Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux

1 Introduction

The analysis of unsteady flow and the heat transfer of conducting fluid is another one central part of our research, practical applications of this unsteady flow are noticed in cooling systems in refrigerator, electronic devices in computers, air conditioners and air heat exchangers. The mathematical modeling which describes the motion of a gas carrying small dust particles in addition to this equations satisfied by small disturbances of a steady laminar flow is derived by Saffman [1]. Prasad et al. [2] studied an unsteady flow of a dusty incompressible fluid between two parallel plates under an impulsive pressure gradient. Andersson et al. [3] studied momentum and heat transfer in a laminar liquid film on an unsteady stretching sheet. Elbashbeshy et al. [4] investigated similarity solution of the laminar boundary layer equations corresponding to an unsteady stretching surface. Singh [5] studied an unsteady flow of a conducting dusty fluid passing through a rectangular channel in the presence of uniform magnetic field and time dependent pressure gradient. Dixit et al. [6] investigated an unsteady flow of a dusty viscous fluid through rectangular ducts. Mitra et al. [7] studied an unsteady laminar flow of an electrically conducting, viscous and incompressible dusty fluid between two largely extended non-conducting parallel plates under magnetic field. Ali et al. [8] studied an unsteady fluid and heat flow induced by a submerged stretching surface while steady motion is slowed down gradually. Unsteady boundary layer stagnation point fluid flow and transfer of heat on the way to a stretching sheet by taking in an account of partial slip conditions is investigated by Bhattacharyya et al. [9]. Vajravelu et al. [10,11] studied the motion and heat transfer characteristics in a second grade fluid over a stretching sheet with given temperature together with the effects of some physical quantities like internal heat generation or absorption. And work due to deformation and the heat transfer in the laminar boundary layer of a viscous fluid on a linearly stretching sheet, continuous surface with variable wall temperature subject to suction or blowing. Gireesha et al. [12] investigated heat transfer characteristics of an incompressible dusty fluid past a vertical linearly stretching sheet. Viscoelastic boundary layer flow and heat transfer characteristic over linearly stretching sheet subjected to viscous dissipation and non-uniform heat source, MHD boundary layer flow and heat transfer characteristics of a viscoelastic fluid over a linearly stretching sheet in the presence of thermal radiation, and non-uniform heat source, boundary layer flow and heat transfer of viscoelastic fluid over linear stretching sheet subjected to viscous dissipation, non-uniform heat source has been studied by Abel et al. [13, 14, 15]. Unsteady boundary layer over linear stretching sheet for certain distributions and surface temperature or surface heat flux is studied by Sharidan et al. [16]. Ishak et al. [17] studied heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. Characterization of heat and mass flux on MHD unsteady Eyring Powell steady dusty nanofluid over an unsteady sheet with re-served Brownian motion and thermophoresis in nanofluid using the Cattaneo Christov heat flux model, and MHD Carreau fluid in a suspension of dusty nanoparticles like graphene and Ag part-icles using Cattaneo Christov heat flux model was studied by Mamatha et al. [18,19].

Dispersion of graphene nanoparticles in dusty fluids has many applications in biocompatibility, bio-imaging processing, biosensors, detection and purification of cancer treatment, and in monitoring stem cell differentiation etc. An unsteady three dimensional flow of Carreau and Casson fluids past a stretching sheet in the presence of homogeneous and heterogeneous reaction by the effect of nonlinear thermal radiation and non-uniform heat source/sink, numerical study on an unsteady Casson nanofluid motion over a rotating cone in a rotating frame filled with water based ferrous nanoparticles, an unsteady Magnetonano fluid flow caused by a rotating cone with temperature dependent viscosity, a surgical implant application and heat and mass transfer of MHD Casson fluid over a moving wedge with slip, nonlinear stretching sheet subjected to uniform heat source/sink and chemical reaction studied by Raju et al. [20,21,22, 23]. Krishnamurthy et al. [24] investigated the effect of nonlinear thermal radiation on slip flow and heat transfer of melting of dusty fluid suspended with Cupper nanoparticles immersed in a porous medium over a stretching sheet. Two dimensional stretched flow of Jeffrey material in view of Cattaneo-Christov double diffusion model was addressed by Hayat et al. [25]. Haq et al. [26]. Studied the process of natural convection flow in moderately heated trapezoidal cavity filled with nano tubes in the incidence to single wall Carbon tubes. Instantaneous effect of non vertical magnetic field and prescribed surface temperature on boundary layer flow of nano fluid over stretching sheet has been analyzed by Rashid et al. [27] and reported that flow character of nanofluid are determined by stream line pattern. Motion of nanofluid and heat transfer characteristics between two horizontal plates in a revolving system was studied by Sheikholeslami et al. [28], and they found that, nanoparticles volume fraction and Raynolds number are increased with increasing suction, injection and heat transfer rate. Soomro et al. [29] studied entropy initiation study for the magnetohydrodynamic combined convection flow of water functionalized Carbon nanotubes along with non-vertical stretching surface. Mutual effect of non-vertical nanofluid and velocity slip conditions are analyzed for nanofluid over moving flat plate has been analyzed by Haq et al. [30]. Rehaman et al. [31] investigated on heat and mass transport enrichment of water-based nanofluid for three dimensional MHD steady point flow affected by an exponentially stretching surface Some of important research analysis considered in our study is given in [32, 33, 34, 35, 36]. In addition to above discussions and applications nanofluid with non-Newtonian properties have vast industrial and technological applications such as surgical implantation, advanced powder technology, petroleum industry, aerosol engineering, wood coating, etc. The study of Carreau fluid has received considerable attention in the recent period, since the traditional Newtonian fluid not able to describe precisely the behavior of fluid flow with suspended particles. Hence, the purpose of present study is to investigate the heat source or sink on an unsteady radiative MHD Carreau fluid in a suspension of dust and graphene nanoparticles with CCHF (Cattaneo-Christov heat flux). The arising set of physical system is solved numerically.

2 Formulation of the problem

In this study, we considered unsteady electrically conducting MHD Carreau and Casson fluids in a suspension of dust and graphene nanoparticles. To enhance heat transport phenomena we also considered thermal radiation and Cattaneo-Christov heat flux model. The flow configuration is shown in the Fig. 1. The x-axis is along the shrinking surface towards the direction of motion of fluid and y-axis is normal to the sheet. The flow is bounded in the region y>0. It is assumed that the sheet is shrunk by two equal and opposite forces with the velocity u_w(x)=bx/(1−ct). The flow field is exposed by the external uniform magnetic field of strength B₀ along the y-axis which generates magnetic field effect in perpendicular (x)direction. The simulation is performed by mixing of graphene +water into dusty nano fluid. We considered Nano and dust particles are to be spherical in shape. Throughout the flow problem number density of the dust particles are considered to be constant and volume fraction of dust particles is neglected. The drag force is taken into consideration for the fluid and particle interaction. Stokes linear drag theory is used to model the drag force. The wall and the ambient temperatures are T_w(x,t) and T_∞ respectively.

Fig. 1

Physical model representing the flow problem

The constitutive equation for a Carreau fluid is given by [20];

Where τ – extra stress tensor, η₀ – zero-shear rate viscosity, η_∞ – infinite --shear rate viscosity, Λ – material time constant, and n – dimensionless power law index. The shear rate y˙¯ is stated as

Here Π – second invariant strain tensor.

The rheological model for an istropic flow of Casson fluid is given by

In the above equation Π=e_ije_ij and e_ij is the (i,j)th component of the deformation rate, Πthe product of the component of deformation rate with itself, Π_c is a critical value of this product based on the non-Newtonian model, Π_B is the plastic dynamic viscosity of non-Newtonian fluid, and is the yield stress of the fluid. The anonymous researchers has suggested the value of n = 1. However, in many applications this value is n> 1.

According to the above assumptions, the physical governing system can be expressed as (Krishnamurthy et al. [24] and Hayat et al. [25]):

Based on the physical problem the boundary conditions are

The thermo physical properties of nanofluids are given by

The radiative heat flux is given by

To convert the governing equations into a set of similarity equations, we introduce the following transformation as,

By applying equation (10) into equations (1) and (2) are automatically satisfies the continuity. Equations (3) – (9) are transformed as follows.

The transformed boundary conditions are

Where A, We, M, K, α, β_ν, Pr, Λ, Q_H, l, β_T, Ec, R, γ, τ_ν are specified as

For the purpose of engineering interest, skin friction coefficient (C_f), local Nusselt number (Nu_x) are defined as;

Where Rex=Uwxυf is local Reynolds number.

3 Method of solution

The nonlinear differential equations (ODEs) (11) – (14) with the boundary constraints (15) are solved numerically using Runge-Kutta Feldberg method. Initially, the set of nonlinear ODEs converted to 1_st order differential equations, by using the following process:

With boundary conditions as

we guess the values of y₃(0), y₄(0), y₅(0), y₇(0) which are not given at the initial conditions. The equations (20) – (24) are integrated by taking the help of Runge-Kutta Feldberg method with the successive iterative step length is 0.01.

The correctness of the supposed values is checked by equating the calculated values y₂, y₄, y₅, y₆, y₈ at ζ=ζ_max with their given values at ζ=ζ_max. If there is any difference exist the process is continued upto the required good values. Alternatively, we are using the Runge-Kutta method to get the accurately found the initial values of y₂, y₄, y₅, y₆, y₈ and then integrate (20) – (24) by using the Runge-Kutta method. This process is repeated until the settlement between the designed value and the condition given at is within the specified degree of accuracy 10₋₅.

4 Result and Discussion

Non-dimensional governing equations (11) – (14) with the boundary conditions (15) are solved numerically with shooting technique. In order to obtain the results, numerical computations are carried out by considering different values of non-dimensional governing parameters as; p=0.05, ζ=1, l_a=0.3, γ=0.3, R=0.5, ϕ_d=0.01, α=0.2, β_v=0.2, β_v=0.2, Ec=0.1, We=0.2, M=0.5.l=2, A=1, m=0.3, K=0.5, Q_H=0.2, B=0.2. These values are considered as fixed values throughout the study excluding the variations in the respective figures and tables. In figures Solid line indicates the fluid phase profiles and dashed lines refers the dust phase profiles of the flow. Table 1 displays the variation of local Nusselt number for varied values of Pr. The present results are compared with available results in the literature.

The influence on velocity and temperature fluid and dust phase with reference to volume fraction of nanoparticles (ϕ), magnetic field M, thermal relaxation time (γ), porosity (K), heat source parameter (Q_H), unsteadiness parameter (A), thermal radiation parameter (R)and for both the mixtures of (Carreau and Casson) fluid and dust phases are discussed as follows.

Figs. 2–3 depicts the effect of M on both the velocity and temperature profiles. It is found that by increasing the value of M the velocity and temperature profiles are falls down in the case of Carreau fluid, this is due to fact that electric conducting fluid in the presence of magnetic field induces a retarding Lorentz force which works in perpendicular direction to the applied magnetic field. Since M is the ratio of hydromagnetic body force to viscous force, larger value of M gives higher hydromagnetic body force, the same effect follows in the case of Casson fluid phase, but in dust phase the velocity of Casson fluid reduces and temperature profile enhances. Figs. 4 – 5 displays the effect of the volume fraction on velocity and temperature profile, by this graph it is found that, by the increasing value of ϕincreases the temperature profiles of both Casson and Carreau fluid, but the reverse effect can observe in the case of velocity profiles, this is due to fact that the more number of ϕincreases the interaction between particles this interaction enhances temperature profiles and decrements the velocity profiles. Fig. 6 shows the effect of Q_H on temperature profiles, it is found that the heat source parameter enhances the temperature profile as heat source parameter act as heat generator/absorber which gives thermal energy in the direction of flow, in this case the Casson fluid temperature profiles are higher than Carreau fluid, the reverse effect is observed in dust phase. Fig. 7 refers the effect of radiation parameter R on temperature profiles, by this graph it is clear that the radiation parameter strengths the temperature profiles of Casson and Carreau fluid, here we noticed that the temperature profiles of Casson fluid is high compared to Carreau fluid, as radiation produces heat molecules, these molecules develops the thermal boundary layer. Fig. 8 shows the effect of thermal relaxation time γ on temperature profiles, it is clear that the thermal relaxation time enhances the temperature profiles of Carreau and Casson fluid, and the temperature profiles of Casson fluid is higher than Carreau fluid, Figs. 9 – 10 depicts the effect of unsteadiness parameter on both the velocity and temperature, by these graphs it is clear that the unsteadiness parameter strengthens the velocity and temperature profiles, the velocity profiles of Carreau fluid are high compared to Casson fluid but the temperature profiles of Carreau fluid are lower than Casson fluid. Figs. 11 – 12 depicts the effect of porosity on velocity and temperature profiles of Carreau and Casson fluid, from these graphs it is clear that the porosity effect strengthens the velocity of both the Casson and Carreau fluid but the reverse effect can observe in the case temperature profiles.

Fig. 2

Impact of M on velocity profiles.

Fig. 3

Impact of M on temperature profiles.

Fig. 4

Impact of ϕ on velocity profiles.

Fig. 5

Impact of ϕ on temperature profiles.

Fig. 6

Impact of Q_H on temperature profiles.

Fig. 7

Impact of R on temperature profiles.

Fig. 8

Impact of γ on temperature profiles.

Fig. 9

Impact of A on velocity profiles.

Fig. 10

Impact of A on temperature profiles.

Fig. 11

Impact of K on velocity profiles.

Fig. 12

Impact of K on temperature profiles.

Table 1 validates the current solutions with already existing literature. Table 2 is constructed to analyze the numerical values of skin friction coefficient and local Nusselt number for different physical governing parameter the improved values of parameters M and ϕ improves heat transfer rate and skin friction of both Carreau and Casson fluid. The improved values of Q_H, R and γ does not effects the skin friction of both Casson and Carreau fluid i.e skin friction remains unaltered by the improvement of these parameters, but these improved values reduces heat transfer rate of both Casson and Carreau fluid. The parameter increases skin fraction and decreases heat transfer rate of both Casson and Carreau fluid. The parameter K increases skin friction and heat transfer rate of both Casson and Carreau fluid.

5 Conclusions

Considered unsteady radiative magneto hydrodynamic Casson-Carreau fluids in suspension of graphene particle with Cattaneo-Christov model. A simulation is performed by mixing of graphene nanoparticles into the base water. The arising set of governing partial differential equation (PDEs) are transformed into set of ordinary differential equation (ODEs) using similarity transformation and then solved numerically using shooting technique with Runge-Kutta (RK) method. Some important outcomes of our result are mentioned as follows.

1. The effect of porosity decreases the heat transfer for both Carreau and Casson fluid.

2. The effect of unsteadiness parameter increases the heat transfer of both Carreau and Casson fluid.

3. The thermal radiation parameter increases the heat transfer of both Carreau and Casson fluid.

4. Volume fraction increases heat transfer rate and decreases the velocity profile.