Mixed convection flow of couple stress fluid between rotating discs with chemical reaction and double diffusion effects

1 Theoretical Model

Consider the flow of a steady, incompressible, laminar, axisymmetric couple stress fluid with cross diffusion effects in between two parallel rotating disks separated by a distance d. On the lower disc the cylindrical coordinate system (R, φ, Z) is established with the origin at the disk center and rotating with it. Let the condition of lower disk be the reference. The flow diagram of two parallel rotating discs are shown in Fig. (1). The lower disc is placed at Z = 0 whereas the upper disc is placed at Z = d. The upper disc is rotating with Ω₂ and the lower one is rotating with an angular velocity Ω₁. Uniform temperature T₁ and concentration C₁ are maintained on lower disc, while the other is maintained at T₂ and C₂ respectively. The fluid has constant physical properties. The Boussinesq approximation is invoked by considering the buoyancy effects induced by the body forces i.e., the Coriolis and the centrifugal forces and the gravity are linked with the density due to the rotation of the disc and the curvilinear motion of the fluids, these all are taken as variables. The temperature, concentration relation and linear density, ρ = ρ_r[1–β_T(T–T₁–β_C (C – C₁)] is considered for the rotational buoyancy effects, where the subscript r denotes a reference state. β_C and β_C are the coefficients of solutal and thermal expansions.

With the above assumptions and considerations, the governing equations for the flow are given by

Fig. 1

Physical model and coordinate system.

where in the direction R, φ and Z, the velocity components are U, V and W respectively. T is the temperature, C is the concentration, ν is the kinematic viscosity, C_p is the specific heat at constant pressure, C_s is the concentration susceptibility, D_m is the mass diffusivity, g is the acceleration due to gravity, k₁ is the rate of chemical reaction, K_T is the thermal diffusion ratio, P is the pressure, T_m is the mean fluid temperature, η₁ is the coupling material constant and α is the thermal diffusivity. P* = P–P_r is the difference between local pressure and the reference and ∇P_r/ρ_r = Ω × Ω × R + g is the conservative part of the pressure field. ∇12=[∂2∂R2+1R∂∂R−1R2+∂2∂Z2] and ∇22=[∂2∂R2+1R∂∂R+∂2∂Z2].

On the disks, according to the no-slip condition, the radial and axial velocities (U and W) are zero. At the lower disc the tangential velocity V is equal to zero. Though, the tangential velocity at upper disk is R(Ω₂ – Ω₁) due to the relative motion of two disks. Hence the boundary conditions are given by

Introducing the following similarity transformations

where ΔT = T₂ – T₁ and ΔC = C₂ – C₁ are the characteristic temperature and concentration differences respectively. G is the tangential velocity, H is the axial velocity and θ is the temperature function. The transformation is of Karman type but these are extended to the temperature and concentrations. The final dimensionless form of the governing equations (1)-(6) are of the form:

using (9), (8) in (2), (4) and by eliminating pressure term we get

where the superscript ( )′ denotes differentiation with respect to η, the relative importance of thermal diffusion effects from viscous is indicated by the Prandtl number Pr=να whereas the relative importance of solutal diffusion effects from viscous indicated by the Schmidt number Sc=νDm, the characterization of the rotational effect can be dealt by the Reynolds number Re=d2Ω1ν and it is ranging from 50 to 500, B₁ = β_T(T₂ – T₁) is the thermal buoyancy parameter, B₂ = β_C(C₂ – C₁) is the solutal buoyancy parameter, Df=DmKT(C2−C1)νCSCP(T2−T1) is the Dufour number, Sr=DmKT(T2−T1)νTm(C2−C1) is the Soret number, K=k1ν is the chemical reaction parameter, S=1dη1μ is the Couple stress parameter.

Boundary conditions (7) in terms of G, H,θ and ϕ become

The parameter γ=Ω2−Ω1Ω1 denotes the relative rotation rate to the upper disk with respect to the lower disk.

By definition, friction factor is Cf=2μ(∂U∂n)w/ρ(RΩ1)2, in normal direction to the walls the velocity gradient is denoted by (∂U∂n)w. The friction factors at disks 1 and 2 in the direction are given by:

respectively. Where Re* = (RΩ₁)d/ν is the local Reynolds number. The parameters C_f1Re* and C_f2Re* are closely related to the torque of the rotating disks. Heat and mass transfer performance is expressed by Nusselt and Sherwood numbers defined as Nu=(∂T∂n)w/(T2−T1), Sh=(∂C∂n)w/(C2−C1). The heat and mass transfer rates on the two disks are

2 Solution of the problem by HAM

The initial approximations of H(η), G(η), θ(η) and ϕ(η) are chosen as

with the the auxiliary linear operators

such that

where c_i(i = 1, 2,...,12) are constants.

The non-zero convergence control parameters h₁, h₂ and h₃ are introduced in HAM Solution of the problem. To find the range of the values of the auxiliary parameters h₁, h₂, h₃ and h₄, the h-curves are plotted and shown in Figs. 2 - 5 by taking the values of the parameters Re = 50, B₁ = B₂ = 0.01, Pr = 0.71, Sc = 0.22, S = 1 and γ = –1. It is found that the range of auxiliary parameters are –1.4 < h₁ < –0.55, –1.35 < h₂ < –0.65, –1.45 < h₃ < –0.55 and –1.3 < h₄ < –0.7 respectively. The optimum values of the auxiliary parameters are chosen as h₁ = -1.015, h₂ = -0.975, h₃ = -0.95, h₄ = -0.99 by using the average residual errors (at different order of approximations (m) residual errors calculated [32]). The validity of the convergence control parameters is shown through the Table 1.

Fig. 2

h curve for H(η)

Fig. 3

h curve for G(η)

Fig. 4

h curve for θ(η)

Fig. 5

h curve for ϕ(η)

3 Results and discussion

Graphical illustrations for velocity profiles F(η), H(η), G(η), θ(η) and ϕ(η) are presented in Figs. 6 to 20. The effects of the couple stress fluid parameter (S), Soret (S_r) and Dufour (D_f) parameters and the chemical reaction parameter (K) are presented. The physical parameters γ, B₁, B₂ are fixed at –1, 0.01, 0.01 respectively to study the nature of emerging parameters on all the profiles.

Fig. 6

Effect of D_f and S_r on radial velocity profile (F) at S = 1.0, K = 0.1

Fig. 7

Effect of D_f and S_r on tangential velocity profile (G) at S = 1.0, K = 0.1

Figs. 6-8 displays the non-dimensional velocities F(η), G(η), H(η) for various values of D_f and S_r. It can seen from there figures that as there is an increase in Soret number (or decrease of Dufour number) leads to decrease in the velocity of the fluid. Since the values of D_f increases due to either an increase in the temperature difference or decrease in the concentration difference. Therefore the velocity profiles decreases with the increase of the Soret number i.e., the lowest peak of the reverse flow velocity compatible with the lowest Dufour number and highest Soret parameter. The effects of Dufour parameter D_f and Soret parameter S_r on temperature profile is presented in Fig. 9. It is observed that, an increase in S_r (or decrease of D_f) leads to the decrease in the temperature profile of the fluid. The parameter S_r (Soret number) does not enter directly into the energy equation and the effect of the Dufour grimace the heat fluxes and intensify the mass fluxes. Hence the flow heats vigorously with the increase of the Dufour parameter. Fig. 10 demonstrates that the concentration of the fluid increases as Dufour effect decreases (or Soret effect increases). This is because of temperature gradients contribution to diffusion of the species.

Fig. 8

Effect of D_f and S_r on axial velocity profile (H) at S = 1.0, K = 0.1

Fig. 9

Effect of D_f and S_r on temperature profile (θ) at S = 1.0, K = 0.1

Fig. 10

Effect of D_f and S_r on concentration profile (ϕ) at S = 1.0, K = 0.1

Figures 11 to 15 shows that the couple stress fluid parameter S effect on F(η), H(η), G(η), θ(η) and ϕ(η). It is observed from these figures that the velocities in all the directions (radial, tangential and axial) are decreases with an increase in the couple stress parameter S. The temperature of the fluid decreases as S increases. Figure 15 demonstrates that an increase in S leads to the increase in the concentration of the fluid. It is known that comparing with the Newtonian fluids the velocity of non-Newtonian fluid (couple stress fluid) is less.

Fig. 11

Effect of S on radial velocity profile (F) at S_r = 2.0, D_f = 0.03, K = 0.1

Fig. 12

Effect of S on tangential velocity profile (G) at S_r = 2.0, D_f = 0.03, K = 0.1

Fig. 13

Effect of S on axial velocity profile (H) at S_r = 2.0, D_f = 0.03, K = 0.1

Fig. 14

Effect of S on temperature profile (θ) at S_r = 2.0, D_f = 0.03, K = 0.1

Fig. 15

Effect of S on concentration profile (ϕ) at S_r = 2.0, D_f = 0.03, K = 0.1

The chemical reaction K on velocities, temperature and concentration are shown in figs. 16 to 20. It is observed from these figures that, the fluid velocities in all the directions decreases as K increases. The temperature profile and concentration profile of the fluid also decreases when the chemical reaction parameter increases. Since the chemical molecular diffusivity drop down when chemical reaction is high, i.e., lesser diffusion. Hence, they are procured by the transfer of species. The higher K will decrease the concentration species. Therefore with the increase in K suppresses the distribution of the concentration at all the points of the fluid flow. With this, it can be claimed that on the distribution of the concentration, massive effect is higher with heavier diffusing species.

Fig. 16

Effect of K on radial velocity profile (F) at S_r = 2.0, D_f = 0.03, S = 1.0

Fig. 17

Effect of K on tangential velocity profile (G) at S_r = 2.0, D_f = 0.03, S = 1.0

Fig. 18

Effect of K on axial velocity profile (H) at S_r = 2.0, D_f = 0.03, S = 1.0

Fig. 19

Effect of K on temperature profile (θ) at S_r = 2.0, D_f = 0.03, S = 1.0

Fig. 20

Effect of K on concentration profile (ϕ) at S_r = 2.0, D_f = 0.03, S = 1.0

The effects of Dufour and Soret parameters, chemical reaction parameter on the physical parameters like skin-friction coefficient, rate of heat and mass transfers are shown in Table 2. It can be seen from this table that the value of friction factor increases with an increase in the chemical reaction parameter. It is observed that as the reaction parameter increases the heat and mass transfer rates also increases. Finally, the effects of Soret and Dufour effects on friction factor and the heat and mass transfer rates are shown in this table for the fixed values of the reaction parameter. The exploits of these parameters is apparent from the Table 2 and therefore are not explained for briefness.

4 Conclusions

In this present study, the effects of chemical reaction with Soret and Dufour on steady couple stress fluid flow in between rotating parallel discs have been presented. Similarity transforms have been used to transform the governing equations into the system of ordinary differential equations. Homotopy analysis method has been applied to get approximate analytic solutions. Due to rotation of the discs the Coriolis forces and the centrifugal are introduced into the momentum equation. The profiles of the dimensionless velocity, temperature and concentration for the various values of the emerging parameters of the present study. In table form the skin-friction and the heat and mass transfer rates are presented.

From this study it is clear that the fluid velocity, temperature decreases and the concentration of the fluid increases as an increase in S. There is decrease in the velocities, temperature and the rate of mass transfer of the fluid whereas the concentration, skin friction, heat transfer rate increases with the decrease of Dufour number (or increase of Soret number). When there is an increase in the reaction parameter there is decrease in the velocity, temperature and concentration and increase in friction, rates of heat and mass transfer.