profiles for both velocity and temperature are smoothly asymptotic decays to the free stream indicating that excellent convergence (and stability) is obtained with the numerical method. Velocity boundary layer thickness will be increased with a rise in Da and thermal boundary layer thickness reduced. The effect of the Forchheimer inertial drag parameter (Fs) on dimensionless temperature (𝑇) profiles is shown in Figure 4. The Forchheimer drag force is a second-order retarding force simulated in the momentum conservation equation. Increasing Fs values from 0.0 through 0.1, 1.0, 5.0, 10.0, 20.0, and 50.0 causes a strong increase in Forchheimer drag which decelerates the flow, that is, reduces velocities. For higher values of Fs, it is expected that the porous medium flow becomes increasingly chaotic. Temperature (𝑇) however is slightly increased with a rise in Forchheimer parameter. The effects of the Prandtl number (Pr) on velocity profiles are depicted in Figure 5. Pr encapsulates the ratio of momentum diffusivity to thermal diffusivity. Larger Pr values imply a thinner thermal boundary layer thickness and more uniform temperature distributions across the boundary layer. Hence, thermal boundary layer will be much less thick than the hydrodynamic (translational velocity) boundary layer. Smaller Pr fluids have higher thermal conductivities, so that heat can diffuse away from the cone surface faster than for higher Pr fluids (thicker boundary layers). Physically the lower values of Pr correspond to liquid metals (Pr ∼ 0.02, 0.05), Pr = 0.7 is accurate for air or hydrogen and Pr = 7.0 for water. The computations show that translational velocity 𝑈 is therefore reduced as Pr rises from 0.72 through 1.0, 2.0, 5.0, 7.0 and 10.0 since the fluid is increasingly viscous as Pr rises.

(a)

(b)

Figure 3 

(a) Transient velocity profiles at 𝑋=1.0for different Da. (b) Transient temperature profiles at 𝑋=1.0 for different Da.

Figure 4 

Transient temperature profiles at 𝑋=1.0 for different Fs.

Figure 5 

Steady state velocity profiles at 𝑋=1.0 for different Pr.

Figure 6 shows the effect of the Schmidt number (Sc) on the dimensionless concentration (𝐶).We note that the Schmidt number (Sc) embodies the ratio of the momentum to the mass diffusivity. Sc therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. Smaller Sc values can represent for example hydrogen gas as the species diffusing in air, Sc = 2.0 implies hydrocarbon diffusing in air, and higher values to petroleum derivatives diffusing in fluids (e.g. ethyl benzene) as indicated elsewhere [17]. As Sc increases, Figure 6 shows that 𝐶 values are strongly decreased as larger values of Sc correspond to a decrease in the chemical molecular diffusing that is, less diffusion therefore takes place by mass transport. The dimensionless concentration profiles all decay from a maximum concentration to zero in the freestream. Greater Sc values correspond to lower chemical molecular diffusivity of the parent fluid so that less diffusion of the species occurs in the regime. Concentration boundary layer thickness will therefore be reduced. For low Sc fluid greater species diffusion occurs and concentration boundary layer thickness increased. For Sc = 1, the Concentration and velocity boundary layers will have approximately the same thickness that is, species and momentum will be diffused at the same rates. With lower Sc values the decay of concentration from the cone surface is more controlled, for increasing values of Sc the profiles descend more and more steeply and concentration falls faster from the surface to a short distance into the boundary layer regime.

Figure 6 

Steady state concentration profiles at 𝑋=1.0 for different Sc.

The effect of surface heat flux power exponent (𝑚) on the steady state temperature (𝑇) is shown in Figure 7. An increase in the value of 𝑚 reduces the temperature. It is also seen that the time required to reach the steady state temperature is more at lower values of 𝑚. Figure 8 depict the distribution of concentration (𝐶) with radial coordinate (𝑌) for various values of the surface mass flux power law exponent (𝑛). The concentration reduces with the increasing 𝑛 values from 0.0 through 0.25, 0.50, 0.75 and 1.0. Increasing Fs clearly reduces the local Nusselt number as shown in Figure 9.

Figure 7 

Steady state temperature profiles at 𝑋=1.0 for different 𝑚.

Figure 8 

Steady state concentration profiles at 𝑋=1.0 for different 𝑛.

Figure 9 

Effect of Fs on local Nusselt number at 𝑋=1.0.

A slight increase in local Nusselt number accompanies the increment in Pr as shown in Figure 10. The influence of the concentration to thermal buoyancy ratio parameter (𝑁), on dimensionless temperature (𝑇) with radial coordinate (𝑌) is shown in Figure 11. 𝑁=0 indicates that thermal and species buoyancy forces are both absent. For 𝑁>0, thermal and species buoyancy forces aid each other. 𝑁=1 implies that both buoyancy forces are of the same order of magnitude. A rise in 𝑁 from 0.0 through 1.0, 2.0, 3.0 and 5.0 induces a retarding effect on the flow in the porous regime that is, velocities are decreased.Increasing 𝑁 (thermal and concentration buoyancy forces assisting each other) decreases temperatures in the regime that is, cools the boundary layer regime. The effect of semivertical angle of the cone (𝜙) on dimensionless temperature (𝑇) with 𝑌-coordinate is shown in Figure 12. It is observed that a rise in 𝜙 substantially increases the temperature 𝑇 in the boundary layer regime. And more time is required to reach the steady state. Figure 13 the influence of magnetic parameter (𝑀) versus spanwise spatial distributions of velocity 𝑈 are depicted. Application of magnetic field normal to the flow of an electrically conducting fluid gives rise to a resistive force that acts in the direction opposite to that of the flow. This force is called the Lorentz force. This resistive force tends to slow down the motion of the fluid along the cone and causes an increase in its temperature and a decrease in velocity as 𝑀 increases. An increase in 𝑀 from 1 though 2, 3, 4 clearly reduces streamwise velocity 𝑈 both in the near-wall regime and far-field regime of the boundary layer.

Figure 10 

Effect of Pr on local Nusselt number at 𝑋=1.0.

Figure 11 

Steady state temperature profiles at 𝑋=1.0 for different 𝑁.

Figure 12 

Steady state temperature profiles at 𝑋=1.0 for different 𝜙.

Figure 13 

Steady state velocity profiles at 𝑋=1.0 for different values of 𝑀.

5. Conclusions

Numerical solutions have been presented for the buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime. Present results are compared with those of [8] and found to be in excellent agreement. The following conclusions are drawn.(i)Increasing Grashof number boosts the translational velocity in the cone surface regime and decreases temperature throughout the flow regime.(ii)Increasing Darcy number accelerates the flow, that is, increases translational velocities. However, the temperature is reduced with a rise in Darcy number.(iii)An increase in the Forchheimer inertial drag parameter is observed to slightly increase the temperature but reduces both velocity and local Nusselt number.(iv)An increase in Prandtl number is observed to decrease both temperature and velocity, but the concentration is slightly increased. A slight increase in local Nusselt number accompanies the increment in Pr.(v)The concentration is observed to significantly decrease with an increase in Schmidt number.(vi)The temperature is observed to decrease with an increase in buoyancy ratio parameter but decrease with an increase in semivertical angle of the cone. The time taken to reach the steady state increases with increasing 𝜙.

Nomenclature

𝑥,𝑦:	Coordinates along the cone generator and normal to the generator
𝑢,𝑣:	Velocity components along the 𝑥- and 𝑦-directions
𝑔:	Gravitational acceleration
𝑟:	Local radius of cone
𝑡′∶	Time
𝑡:	Dimensionless time
𝑇′:	Temperature
𝑇:	Dimensionless temperature
𝐶′:	Concentration
𝐶:	Dimensionless concentration
𝐷:	Mass diffusion coefficient
𝐾:	Permeability of porous medium
𝑞𝑤:	Heat flux (i.e., heat transfer rate per unit area)
𝑞∗𝑤:	Mass flux (i.e., mass transfer rate per unit area)
𝑘:	Thermal conductivity of fluid
𝐿:	Reference length
𝑋,𝑌:	Dimensionless coordinates along the cone generator and normal to the generator
𝑈,𝑉:	Dimensionless velocity components along the x- and y-directions
𝑏:	Forchheimer geometrical constant
Da:	Darcy number
Fs:	Forchheimer number
Gr𝐿:	Grashof number
𝑀:	Magnetic parameter
𝐵0:	Magnetic field strength
Pr:	Prandtl number
𝑁:	Buoyancy ratio parameter
Sc:	Schmidt number
𝑚:	Power-law index for surface heat flux relation
𝑛:	Power-law index for surface mass flux relation
Nu𝑥:	Local Nusselt number
Nu𝑋:	Dimensionless local Nusselt number
Sh𝑥:	Local Sherwood number
Sh𝑋:	Nondimensional local Sherwood number
𝑅:	Dimensionless local radius of cone.

Greek Symbols

𝜇:	Dynamic viscosity of fluid
𝜈:	Kinematic viscosity of fluid
𝜙:	Semivertical cone angle
𝛼:	Thermal diffusivity
𝛽:	Volumetric thermal expansion coefficient
𝜃:	Dimensionless temperature function
𝜏:	Dimensionless time
𝜏𝑋:	Dimensionless local shear stress function (skin friction).

Subscripts

𝑤:	Condition on the wall
∞:	Free stream condition.