MHD boundary layer flow of Casson nanofluid over a vertical exponentially stretching cylinder under Newtonian heating

 

M. Sreedhar Babu¹, K. Gangadhar², M. Lavanya¹

¹Department of Applied Mathematics, Y.V. University, Kadapa, A.P.

²Department of Mathematics, Acharya Nagarjuna University, Ongole, AP -523001, INDIA

 

ABSTRACT:

In this paper, similarity solution of the steady boundary layer flow and heat transfer of a Casson nanofluid flowing over a vertical cylinder which is stretching exponentially along its radial direction in the presence of MHD and Newtonian heating is investigated. The governing partial differential equations are converted into nonlinear, ordinary, and coupled differential equations and are solved using bvp4c Matlab solver. The effects of important parameters such as Magnetic parameter, Reynolds number, Prandtl number, Lewis number. Natural convection parameter and the conjugate parameter for Newtonian heating are described through graphs. The numerical results are compared with the published data and are found to be in good agreement.”.

 

 

 

1. INTRODUCTION:

Magneto-hydronomics (MHD) boundary layers with heat and mass transfer over flat surfaces are found in many engineering and geophysical applications such as geothermal reservoirs, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of nuclear reactors. Many chemical engineering processes like metallurgical and polymer extrusion processes involve cooling of a molten liquid being stretched into a cooling system. The fluid mechanical properties of the penultimate product depend mainly on the cooling liquid used and the rate of stretching. Some polymer liquids like polyethylene oxide and polyisobuylene solution in cetin, having better electromagnetic properties are normally used as cooling liquid as their flow can be regulated by external magnetic fields in order to improve the quality of the final product.

 

Makinde and Aziz (2010) investigated the magneto hydro dynamic effect on heat and mass transfer embedded in a porous medium with a convective boundary condition. Rahman et al. (2010) investigated the effects of joule heating and magneto-hydro dynamics mixed convection in an obstructed lid-driven square cavity. Olanrewaju et al. (2011) investigated the stagnation point flow of micro polar fluid over a vertical plate with MHD and thermal radiation. Gangadhar et al. (2012) investigated the hydrodynamic effect on heat and mass transfer past a vertical plate in addition to the effects of convective boundary condition and chemical reaction. Gangadhar (2012) investigated the effects of radiation and viscous dissipation on MHD boundary layer flow of heat and mass transfer through a porous vertical flat plate. Mohammed Ibrahim et al. (2015) studied the effect of MHD on oscillatory flow of heat and mass transfer. Rawat et al. (2016) studied the influence of Non Darcy porous medium and MHD on micro polar fluid over a non-linear stretching sheet and they concluded that on increasing the material parameter leads to a decreasing skin-friction coefficient as well as wall couple stress.”

Casson nanofluid in cylindrical geometry has important application in blood flow. Mathematicians as well as medical researcher are widely working on Casson nanofluid model. Rehman and Nadeem (2012) have examined the mixed convection heat transfer in micropolar nanofluid over a vertical slender cylinder. Malik et al. (2014) studied the similarity solution of the steady boundary layer flow and heat transfer of a Casson nanofluid flowing over a vertical cylinder which is stretching exponentially along its radial direction using Runge–Kutta Fehlberg method. Sarojamma and Vendabai (2015) investigated the boundary layer flow of a Casson nanofluid past a vertical exponentially stretching cylinder in the presence of a transverse magnetic Field with internal heat generation/absorption.”

 

In this paper, similarity solution of the steady boundary layer flow and heat transfer of a Casson nanofluid flowing over a vertical cylinder which is stretching exponentially along its radial direction in the presence of MHD and Newtonian heating is investigated. The governing partial differential equations are converted into nonlinear, ordinary, and coupled differential equations and are solved using bvp4c Matlab solver. The effects of important parameters such as Magnetic parameter, Reynolds number, Prandtl number, Lewis number. Natural convection parameter and the conjugate parameter for Newtonian heating are described through graphs. The numerical results are compared with the published data and are found to be in good agreement.”

 

2. MATHEMATICAL FORMULATION

Consider steady two-dimensional mixed convection boundary layer flow of a viscous incompressible electrically conducting Casson nanofluid past a vertical circular cylinder of radius a. The cylinder is assumed to be stretched exponentially along the radial direction with velocity U_w. the temperature at the surface of the cylinder is assumed to be Tw and the uniform ambient temperature is taken to be T ͚ such that the quantity Tw – T ͚ > 0 for assisting flow, while Tw – T ͚ < 0 for opposing flow, respectively.”

 

 

 

3         SOLUTION OF THE PROBLEM

The set of equations (2.7) to (2.10) were reduced to a system of first-order differential equations and solved using a MATLAB boundary value problem solver called bvp4c. This program solves boundary value problems for ordinary differential equations of the form, by implementing a collocation method subject to general nonlinear, two-point boundary conditions. Here p is a vector of unknown parameters. Boundary value problems (BVPs) arise in most diverse forms. Just about any BVP can be formulated for solution with bvp4c. The first step is to write the ODEs as a system of first order ordinary differential equations. The details of the solution method are presented in Shampine and Kierzenka (2000).”

 

4         RESULTS AND DISCUSSION

Numerical values have been assigned to the governing parameters, the velocity, the temperature and concentration as well as skin friction coefficient, local Nusselt number and local Sherwood number, to get a clear understanding of the physical problem encountered. The figs 1-24 show the numerical calculations.

 

Figures 1-3 present the variation in the velocity, temperature and concentration profiles with the effect of magnetic parameter 𝑀 respectively. The velocity profiles decrease with the raising of magnetic parameter 𝑀. This is due to magnetic field opposing the transport phenomena, since the variation of magnetic parameter 𝑀 causes the variation of Lorentz forces. The Lorentz force is a drag like force that produces more resistance to transport phenomena and that causes reduction in the fluid velocity. The effect of magnetic field is more in shear-thinning fluids than shear thickening fluids. The effect of magnetic fields increases the temperature and concentration profiles (Figures 2and3).”

 

Figures 4-6 establishes the different values of Casson fluid parameter (β) on velocity, temperature and concentration distributions, respectively. It is noticed that, with the hype in the values of β from 0.1 to 1.0 then the resultant dimensionless velocity, temperature and concentration increases consequently increases the momentum, thermal and concentration boundary layer thickness. Figures 7-9 demonstrates the different values of natural convection parameter (λ) on velocity, temperature and concentration distributions, respectively. It is noticed that, with the hype in the values of λ from 0 to 1.2 then the velocity increases consequently increases the momentum boundary layer thickness but the temperature and concentration of the fluid diminishes.”

 

Figures 10-12 establishes the different values of Reynolds number (Re_x) on velocity, temperature and concentration distributions, respectively. It is noticed that, with the hype in the values of Re_x from 0.5 to 3 then the resultant dimensionless velocity, temperature and concentration decreases consequently decreases the momentum, thermal and concentration boundary layer thickness. The effect of thermophoresis parameter (Nt) on the temperature and nanoparticle volume fraction is shown in figures 13 and 14 respectively. It is conformed that enhances the values of Nt from 0.1 to 1, the temperature and concentration distributions of the fluid increases. This leads to enhance the thermal and concentration boundary layer thickness. The effect of Brownian motion parameter (Nb) on the temperature and concentration distributions is shown in figures 15 and 16 respectively. It is conformed that enhances the values of Nb from 0.1 to 1.0, the temperature increases but concentration of the fluid decreases. In the system nanofluid, the Brownian motion takes a place in the presence of nanoparticles. When hype the values of Nb, the Brownian motion is affected and the concentration boundary layer thickness reduces and accordingly the heat transfer characteristics of the fluid changes.”

Figures 17 and 18 depict the various values of conjugate parameter for temperature (γ) on the temperature and concentration distributions, respectively. It is observed that raising the values of γ from 0.1 to 1.2 the resultant temperature and concentration of the fluid increases consequently the thickness of thermal and concentration boundary layer enhances.

I

n figures 19 and 20 demonstrated the effect of Prandtl number (Pr) verses temperature and concentration distributions, respectively. It is noticed that thermal and concentration boundary layer thickness reduces with hype the value of Pr. This is because Prandtl number is the ratio of momentum diffusivity to the nanofluid thermal diffusivity. In figure 21 show that the effect of Lewis number (Le) on concentration distribution respectively. It is noticed that concentration boundary layer thickness decrease with raising the values of Le.”

 

The variation of magnetic parameter (M), local Reynolds number (Rex), natural convection parameter (λ) and Casson fluid parameter (β) on skin-friction coefficient is demonstrated in figure 22 respectively. On increasing the magnetic parameter, local Reynolds number and Casson fluid parameter the resultant skin-friction coefficient increases but skin friction coefficient decreases with an increasing the natural convection parameter. The different values of the magnetic parameter (M), local Reynolds number (Rex), natural convection parameter (λ) and Casson fluid parameter (β) on local Nusselt number is depicted in figure 23 respectively. On increasing the magnetic parameter and Casson fluid parameter the resultant local Nusselt number increases whereas the local Nusselt number decreases with an increasing the local Reynolds number and mixed convection parameter. The different values of the magnetic parameter (M), local Reynolds number (Re_x), natural convection parameter (λ) and Casson fluid parameter (β) on local Sherwood number is depicted in figure 24 respectively. On increasing the magnetic parameter and Casson fluid parameter the resultant local Sherwood number decreases whereas the local Sherwood number increases with an increasing the local Reynolds number and mixed convection parameter.”

 

In order to standardize the method used in the present study and to decide the accuracy of the present analysis and to compare with the results available (Malik et al. (2013), Sarojamma and Vendabai (2015)) relating to the local skin-friction coefficient and local Nusselt number and found in an agreement (Figures 25 and 26).

 

5         CONCLUSIONS

In this paper, similarity solution of the steady boundary layer flow and heat transfer of a Casson nanofluid flowing over a vertical cylinder which is stretching exponentially along its radial direction in the presence of MHD and Newtonian heating is investigated. Using similarity transformations, the governing equations are transformed to self-similar ordinary differential equations which are then solved using Bvp4c MATLAB solver. From the study, the following remarks can be summarized.”

1.        On increasing the magnetic parameter, the resultant dimensionless velocity distribution decreases within the boundary layer but the dimensionless temperature and concentration distributions increases within the boundary layer.

2.        The local skin friction coefficient and local Nusselt number increases by increasing the magnetic parameter whereas the local Sherwood number decreases with an increasing the magnetic parameter.

3.        The dimensionless velocity, temperature and concentration distributions increase by increasing the Casson fluid parameter.

4.        On increasing the Casson fluid parameter, the skin friction coefficient and local Nusselt number increases but the local Sherwood number decreases with an increasing the Casson fluid parameter.

5.        By increasing the mixed convection parameter then the resultant dimensionless velocity distribution increases whereas the dimensionless temperature and concentration distributions decreases with an increasing the natural convection parameter. On increasing the natural convection parameter the resultant skin friction coefficient and local Nusselt number decreases but the local Sherwood number increases.

 

 

 

Figure 1. Dimensionless velocity distribution for different values of M for fixed β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 2. Dimensionless temperature distribution for different values of M for fixed β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 3. Dimensionless concentration distribution for different values of M for fixed β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 4. Dimensionless velocity distribution for different values of β for fixed M = 0.5, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 5. Dimensionless temperature distribution for different values of β for fixed M = 0.5, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.            

(b)

Figure 6. Dimensionless concentration distribution for different values of β for fixed M = 0.5, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 7. Dimensionless velocity distribution for different values of λ for fixed M = 0.5, β = 0.3, Re_x = 5, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 8. Dimensionless temperature distribution for different values of λ for fixed M = 0.5, β = 0.3, Re_x = 5, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 9. Dimensionless concentration distribution for different values of λ for fixed M = 0.5, β = 0.3, Re_x = 5, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

    Figure 10. Dimensionless velocity distribution for different values of Re_x for fixed M = 0.5, β = 0.3, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 11. Dimensionless temperature distribution for different values of Re_x for fixed M = 0.5, β = 0.3, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 12. Dimensionless concentration distribution for different values of Re_x for fixed M = 0.5, β = 0.3, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 13. Dimensionless temperature distribution for different values of Nt for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 14. Dimensionless concentration distribution for different values of Nt for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 15. Dimensionless temperature distribution for different values of Nb for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Le = 2.

 

Figure 16. Dimensionless concentration distribution for different values of Nb for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Le = 2.

 

Figure 17. Dimensionless temperature distribution for different values of γ for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, Nb = 0.2, Le = 2.

 

Figure 18. Dimensionless concentration distribution for different values of γ for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, Nb = 0.2, Le = 2.

 

Figure 19. Dimensionless temperature distribution for different values of Pr for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 20. Dimensionless concentration distribution for different values of Pr for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 21. Dimensionless concentration distribution for different values of Le for fixed M = 0.5, β = 0.3, Re_x = 5, λ = 0.2, ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2.

 

Figure 22. Local skin-friction coefficient for different values of M, Re_x, λ and β for fixed ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2 and Le = 2.

 

Figure 23. Local Nusselt number for different values of M, Re_x, λ and β for fixed ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

Figure 24. Local Sherwood number for different values of M, Re_x, λ and β for fixed ϕ = 0.05, Nr = 1, Pr = 2, Nt = 0.2, γ = 0.5, Nb = 0.2, Le = 2.

 

25 Validation of –f ′′(0) results obtained by  Malik et al (2013) and, Sarojamma and Vendabai (2015) for different values of λ and β.

 

25 Validation of –θ ′(0) results obtained by  Malik et al (2013) and, Sarojamma and Vendabai (2015) for different values of λ and β.

 

 

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Accepted on 24.03.2017           © RJPT All right reserved