Impact of outer velocity on flow heat and mass transfer of Casson nanofluid over a non-linear stretc
* Author to whom correspondence should be addressed.
Journal of Thermal Engineering 2021, Vol. 7, Issue 6, pp. 1353-1365; doi.org/10.18186/thermal.990615
Abstract
Keywords: Non-linear stretching sheet; Brownian motion; Thermophoresis; Casson fluid; MHD
Introduction
Flow behavior over stretching surfaces has attracted many authors due to its wide area of industrial and manufacturing applications like artificial fibers, manufacturing of metallic sheets, petroleum industries, metal spinning, polymer processing etc. The flow caused by the stretching
surface was initially investigated by Crane [1]. After that, Gupta and Gupta [2] and Dutta et al. [3] gained new heights of Crane [1] work by comprising the analysis of mass and heat transportation using many different aspects. Yoon et al. [4] reported the comparison between theoretical and experimental results by considering a thin metal stretching
*Corresponding author. *E-mail address: vini252011makkar@gmail.com, vikaspoply@gmail.com This paper was recommended for publication in revised form by Regional Editor Regional Editor Baha Zafer Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
sheet. Sarma and Rao [5] investigated the flow behavior in viscoelastic fluid over a stretching sheet. The nanofluid also comes in existence when we add a small quantity of nanosized particles which are also called ultrafine particles to the base fluid. Recently, study of nanofluid has much more importance due to its uses, properties and applications in different research areas. Choi and Eastman [6] introduced the concept of nanofluid. The main characteristic of nanofluid is to increase the thermal conductivity of base fluid [7]. In recent years, MHD flow has gained attention due to its controllable heat transfer rate and great utilizations in various industrial areas and manufacturing processes. Hayat et al. [8] studied the impact of MHD on heat and flow transportation over stretching surfaces. On stretching surfaces, a stagnation point micropolar MHD nanofluid flow was numerically investigated by Anwar et al. [9]. After that Shawky et al. [10] studied Williamson nanofluid towards a stretching sheet and he reported that the increment in non-Newtonian parameter increases both the Skin friction coefficient as well as the rate of heat transfer. Vajravelu and Cannon [11] examined the flow behavior of fluid on non-linear stretching sheet. Also, a mixed convection magnetohydrodynamic nanofluid flow along with entropy analysis on a non-linear stretching sheet has been represented by Matin et al. [12]. Das [13] studied the partial slip effect on nanofluid past a permeable non-linearly stretching sheet. Powell-Eyring MHD nanofluid flow having variable thickness past a non-linear stretched surface was interpreted by Hayat et al. [14]. Jain and Choudhary [15] investigated the Soret and Dufour effects with MHD flow using chemical reactions. Siddheshwar and Mahabaleshwar [16] studied the problem of heat and flow transportation on non-linear stretching sheet with suction/injection. The yield stress is displayed by a non-Newtonian Casson fluid. Fluid acts as a solid, if the yield stress is more than the applied shear stress. On the other hand, fluid behaves as a liquid if the yield stress is less than the applied shear stress. Recently, Casson fluid has gained more attention due to its vast applications and uses in many industrial and medical sectors [17]. Nadeem et al. [18] analyzed the effect of MHD flow in Casson fluid over a porous stretching sheet. Kameswaran et al. [19] investigated the dual solution in a Casson fluid flow past a shrinking/stretching sheet. Krishna et al. [20] studied the chemical reaction effect on MHD Casson fluid flow taking porous medium on a stretched surface. Hakeem et al. [21] used impermeable stretching surface with Casson fluid to determine the inclined Lorentz force effect on heat and flow transportation. Murthy [22] investigated the unsteady MHD Casson fluid flow with partial slip on exponentially stretching sheet. Recently, various aspects on the rate of flow, heat and mass characteristics are analyzed by numerous authors in [23]–[29]. Furthermore, the non-linear effect over stretching surfaces under different physical circumstances has also been
examined by researchers in [30]–[35]. No study has been done so far to analyze the combined effect of outer velocity with non-linear stretching in a Casson nanofluid. In present study, we analyzed the combined effect of outer velocity with non-linear stretching sheet in a MHD Casson nanofluid. In this analysis, we observe that an inverted boundary layer is formed in velocity profiles when we take the outer velocity parameter greater than one. Also, outer velocity has a great impact on the rate of heat transportation and nanoparticle concentration. The outcomes of Current study reveals that as non-linearity in the stretching sheet increases, velocity decreases for outer velocity less than one and increases for outer velocity greater than one.
Materials And Methods
2-D incompressible boundary layer flow with Casson nanofluid towards a non-linear stretching sheet has been considered. This non-linear behavior produces a flow and sheet is stretched with stretching velocity uw = axN, where a, x and N denotes a constant, stretching surface coordinate and non-linear stretching parameter respectively. Figure 1 depicts the physical model of the present problem. In this physical configuration system, Tw, Cw, T∞ and C∞ represents temperature at the sheet, nanoparticle volume fraction at surface, ambient temperature attained and ambient nanoparticle volume fraction serially. The induced magnetic field is very small comparative to apply magnetic field so is neglected. The basic governing equations for nanofluid in terms of Cartesian coordinates are described as: ∂u ∂u + = 0 ∂x ∂y u
∂u ∂u ∂U 1 ∂ 2u σ B2 +v = U + ν 1 + 2 − (u − U ) (2) ∂x ∂y ∂x β ∂y ρ
2 ∂C ∂T ∂T ∂T ∂T +v = αm ∇2T + τ DB + ( DT / T∞ ) ∂x ∂y ∂y ∂y ∂y
Initial boundary conditions are given as: = v 0,= uw ax N= , C Cw= , T Tw= at y 0 u= → U bx N , C → C∞ , T → T∞ as y → ∞
Here vertical and horizontal velocities are represented by υ and u, respectively. Also ν, β, σ, B, ρf, U denotes the
kinematic viscosity, Casson fluid parameter, velocity slip parameter, magnetic field intensity, density of base fluid, km outer velocity, αm = is the thermal diffusivity, α is a (ρ c) f (ρ c)p positive constant, τ = defines a proportion of heat (ρ c) f capacities of nanofluid to base fluid, ρp is density of particles, DT is thermophoretic diffusion coefficient and DB is Brownian diffusion coefficient. Equations (1) to (4) with boundary conditions (5) are transformed using similarity variables a ( N + 1) N2−1 = x ,u ax N f 0′ (ξ ) , ξ y= 2ν
The transformed boundary conditions are = f 0 (0) 0,= f 0′(0) 1,= φ0 (0) 1,= θ0 (0) 1 (10) and f 0′(ξ ) → λ , φ0 (ξ ) → 0, θ 0 (ξ ) → 0 as ξ → ∞ where ( )′ denotes derivative w.r.t ξ. Here λ, M, Nt, Nb, Sc and Pr represents the outer velocity parameter, magnetic parameter, thermophoresis parameter, Brownian motion parameter, Schmidt number and Prandtl number respectively and these fluid parameters are given by:
( ρ c ) p DT (Tw − T∞ ) b σ B2 = ,M = , Nt , N −1 a ( ρ c ) f T∞ν ax
( ρ c ) p DB (Cw − C∞ ) ν ν Nb , Sc = and Pr = = (ρ c) f ν DB α
a ( N + 1) N2−1 N −1 v= x f 0 (ξ ) + − ξ f 0′ (ξ ) (6) Also, local Sheerwood number and local Nusselt numN 2 + 1 ber are respectively defined as: T − T∞ C − C∞ θ 0 (ξ ) = = ,φ0 (ξ ) Tw − T∞ Cw − C ∞ xqm xqw = Shx = and Nux (12) DB (Cw − C∞ ) k(T f − T∞ ) Substituting equation (6) into equations (2) to (4), we have where qm and qw are mass flux and heat flux at the stretching surface serially 1 2N 2 2 2 1 + f 0′′′+ f 0 f 0′′− f 0′ − λ f 0′ N +1 β N −1 (7) a ( N + 1) qm = − DB (Cw − C∞ )x 2 φ0′(0) (13) 2 ′ − − = M f 0 λ ( ) 2ν 0 N +1
Numerical Procedure
The system of non-linear differential equations (7)–(9) are not solved analytically due to extremely non-linear behaviour. To solve this system of non-linear differential equations (7)–(9) together with boundary conditions (10), we adopt Runge Kutta Fehlberg approach (numerical procedure) with shooting technique. The coupled differential equations (7)–(9) has been reduced to a system of first order differential equations and are described as:
2 2 − y1 y3 + M y 22 − λ 2 + M ( y2 − λ ) N +1 N +1 1 1 + β (17)
1 Nt y7′ = − y1 y7 Sc + Pr y y + Nby5 y7 + Nty52 Nb 1 5 2
The relevant boundary conditions are reduced to = y1 0,= y2 1,= y6 1,= y 4 1 at= ξ 0 and = y2 λ ,= y6 0,= y 4 0 as ξ → ∞
′ ′′ ′ f 0′′′ = y1 f= f= f= 0 , y2 0 , y3 0 , y3 ′ ′ θ 0′′ = y 4 θ= θ= 0 , y5 0 , y5 ′ ′ φ0′′ = y 6 φ= φ= 0 , y7 0 , y7
The above system of differential equations is solved using shooting technique with RKF method. Figure 2 display the flow chart of numerical procedure adopted above.
Results And Discussion
In current study, non-linear differential equations (7)– (9) with boundary conditions (10) are computed numerically. The variation of fluid parameters like magnetic parameter M, Prandtl number Pr, thermophoresis parameter Nt, Brownian motion parameter Nb, Schmidt number Sc, non-linear stretching parameter N and Casson fluid parameter β on velocity distribution, fluid temperature and nanoparticle concentration are computed and explained through graphs and tables. Table 1 represents the validation of our outcomes for –θ′0(0) with Nb = 0, Nt = 0 and for various entries of N and Pr. In present problem, an excellent validation of the numerical procedure are done with those of Rana and Bhargava [33]
Table 1. Values of –θ′0(0) with Nb = 0 and Nt = 0, a comparison Pr
results
and Cortell [32] by neglecting the entries of Nt and Nb. Table 2 shows the influence of various fluid parameters Schmidt number Sc, magnetic field parameter M, Prandtl number Pr, Brownian motion parameter Nb, thermophoresis parameter Nt, non-linear stretching parameter N and Casson fluid parameter β on velocity, temperature and concentration for two distinct values of outer velocity λ. Figures 3 and 4 show the effect of β on velocity profiles for λ = 0.5 and λ = 1.2 respectively. As Casson parameter β increases, fluid yield stress decreases that hurdles the free movement of fluid particles and consequently width of boundary layer decreases that will declines the fluid velocity for λ = 0.5 as shown in Figure 3. Thusly, Skin friction coefficient exceeds for large values of β as computed in Table 2. On contrary, inverse trend is observed in case of λ = 1.2 that can be seen in Figure 4. Further, the impact of Casson fluid parameter β on fluid temperature and nanoparticle concentration is depicted in Figures 5–8. Figures 5 and 6 represents the temperature profile for two distinct values of outer velocity λ (i.e. for λ =
0.5. and λ = 1.2). With an increase in the value of Casson
fluid parameter β, fluid velocity goes on decreasing for λ =
0.5. that will decrease the rate of heat transfer and thus fluid
Figure 3. Impact of Casson fluid parameter β on velocity profile f′0(ξ), when λ = 0.5.
Figure 4. Impact of Casson fluid parameter β on velocity profile f′0(ξ), when λ = 1.2. temperature enhances for λ = 0.5 that can be visualized in Figures 5. Hence, Nusselt number declines for larger values of β can be seen in Table 2. Also, a very small decrease in fluid temperature is noticed for λ = 1.2 as seen in Figure 6. Figures 7 and 8 display the impact of Casson fluid parameter β on nanoparticle concentration for λ = 0.5 and λ = 1.2 respectively. Figure 7 shows that an intensification in the value of β results to increase the nanoparticle concentration for outer velocity λ = 0.5 which increase the value of local Sheerwood number. On the other hand, reverse trend is observed for nanoparticle concentration when λ = 1.2 as visualized in Figure 8. Also, Figures 9 and 10 demonstrate the temperature profiles for various values of Schmidt number Sc. With an increase in the values of Sc, thermal diffusivity enhances which results in increase in the thickness of boundary layer and therefore fluid temperature increases for both values of λ and thusly, Nusselt number decreases that can be seen from Table 2. Furthermore, Figures 11 and 12 represents the effect of Sc on nanoparticle concentration for λ = 0.5 and λ = 1.2 respectively. As Sc increases, mass diffusivity decreases. Thus, nanoparticle concentration reduces as illustrated in the Figures 11 and 12 for both λ = 0.5 and λ =
1.2. respectively.
Figures 13 and 14 depicts the variation in velocity distribution under the influence of magnetic parameter M for two distinct cases of outer velocity λ = 0.5 and λ = 1.2 respectively. Here, the presence of M will generates Lorentz force which hurdles the free movement of fluid particles. Thus magnetism can be used to control the velocity of nanofluid flow over stretching sheet. An increase in the value of M causes the declination of nanofluid velocity shown in Figure 13. Whereas reverse nature is observed for λ = 1.2 as shown in Figure 14 due to an inverted boundary layer formation for λ > 1. Figure 15 demonstrate variation of magnetic parameter M on temperature field for λ = 0.5. From Figure 15, it is clear that an enlargement in magnetic
Table 2. Results of f″0(0), –θ′0 (0) and –ϕ′0 (0) for distinct fluid parameters when λ = 0.5 and λ = 1.2 Sc 1 1.5
Figure 5. Impact of Casson fluid parameter β on temperature profile θ0(ξ), when λ = 0.5.
Figure 6. Impact of Casson fluid parameter β on temperature profile θ0(ξ), when λ = 1.2.
field produces a slight enhancement in fluid temperature for λ = 0.5. After that, Figure 16 expresses the variation in nanoparticle concentration with change in the value of M. For larger value of magnetic parameter M nanoparticle concentration increases for λ = 0.5 as represented in Figure 16. Physically, this can be explained that as M increases, fluid velocity decreases due to this rate of mass transfer decreases and hence concentration increases. Also from Table 2, we observed that as M increases, Sheerwood number decreases which represent the reduction in the convective mass transfer and hence concentration increases.
Impact of non-linear stretching parameter N on profiles of velocity, temperature and concentration for different values of outer velocity are represented by the Figures 17–22. With an increase in the value of non-linear stretching parameter N fluid velocity decreases (as shown in Figure 17 for λ < 1 and increases for λ > 1 due to invested boundary layers formations as seen in Figure 18. Consequently, momentum boundary layer thickness decreases with increasing non-linear stretching parameter N for both values of λ. Thus, the presence of nonlinear stretching parameter leads to slight rise the thickness of thermal boundary
Figure 7. Impact of Casson fluid parameter β on concentration profile ϕ0(ξ), when λ = 0.5.
Figure 10. Impact of Schmidt number Sc on temperature profile θ0(ξ), when λ = 1.2.
Figure 8. Impact of Casson fluid parameter β on concentration profile ϕ0(ξ), when λ = 1.2.
Figure 11. Impact of Schmidt number Sc on concentration profile θ0(ξ), when λ = 0.5.
Figure 9. Impact of Schmidt number Sc on temperature profile θ0(ξ), when λ = 0.5.
Figure 12. Impact of Schmidt number Sc on concentration profile θ0(ξ), when λ = 1.2.
Figure 13. Impact of magnetic parameter M on velocity profile f′0(ξ), when λ = 0.5.
Figure 16. Impact of magnetic parameter M on temperature profile ϕ0(ξ), when λ = 1.2.
Figure 14. Impact of magnetic parameter M on velocity profile f′0(ξ), when λ = 1.2.
Figure 17. Impact of non-linear stretching parameter N on velocity profile f 0′(ξ), when λ = 0.5.
Figure 15. Impact of magnetic parameter M on temperature profile θ0(ξ), when λ = 0.5.
Figure 18. Impact of non-linear stretching parameter N on velocity profile f 0′(ξ), when λ = 1.2.
Figure 19. Impact of non-linear stretching parameter N on temperature profile θ0(ξ), when λ = 0.5.
Figure 21. Impact of non-linear stretching parameter N on temperature profile ϕ0(ξ), when λ = 0.5.
Figure 20. Impact of non-linear stretching parameter N on temperature profile θ0(ξ), when λ = 1.2.
Figure 22. Impact of non-linear stretching parameter N on temperature profile ϕ0(ξ), when λ = 1.2.
layer and hence fluid temperature increases slightly for λ < 1 as shown in Figure 19. On the other hand, reverse impact is noticed for outer velocity parameter greater than one as visualized in Figure 20. Due to the presence of outer velocity, the variation in the temperature profile are less as shown in Figure 19 and 20. The impacts of non-linear stretching parameter N on profiles of nanoparticle concentration are represented by Figures 21 and 22 for two cases of λ. As N increases, nanoparticle concentration increases for λ < 1 and decreases for λ > 1 as depicted in Figures 21 and 22. Figures 23 and 24 demonstrate the impact on fluid temperature for λ = 0.5 and λ = 1.2 with variation of Brownian motion parameter Nb. A casual movement of suspended particles is the Brownian motion which comes to be in existence when liquid elements strike one another and that results an increase in temperature at the surface as seen in Figures 23 and 24 for both cases of λ. The striking of atoms or molecules of the fluid particles with each other will create an arbitrary motion called Brownian motion of suspended (Pendolus) particles and that will enhance the width of
boundary layer. Consequently, for larger Nb, temperature gradient falls as visualized in Table 2. A small increment is noticed in case of temperature profile for higher values of Nb. An impact of Nb on ϕ0(ξ) for various values of λ is represented by the Figures 25 and 26. Higher value Nb of will increase the fluid particles collision by which nanoparticle concentration declines for considered cases of λ. Table 3 represents the impact of outer velocity parameter λ on velocity profile f″0(0), temperature profile –θ′0(0) and concentration profile –ϕ′0(0) by taking fixed entries of fluid parameters as M = 0.1, Pr = 5, Nb = 0.1, Nt = 0.3, Sc = 1, N = 1 and β = 0.1. Here, it is observed that velocity profile increases and consequently, Skin friction coefficient increases. Intensification in the value of outer velocity λ causes the value of fluid temperature as well as nanoparticle concentration to declines and their respective local Nusselt number and local Sheerwood number increases. Also, Figures 27 and 28 depicts the fluid temperature under the influence of thermophoresis parameter Nt for λ =
0.5. and λ = 1.2. Consequently, temperature gradient falls
Figure 23. Impact of Brownian motion parameter Nb on temperature profile θ0(ξ), when λ = 0.5.
Figure 25. Impact of Brownian motion parameter Nb on concentration profile ϕ0(ξ), when λ = 0.5.
Figure 24. Impact of Brownian motion parameter Nb on temperature profile θ0(ξ), when λ = 1.2.
Figure 26. Impact of Brownian motion parameter Nb on concentration profile ϕ0(ξ), when λ = 1.2.
Table 3. Values of f″0(0), –θ′0(0) and –ϕ′0(0) for different λ with fixed entries of various fluid parameters as M = 0.1, Pr = 5, Nb = 0.1, Nt = 0.3, Sc = 1, N = 1 and β = 0.1 λ
for larger value of Nt and that will decline the conduction of nanoparticles. Thus, ultrafine particles reallocate from warmer to colder place that ultimately raise the boundary layer thickness and thereby temperature goes high for both values of λ. Hence, Nusselt number comes down that can be visualized in Table 2. On the other hand, Figures 29 and 30 illustrate nanoparticle concentration for different λ under the consequence of Nt and simultaneously nanoparticle
volume fraction field shows similar effect as that in case of temperature for each λ.
Conclusion
This study represents influence of outer velocity with mass and heat transportation on Casson nanofluid past a nonlinear stretching sheet. Present results are compared with Rana and Bhargava [33] and Cortell [32] as a limiting case of our study. Basic equations are solved by applying shooting technique. Main fallout of the present study incorporates the significance of non-linear stretching parameter and outer velocity in addition to MHD on profile of velocity, fluid temperature and nanoparticle concentration of heat, mass and flow transportation of Casson nanofluid. Key findings of current analysis for two distinct entries of outer velocity λ (i.e. λ = 0.5 and λ = 1.2) are summarized as: 1. For λ = 0.5; it is observed that velocity profile decreases for increasing value of non-linear stretching parameter N, Casson parameter β,
Figure 27. Impact of Brownian motion parameter Nt on temperature profile θ0(ξ), when λ = 0.5.
Figure 29. Impact of Brownian motion parameter Nt on concentration profile ϕ0(ξ), when λ = 0.5.
Figure 28. Impact of Brownian motion parameter Nt on temperature profile θ0(ξ), when λ = 1.2.
Figure 30. Impact of Brownian motion parameter Nt on concentration profile ϕ0(ξ), when λ = 1.2.
and magnetic parameter M. Raise in temperature is noticed for Schmidt number Sc, Casson fluid parameter β, magnetic parameter M, non-linear stretching N, Brownian motion parameter Nb and thermophoresis parameter Nt. Also, nanoparticle concentration increases for Casson fluid parameter β, magnetic parameter M, non linear stretching parameter N and thermophoresis parameter Nt whereas decreases for Schmidt number Sc Brownian motion parameter Nb. 2. For λ = 1.2; velocity profile enhances with enhancement in the entries of Casson fluid parameter β, magnetic parameter M and non-linear stretching parameter N. On the other hand, fluid temperature increases with increase in Schmidt number Sc Brownian motion parameter Nb, thermophoresis parameter Nt and decreases with the increase of Casson fluid parameter β, magnetic parameter M and non-linear stretching parameter N. Also, concentration profile rises for thermophoresis
parameter Nt while declines for higher values of Schmidt number Sc, Casson fluid parameter β, magnetic parameter M, non-linear stretching parameter N and Brownian motion parameter Nb.
3. Boundary layer thickness decreases with an
increase in. Present study finds great application in insulation of wires, manufacturing of tetra packs, production of glass fibers, fabrication of various polymers and plastic packs, rubber sheets etc. The quality of desired product depends on the rate stretching, external magnetic field and composition of material used.
Nomenclature
Horizontal velocity, ms–1 Vertical velocity, ms–1 Cartesian coordinates Magnetic field intensity, T
Stream velocity Temperature, K Concentration, Kgm–3 Nanoparticle volume fraction, Kgm–3 Mass flux, Kgm–2 s–1 magnetic field parameter, Tesla Schmidt number Prandtl number Nusselt number Thermophoresis parameter Ambient temperature attained, K Sheerwood number Brownian diffusion coefficient Brownian motion parameter, m2 s–1 Stretching velocity Temperature at the sheet, K Hass flux, W–2 m Ambient nanoparticle volume fraction, Kgm–3 Thermophoresis diffusion coefficient, m2 s–1
Greek symbols ν Kinematic viscosity, m2 s–1 β Casson fluid parameter σ Electrical conductivity λ Outer velocity parameter ξ Similarity variable ρf Density of base fluid, Kgm–3 k αm = m Thermal diffusivity (ρ c) f (ρ c)p τ= Ratio of heat capacities (ρ c) f θ0(ξ) Non-dimensional temperature ϕ0(ξ) Non-dimensional nanoparticle concentration Subscripts p ∞ f w
Data Availability Statement
No new data were created in this study. The published publication includes all graphics collected or developed during the study.
Ethics
There are no ethical issues with the publication of this manuscript.
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MAKKAR, V.; POPLY, V. Impact of outer velocity on flow heat and mass transfer of Casson nanofluid over a non-linear stretc. Journal of Thermal Engineering 2021, Vol. 7, pp. 1353-1365. https://doi.org/10.18186/thermal.990615