Cattaneo -Christov heat flux model for magnetohydrodynamic flow in a suspension of dust particles towards a stretching sheet

2.1 Flow Analysis

In the present analysis, we consider the flow of an electrically conducting dusty fluid which is incompressible viscous in nature with an electrical conductivity over a linearly stretching sheet. The flow is two dimensional where stretching surface is chosen along the x-axis and y-axis is normal to it flow is assumed to be confined with respect to the region y > 0. The sheet is stretched by keeping origin fixed and applying two equal and opposite forces along the x-axis. A magnetic field of uniform strength B₀ is applied along the y-axis. The fluid is assumed to be Newtonian and its properties are constant and the chemical reaction is assumed to take place during the flow. The volume fraction and interparticle collision of dust particles are assumed to be negligible. The induced magnetic field is neglected by considering the Reynolds number to be small .It is assumed that dust particles are uniformly distributed throughout the fluid, equal in size and has spherical shape. Only because of drag force there is interaction, heat and mass transfer between the fluid and dust phases. To model the drag force stokes linear drag concept is employed. U_w (x) = ax is the stretching sheet velocity where as a > 0 is the stretching rate, T_w represents the temperature of the fluid at the sheet whereas T_∞ the ambient temperature so that T_w > T_∞. In view of the above assumptions, the boundary layer flow along stretching sheet describing the conservation of mass and momentum for both clean fluid and dusty fluid can be expressed as follows [7, 23]:

where (u, v) and (u_p, v_p) denote velocity components of the fluid and dust particle phases along x and y directions. Furthermore, are the density of the fluid, dynamic viscosity of the fluid, density of the particle phase, number density of dust particles, B₀ is the uniform magnetic field, is the density ratio, K is the Stokes drag coefficient (K = 6∏ μr) and r is the radius of dust particle.

The boundary conditions applicable to the flow problem are

By employing the generalized Bernoulli’s equation, in the free–stream U(x) = bx the equation (2) transforms to

Substituting (7) into (2)

Introducing the following similarity transformations:

where ψ is the stream function defined by the relation:

One can observe continuity equation is satisfied and equations (8), (3), (5), (9) take the following forms:

The transformed boundary conditions are:

Here prime indicates order of differentiation with respect to ζ and M = σB02ρa is the magnetic field parameter, l=mNρ is the dust particles mass concentration parameter, λ=ba is the ratio of the free stream velocity parameter to the stretching sheet parameter, τ=mk is the relaxation time of the dust particles, i.e., the time required by a dust particle to adjust its velocity relative to the fluid, β=1aτ is the fluid particle interaction parameter, ρr=ρpρ is the relative density.

2.2 Heat transfer analysis

The boundary layer heat equations for both fluid and particle phase in the presence of Cattaneo –Christov heat flux are given by

Here T and T_p represents temperature of the fluid and dust particles, C_m and C_p are specific heat of dust particles and fluid, τ_v is the relaxation time of dust phase, τ_T indicates the thermal equilibrium time i.e., time required by the dust cloud to adjust it temperature to that of fluid, k is the thermal conductivity of the fluid and λ is the thermal relaxation time.

The solutions of the equations (15) and (16) depends on the nature of the prescribed boundary condition

1. Prescribed Wall temperature

2. Prescribed heat flux

3.1 Momentum Boundary layer analysis

Figures 2a and 2b illustrates the influence of non-dimensional fluid particle interaction parameter (β) on the hydrodynamic boundary layer of both fluid and dust phase. It is found that accelerating values of fluid particle interaction parameter β decay the velocity profiles of fluid and dust phase. Moreover, it is observed that the fluid particle interaction parameter effectively reduces the fluid velocity in PWT case than PHF case. Whereas in case of dust phase velocity profiles behave similar in both the PWT and PHF case. Figs 3a and 3b displays the influence of magnetic parameter on velocity profiles of fluid and dust phase. It is observed that velocity profiles decay with the increasing values of magnetic field parameter in both fluid and dust phase. Physically, fluid motion is restricted with the increasing values of M due the drag force developed by Lorentz force. Nature of velocity profiles is similar in both PWT and PHF case in case of dust fluid. Figures 3a, 4b presents the effect of λ –the ratio of free stream velocity parameter to stretching velocity parameter. We observe that dust phase velocity is an increasing function of λ but an opposite outcome is observed in fluid phase. But the nature of velocity profiles is similar in both PWT and PHF case for fluid and dust phase.

Fig. 1

The physical geometry of flow configuration

Fig. 2

a, b) Velocity profiles with varying fluid particle interaction parameter β.

Fig. 3

a, b) Velocity profiles with varying magnetic parameter (M).

Fig. 4

a, b) Velocity profiles with varying parameter (λ) on the fluid and dust phase.

3.2 Thermal boundary layer analysis

The effect of fluid particle interaction parameter on fluid and dust phase temperature profiles is plotted in Figures 5a and 5b. An increase in β may be regarded as increase in temperature profiles in both fluid and dust phase. Further, for large values of β temperature profiles are similar in both PHF and PWT case in case of dust and fluid phase. With increase in β there is a sharp growth in wall slope of temperature function in both the PHF and PWT case for both the fluids (Clean and Dust phase).

Fig. 5

a, b) Temperature profiles with varying fluid particle interaction parameter (β).

Figures 6a, 6b the effect of magnetic parameter on thermal boundary layers is displayed. It is observed that accelerating values of Magnetic field parameter enhance the temperature field in case of both kinds of fluid (Clean and Dust fluid). This is due to Lorentz force. Moreover we can observe magnetic field effectively controls the thermal field of the flow in PWT case when compared with PHF case. Figures 7a, 7b portray the effect of thermal relaxation time on temperature profiles of fluid and dust phase. We notice increasing values of thermal relaxation time corresponds to low temperature. Increasing values of thermal relaxation time enhance the gap between the fluid particles due to which they require large time to bring heat to their neighboring particles. One can noticed that the influence of thermal relaxation time is more effective over a PHF case than PWT case. The influence of Eckert number on temperature profiles of fluid and dust phase are displayed in Figures 8a and 8b. A rise in values of Ec enhances the thermal boundary layer in case of fluid and dust phase. Eckert number relates the relationship between Kinetic energy in the flow to the enthalpy. It embodies in converting Kinetic energy to internal energy by work done against the viscous fluid stresses. It is clear that rising viscous dissipation enhances in the temperature and thermal boundary layer thickness. It is also observed that temperature in PWT case is higher than PHF case. The effect of λ on temperature profiles are plotted in Figures 9a and 9b for both the fluids (Clean and Dust) and boundary conditions (PWT and PHF cases). Deterioration in both temperature profiles can be observed with increasing values of λ. This may happen due to very low values of free stream velocity to stretching ration velocity.

Fig. 6

a, b) Temperature profiles with varying magnetic parameter (M).

Fig. 7

a, b) Temperature profiles with varying thermal relaxation time (y).

Fig. 8

a, b) Temperature profiles with varying Eckert number (Ec).

Fig. 9

a, b) Temperature profiles with varying parameter (λ).

Table 2 demonstrates the variation of local Nusselt number and friction coefficients for various values of non-dimensional governing parameters. The particle interaction parameter increases the friction factor in both cases (PWT and PHF) and suppresses the local Nusselt number, this is due to dominance of higher particle interaction. Whereas, similarly opposite phenomena was observed in the presence of thermal relaxation parameter. The cause for this trend is when we incorporate thermal relaxation the surrounding particles are highly motivated. Due to this we saw enhancement in rate of heat transfer. Both the friction factor and local Nusselt number are minimized with the rising values of the Eckert number and magnetic field parameters. The opposite behavior of M was observed in λ. The rising values of λ increases the both Nusselt and friction factor values. Table 1 shows the validation of present solutions with already existing studies under some limited condition of Grubka and Bobba [8], Abel and Mahesha [12] and Ramesh et al. [23]. It is found that raising the values of prandtl number improves the local nusselt number.