Hall and ion-slip effects on mixed convection in a chemically reacting fluid between rotating and stationary disks

1 Introduction

The steady flow of fluid and heat transfer between two rotating disks has attracted the attention of many researchers due to its prodigious applications in machinery works such as design of thrust bearings, radial diffusers, gas-turbines and compressor disks, rotating-disk contractors, etc. The rotating-disk configuration is simple in geometry while the flow physics of rotating disks is highly complicated in nature. The complexities reflected in bifurcation phenomena and multiplicity of the flow states are stemmed from the non-linearity of the rotating fluids. Extensive research can be found in the literature to study the flow and heat transfer between disks with different analysis. To mention few, the readers can see Chin and Litt [1], Wang and Watson [2], and Singh and Rajans [3]. Roberts and Shipman [4] analyzed numerically the multiple-cell solution for the fluid flow between rotating and stationary disk by the continuation method. Malik et al. [5] investigated experimentally the transition in rotating disk flow, in which the growth rates of the boundary layer disturbances has been measured using a hot-wire probe including the effects of Coriolis forces and streamline curvature. The behaviour of spiral vortices generated in a boundary layer transition of a rotating disk has been examined by Kohama [6]. Anaturk and Szeri [7] investigated the fluid flow between infinite rotating discs. Ewa and Chyi [8] studied the induced mixed convection flow in a rotor-stator system and noticed that thermal buoyancy effects influenced by the rotational forces in a non-isothermal temperature field do change the onset condition and the related critical parameters for this rotating mixed convection.

When the flow is under the influence of magnetic field on rotating disk, the practical applications may be mentioned in areas such as computer storage devices, lubrication, crystal growth processes, viscometry and rotating machinery. Considerable efforts have been directed toward the steady flow between two rotating discs considering the magnetic effect. Flow due to a rotating disk with Hall effect has been reported by Hassan and Attia [9]. Attia [10] scrutinized unsteady MHD flow near a rotating porous disk with uniform suction or injection. Sattar and Maleque [11] studied flow between a rotating discs for transient convective flow with magnetic field and heat absorption effects. Attia [12] discussed the hydro-magnetic flow of a viscous incompressible electrically conducting fluid due to an infinite rotating disk considering Hall and ion slip effects. In the presence of Hall and variable properties effects, the MHD convective flow due to rotating porous disc has been analyzed by Maleque and Sattar [13].

When heat and mass transfer take place simultaneously in a moving fluid, the relation between the fluxes and the driving potentials are of intricate nature. It has been noticed that an energy flux can be generated not only by temperature gradients but also by concentration gradients. Thermal diffusion or Soret effect, corresponds to species differentiation developing in an initial homogeneous mixture subjected to a temperature gradient. The heat flux induced by a concentration gradient is called Dufour or diffusion-thermo effect. In most of the studies related to heat and mass transfer process, Soret and Dufour effects are neglected on the basis that they are of a smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. But these effects are considered as second order phenomena and may become significant in areas such as hydrology, petrology, geosciences, etc. Many researchers studied the Soret and Dufour effects on convection flow in rotating disks with different analysis [Osalusi et al. [14]; Devi and Devi [15]; Rashidi et al. [16]; Freidoonimehr et al. [17]].

Motivated by all these works, the present study extends the works of Ewa and Chyi [8] by considering the complex interactions of chemically reacting fluid in the presence of hydro-magnetic, Hall and ion-slip, and cross-diffusion effects. From the literature survey, it seems that this type of analysis has not been investigated so far. Hence, the effects of Hall, ion-slip and cross-diffusion in a chemically reacting fluid between two disks have been discussed numerically using spectral quasi-linearisation method. The velocity, temperature and concentration distributions together with the stress, rate of heat and mass transfers are discussed numerically for various physical parameters and displayed through graphs and tables in detail.

2 Mathematical formulation of problem

Consider the motion of a chemically reacting fluid between two disks with Hall and ion slip effects. The lower disk rotates at a rate Ω₁ whereas the upper disk is stationary, located at a distance a and a cylindrical coordinate system (r, φ, z) is fixed on the lower disk and rotating with it. The flow description and geometrical coordinates are shown in Figure 1.

Fig. 1

Geometry of the flow

The following assumptions are made to formulate the problem:

1. The flow is laminar, steady and axisymmetric.

2. The fluid is incompressible, Newtonian and electrically conducting.

3. A uniform magnetic field B₀ is acting perpendicular to the disks and also assumed that the magnetic Reynolds number is small, so that the induced magnetic field is neglected.

4. The Boussinesq approximation is invoked into the physical model in order to consider the buoyancy effects induced by the involved body forces, i.e., (i) the density associated with the terms of gravity, the centrifugal and the Coriolis forces due to the disk rotation, and the curvilinear motion of the fluids are all considered as variable, and (ii) the thermal and concentration buoyancy effect holds by linear density, temperature and concentration relation ρ = ρ_r[1−(T−T₁)–(C–C₁)] is used.

5. The lower disk is maintained at a temperature of T₁ while the upper disk is maintained at a temperature T₂. Further, C₁ and C₂ are concentrations at lower and upper disks respectively.

6. The energy flux generated by temperature and concentration gradients are taken into consideration.

7. A first order chemical reactions is incorporated into the model.

The equations that govern the motion (Ewa and Chyi [8]) with the above assumptions are given by:

where (u, v, w) are the velocity components in (r, φ, z) respectively, T and C are the temperature and concentration, β_T is the thermal-expansion coefficient, β_C is the coefficient of solutal expansion, ρ is the density, g is the acceleration due to gravity, α = 1 + β_iβ_h, where β_i and β_h are the ion-slip and Hall parameters respectively, σ is the electrical conductivity, κ is the thermal diffusivity, C_p is the specific heat at constant pressure, C_s is the concentration susceptibility, P^* = P−P_r is the difference between local pressure and the reference pressure, T_m is the mean fluid temperature, T₁ and T₂ are the temperatures in the lower and upper disks respectively, D_m is the mass diffusivity and K_T is the thermal diffusion ratio.

Boundary conditions are given by

Introduce the following similarity variables

Using the similarity relation f′−2Re12h=0 obtained from the continuity equation together with (8), the flow governing equations become

where Pr=να is the Prandtl number, Df=DmKT(C2−C1)CsCp(T2−T1) is the Dufour number, Sr=DmKT(T2−T1)νTm(C2−C2) is the Soret number, Sc=νDm is the Schmidt number, K=K1a2ν is the chemical reaction parameter, B₁ and B₂ are the thermal and solutal buoyancy parameters respectively, Ha is the Hartmann number and Re is the Reynolds number.

The boundary conditions will have the form

Where prime indicates differentiation with respect to η.

The non-dimensional Nusselt and Sherwood numbers respectively given by

The non-dimensional radial and tangential shear stress respectively given by

3 Numerical Solution using SQLM

Here, we describe the spectral quasi-linearization method (SQLM) for solving the non-linear system of Eqs. (9)–(12) along with the boundary conditions (13)–(14). The QLM was proposed by Bellman and Kalaba [18] as a generalization of the Newton-Raphson method and it is used for solving nonlinear boundary value problems.

Assume that the solutions f_r, G_r, θ_r and ϕ_r of Eqs. (9)–(12) at the (r+1)^th iteration are f_r+1, G_r+1, θ_r+1 and ϕ_r+1. If the solutions at the previous iteration is sufficiently close to the present iteration, the nonlinear components of the Eqs. (9)–(12) can be linearised using one term Taylor’s series for multiple variables so that the Eqs. (9)–(12) give the following iterative sequence of linear differential equations:

where the coefficiets a_s1,r (s1 = 1, 2,…,8), b_s2,r (s2 = 1, 2,..,6), c_s3,r(s3 = 1, 2), d_s4,r (s4 = 1, 2) and R_s5,r (s5 = 1,..,4) are known functions (from previous calculations) and are defined as

subject to boundary conditions:

The above system (18) to (21) constitute a linear system of coupled differential equations with variable coefficients and can be solved iteratively using any numerical method for r = 1, 2, 3, …. In this work, the Chebyshev pseudo spectral method is used to solve the QLM scheme (18) to (21) [For more details, one can refer the works of Motsa et al. [19, 20] and RamReddy et al. [21, 22]].

The initial guesses to start the SQLM scheme for the system of equations (18) - (21) are chosen as functions that satisfy the boundary conditions as follows:

Starting from a given set of initial approximations f₀, G₀, θ₀, ϕ₀, the iteration schemes (18) to (21) can be solved iteratively for f_r+1(η), G_r+1(η), θ_r+1(η), ϕ_r+1(η) when r = 0, 1, 2, …. To solve equations (18) to (21), we discretise the equation using the Chebyshev spectral collocation method.

The basic idea behind the spectral collocation method is the first appearance of a differentiation matrix D which is applied to approximate the differential coefficients of the unknown variables, for example, f(η) at the collocation points as the matrix vector product

at the Gauss-Lobatto collocation points

where N + 1 is the number of collocation points (grid points), D=2Dη∞ is the differentiation matrix and its entries are clearly defined in Canuto et al. [23], and F = [f(τ₀), f(τ₁), …, f(τ_N)]^T is the vector function at the collocation points. Similar vector functions corresponding to G, θ and ϕ are denoted by G, Θ and Φ respectively. Higher order derivatives are obtained as powers of D, that is

where p is the order of the derivatives, and τ is a variable used to map the truncated interval [0, 1] to the interval [−1, 1] on which the spectral method can be implemented.

Substituting Eqs. (24)–(26) into Eqs. (18)–(21) leads to the matrix equation

subject to the boundary condition

In (27) A is a (4N+4)×(4N+4) square matrix, X and R are (4N+4)× 1 column vectors defined by

where

where I is an identity matrix, 0 is a zero matrix and diag[] is a diagonal matrix, all are of size (N+1) × (N+1), where N is the number of grid points, F, G, Θ and Φ are the values of the functions f, G, θ and ϕ when evaluated at the grid points. The subscript r denotes the iteration number.

4 Results and discussion

The nonlinear differential equations (9–12) together with the boundary conditions (13)–(14) have been solved numerically by using spectral quasilinearisation method and results have been obtained for different parameters and presented through graphs and tables. We fix Pr = 0.71, Sc = 0.22, Re = 1000, s = −1, B₁ = 0.1 and B₂ = 0.1 throughout the study unless otherwise mentioned and hence assume the Batchelor-type flow. In order to assess the accuracy of our method, the results (i, e., radial shear stress, tangential shear stress and Nusselt number with varying B₁) are compared with Shooting method as a special case by taking Re =500, B₂ = 0.0, Ha = 0.0, Df = 0.0, Sr = 0.0 and K = 0.0, and it found a good agreement as shown in Table 1.

In a disk system of high rotation rate, the centrifugal force plays a critical role in the flow and heat transfer characteristics. Inspection of Table 2 reveals that, enhancing the ion-slip parameter, fixing the other parameters, the tangential shear stress τ_t and radial shear stress τ_r enhance in magnitude whereas heat transfer rate Nu and mass transfer rate Sh reduces. Similar results observed with the Hall parameter. Increasing the Hartmann number decreases in magnitude τ_r but for a higher Hartmann number Nu, Sh rise in magnitude. Enhancing the chemical reaction parameter reduces τ_r and Nu whereas τ_t and Sh grow in magnitude, similar result found with increasing Soret number.

Physically, when the lower disk rotates uniformly about an axis perpendicular to the plane the layer near the disk carried by friction is thrown outwards owing to the action of centrifugal forces. This is compensated by particles which flow in an axial direction towards the disk to be in tern carried and injected centrifugally. Generally, the radial velocity show reverse flow near the disks where as almost constant towards the middle area of the flow. The tangential flow show a thin boundary layers near the boundaries. The concentration boundary layer is found to be generally thick.

It can be observed that the magnetic field normal to the flow in an electrically conducting fluid introduces a Lorentz force, which acts against the flow. The peak velocity decreases with the increase in Ha due to this retarding effect seen around the boundaries this effect is dominated towards the middle due to the rotational effect. In Fig. 2 the effect of Hartmann number Ha shown. For higher values of Ha, the axial and tangential velocity components increases which are shown in Fig. 2(a) and Fig. 2(c) and also Fig. 2(b) in the central zone, may serves to act as a regulatory mechanism for the flow. It is also noted that near the disks, the velocity profile is high indicating the existence of boundary layer in that region. In Fig. 2(d) and Fig. 2(e) the effect of Hartmann on temperature and concentration presented, it is seen that both temperature and concentration fall down with raise in Hartmann number.

Fig. 2

Effect of Hartmann number on (a) axial velocity, (b) radial velocity, (c) tangential velocity, (d) temperature, and (e) concentration profiles.

Fig. 3 and Fig. 4 display the effects of ion and Hall parameters respectively. As β_i increases the effective conductivity also augments, in turn, decreases the damping force on the velocity component in the direction of the flow, hence reduces the axial and tangential velocity components which are shown in Figs. 3(a) and 3(c). Further, in Fig. 3(b) the radial velocity show reverse flow near the disks whereas towards the center no significant effect has been noticed. The reverse effect to that of ion-slip is observed for higher values of β_h for the case of the axial velocity in Fig. 4(a)). From the Fig. (c), it is found that the tangential velocity decreases with raise of β_h. Both temperature and concentration augments with β_i enhancement and reduces with β_h as depicted in Figs. 3(d)–3(e) and Figs. 4(d)–4(e) respectively.

Fig. 3

Effect of ion-slip parameter on (a) axial velocity, (b) radial velocity, (c) tangential velocity, (d) temperature, and (e) concentration profiles.

Fig. 4

Effect of Hall parameter on (a) axial velocity, (b) radial velocity, (c) tangential velocity, (d) temperature, and (e) concentration profiles.

As it can be seen from Figs. 5(a)–5(c) that no significant effect on velocity profiles varying the chemical reaction parameter. Fig. 5(d) depict that increasing the chemical reaction parameter, the temperature enhances slightly. In Fig. 5(e) the concentration diminishes with rise in chemical reaction parameter. This is due to the fact that the chemical reaction in this system results in consumption of the chemical and hence results in decrease of concentration profile.

Fig. 5

Effect of chemical reaction parameter on (a) axial velocity, (b) radial velocity, (c) tangential velocity, (d) temperature, and (e) concentration profiles.

Figs. 6(a)–6(e) present the influence of Soret and Dufour on temperature and concentration. Increasing Soret number (or simultaneous decrease in Dufour number) augments the temperature as seen in Fig. 6(d) whereas the reverse effect observed on concentration Fig. 6(e). From the definition, Soret number can be defined as the effect of temperature on concentration and Dufour effect can be defined as the effect of concentration on temperature. This shows that, diffusive species with higher Soret values decelerates the concentration profiles whereas temperature species with lower Dufour values has the tendency of enhancing the temperature profiles. Thus, it is concluded from Figs. 6(d)–6(e) that the temperature and concentration distributions are more influenced with the values of Soret and Dufour parameters.

Fig. 6

Effect of Soret and Dufour number on (a) axial velocity, (b) radial velocity, (c) tangential velocity, (d) temperature, and (e) concentration profiles.

5 Conclusion

In this paper, we have sought to determine numerically using spectral quasi-linearisation method how the presence of Hall, ion-slip, magnetic, chemical reaction, Soret and Dufour parameter affects the flow. It is found that increasing Hartmann number decreasing the temperature and concentration profiles. Ion-slip parameter augments the temperature and concentration where as the Hall parameter reduces both the temperature and concentration profiles. By rising the chemical reaction parameter, the temperature rises slightly whereas the concentration diminishes. The higher the Hall parameter magnifies the boundary layer near the boundary of the disks. The raise in Hall or ion-slip parameters shows enhancement in τ_t and τ_r, reduction in Nu, Sh. With the higher Hartmann number, τ_r lower in magnitude and Nu, Sh grow in magnitude but, except τ_t. Rising the chemical reaction or the Soret number reduces the radial shear stress and Nusselt number whereas amplifies the tangential shear stress and Sherwood number.