The influences of thermal radiation and nanoparticles on free convection flow and heat transfer of Casson nanofluids over a vertical plate are investigated. The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are converted to systems of nonlinear ordinary differential equations through similarity transformations. The resulting systems of fully coupled nonlinear ordinary differential equations are solved using the differential transformation method with Padé-approximant technique. The accuracies of the developed analytical methods are verified by comparing their results with the results of past works as presented in the literature. Thereafter, the analytical solutions are used to investigate the effects of thermal radiation, Prandtl number, nanoparticle volume fraction, shape, and type on the flow and heat transfer behaviour of various nanofluids over the flat plate. It is observed that both the velocity and temperature of the nanofluid as well as the viscous and thermal boundary layers increase with increase in the thermal radiation parameter. The velocity of the nanofluid decreases and the temperature of the nanofluid increase, respectively, as the Prandtl number and volume fraction of the nanoparticles in the base fluid increase. The decrease in velocity and increase in temperature are highest in lamina-shaped nanoparticle and followed by platelet-, cylinder-, brick-, and sphere-shaped nanoparticles, respectively. Using a common base fluid to all the nanoparticle types, it is established that the decrease in velocity and increase in temperature are highest in TiO_{2} and followed by CuO, Al_{2}O_{3}, and SWCNT nanoparticles, in that order. It is hoped that the present study will enhance the understanding of free convection boundary layer problems of Casson fluid under the influences of thermal radiation and nanoparticles as applied in various engineering processes.
1. Introduction
The importance and the wide applications of free convection flow and heat transfer in extrusion, melt spinning, glass-fibre production processes, food processing, mechanical forming processes etc. have in recent times aroused various renewed research interests and explorations. In the study of free convection and heat transfer problems, the analysis of incompressible laminar flow of viscous fluid in a steady-state, two-dimensional free convection boundary layer has over the years been a common area of increasing research interests following the experimental investigations of Schmidt and Beckmann [1] and the pioneering theoretical work of Ostrach [2]. In their attempts to study the laminar free convection flow and heat transfer problem in 1953, Ostrach [2] applied method of iterative integration to analyze free convection over a semi-infinite isothermal flat plate. The author obtained the numerical solutions for a wide range of Prandtl numbers from 0.01 to 1000 and validated their numerical results using experimental data of Schmidt and Beckmann [1]. Five years later, Sparrow and Gregg [3] presented a further study on numerical solutions for laminar free convection from a vertical plate with uniform surface heat flux. Considering the fact that the major part of low Prandtl number boundary layer of free convection is inviscid, Lefevre [4] examined laminar free convection of an inviscid flow from a vertical plane surface. In a further work, Sparrow and Gregg [5] developed similar solutions for free convection from a nonisothermal vertical plate. Meanwhile, a study on fluid flow over a heated vertical plate at high Prandtl number was presented by Stewartson and Jones [6]. Due to the disadvantages in the numerical methods in the previous studies [2, 3], Kuiken [7] adopted method of matched asymptotic expansion and established asymptotic solutions for large Prandtl number free convection. In the subsequent year, the same author applied the singular perturbation method and analyzed free convection flow of fluid at low Prandtl numbers [8]. Also, in another work on the asymptotic analysis of the same problem, Eshghy [9] studied free convection boundary layers at large Prandtl number while Roy [10] investigated free convection boundary layer problem for a uniform surface heat flux at high Prandtl number. With the development of asymptotic solution, a combined study of the effects of small and high Prandtl numbers on the viscous fluid flow over a flat vertical plate was submitted by Kuiken and Rotem [11]. In the succeeding year, Na and Habib [12] utilized parameter differentiation method to solve the free convection boundary layer problem. Few years later, Merkin [13] presented the similarity solutions for free convection on a vertical plate while Merkin and Pop [14] used finite difference method to develop numerical solutions for the conjugate free convection problem of boundary-layer flow over a vertical plate. Also, Ali et al. [15] submitted a study on numerical investigation of free convective boundary layer in a viscous fluid.
The various analytical and numerical studies of the past works have shown that the boundary layer problems are very difficult to solve. This is because, besides having very thin regions where there is rapid change of the fluid properties, they are defined on unbounded domains. Although, approximate analytical methods are being used to solve boundary layer problems, they converge very slowly for some boundary layer problems, particularly for those with very large parameters. The numerical methods used in the flow process also encounter some problems in resolving the solutions of the governing equations in the very thin regions and in some cases where singularities or multiple solutions exist. Moreover, in numerical analysis, it is absolutely required that the stability and convergence analysis is carried out so as to avoid divergence or inappropriate results. Such analysis in the mathematical methods increases the computation time and cost. Therefore, in the quest for presenting symbolic solutions to the flow and heat transfer problem using one of the recently developed semianalytical methods, Motsa et al. [16] adopted homotopy analysis of free convection boundary layer flow with heat and mass transfer. In another work, the authors used spectral local linearization approach for solving the natural convection boundary layer flow [17]. Ghotbi et al. [18] investigated the application of homotopy analysis method to natural convection boundary layer flow. Although, homotopy analysis method (HAM) is a reliable and efficient semianalytical technique, it suffers from a number of limiting assumptions such as the requirements that the solution ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. Also, the use of HAM in the analysis of linear and nonlinear equations requires the determination of auxiliary parameter which will increase the computational cost and time. Furthermore, the lack of rigorous theories or proper guidance for choosing initial approximation, auxiliary linear operators, auxiliary functions, and auxiliary parameters limits the applications of HAM. Moreover, such method requires high skill in mathematical analysis and the solution comes with large number of terms. Nonetheless, various analyses of nonlinear models and fluid flow problems under the influences of some internal and external factors using different approximate analytical and numerical methods have been presented in the literature [19–47]. Also, the relative simplicity coupled with ease of applications of the differential transformation method (DTM) has made the method to be more effective than most of the other approximate analytical methods. The method was introduced by Ζhou [48] and it has fast gained ground as it appeared in many engineering and scientific research papers. This is because, with the applications of DTM, a closed form series solution or approximate solution can be provided for nonlinear integral and differential equations without linearization, restrictive assumptions, perturbation, and discretization or round-off error. It reduces complexity of expansion of derivatives and the computational difficulties of the other traditional or recently developed methods. Therefore, Lien-Tsai and Cha’o-Kuang [49] applied the differential transformation method to provide approximate analytical solutions to the Blasius equation. Also, Kuo [50] adopted the same method to determine the velocity and temperature profiles of the Blasius equation of forced convection problem for fluid flow passing over a flat plate. An extended work on the applications of differential transformation method to free convection boundary layer problem of two-dimensional steady and incompressible laminar flow passing over a vertical plate was presented by the same author [51]. However, in the later work, the nonlinear coupled boundary value governing equations of the flow and heat transfer processes is reduced to initial value equations by a group of transformations, and the resulting coupled initial value equations are solved by means of the differential transformation method. The reduction or the transformation of the boundary value problems to the initial value problems was carried out due to the fact that the developed systems of nonlinear differential equations contain an unbounded domain of infinite boundary conditions. Moreover, in order to obtain the numerical solutions that are valid over the entire large domain of the problem, Ostrach [2] estimated the values of f″0andθ′0 during the analysis of the developed systems of fully coupled nonlinear ordinary differential equations. Following Ostrach’s approach, most of the subsequent solutions provided in the literature [3, 9, 10, 12, 14, 15, 50, 51] were based on the estimated boundary conditions given by Ostrach [2]. Additionally, the limitations of power series solutions to small domain problems have been well established in the literature. Nevertheless, in some recent studies, the use of power series methods coupled with Padé-approximant technique has shown to be very effective way of developing accurate analytical solutions to nonlinear problems of large or unbounded domain problems of infinite boundary conditions. The application of Padé-approximant technique with power series method increases the rate and radius of convergence of power series solution. Therefore, in a recent work, Rashidi et al. [52] applied differential transformation method coupled with the Padé-approximant technique to develop a novel analytical solution for mixed convection about an inclined flat plate embedded in a porous medium.
Casson fluid is a non-Newtonian fluid that was first introduced by Casson in 1959 [53]. It is a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [54]. If the yield stress is greater than the shear stress, then it acts as a solid, whereas if the yield stress is less than the applied shear stress, then the fluid would start to move. The fluid is based on the structure of liquid phase and interactive behaviour of solid of a two-phase suspension. It is able to capture complex rheological properties of a fluid, unlike other simplified models like the power law [55] and second-, third-, or fourth-grade models [56]. Some examples of Casson fluid are jelly, honey, tomato sauce, and concentrated fruit juices. Human blood is also treated as a Casson fluid in the presence of several substances such as fibrinogen, globulin in aqueous base plasma, protein, and human red blood cells. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model. It has various applications in fibrinogen, cancer homeotherapy, protein, and red blood cells, forming a chain-type structure. Due to these applications, many researchers are concentrating on characteristics of Casson fluid. Application of Casson fluid for flow between two rotating cylinders is studied in [57]. The effect of magnetohydrodynamic (MHD) Casson fluid flow in a lateral direction past linear stretching sheet was explained by Nadeem et al. [58].
The role of thermal radiation is very important in some industrial applications, such as glass production and furnace design, and also in space technology applications such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics, and space craft reentry aerodynamics which operates at high temperatures, in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, and solar radiations. The effect of thermal radiation on magnetohydrodynamic flow was examined by Raptis and Perdikis [59] while Seddeek [60] investigated the impacts of thermal radiation and variable viscosity on magnetohydrodynamics in free convection flow over a semi-infinite flat plate. In another study, Mabood et al. [61] analyzed unsteady stretched flow of Maxwell fluid in the presence of nonlinear thermal radiation and convective condition while Hayat et al. [62] addressed the effects of nonlinear thermal radiation and magnetohydrodynamics on viscoelastic nanofluid flow. Farooq et al. [63] addressed the effects of nonlinear thermal radiation on stagnation point flow. Also, Shehzad et al. [64] presented a study on MHD three-dimensional flow of Jeffrey nanofluid with internal heat generation and thermal radiation.
The previous studies on fluid flow over stretching under investigation are based on viscous fluid flow as shown in the above-reviewed works. To the best of the author’s knowledge, a study on the influence of thermal radiation and nanoparticle shape, size, and type on the free convection boundary layer flow and heat transfer of Casson nanofluids over a vertical plate at low and high Prandtl numbers using the differential transformation method coupled with Padé-approximant technique has not been investigated. Therefore, the present study focuses on the application of differential transformation method coupled with Padé-approximant technique to develop approximate analytical solutions and carry out parametric studies of the effects of thermal radiation and nanoparticles on free convection boundary layer flow and heat transfer of nanofluids of different nanosize particles over a vertical plate at low and high Prandtl numbers. Another novelty of the present study is displayed in the development of approximate analytical solutions for the free convection boundary layer problem without the use of the estimated boundary conditions f″0andθ′0 during the analysis of the problem.
2. Problem Formulation and Mathematical Analysis
Consider a laminar free convection flow of an incompressible Casson nanofluid over a vertical plate parallel to the direction of the generating body force, as shown in Figure 1. The rheological equation for an isotropic and incompressible Casson fluid, reported by Casson [65], is(1)τ=τ0+μσ˙or(2)τ=2μB+py2πeij,π>πc=2μB+py2πceij,πc<π,where τ is the shear stress; τ0 is the Casson yield stress; μ is the dynamic viscosity; σ˙ is the shear rate; π=eijeijinwhicheij is the i,jth component of the deformation rate and π is the product of the component of deformation rate with itself; πc is a critical value of this product based on the non-Newtonian model; μB is the plastic dynamic viscosity of the non-Newtonian fluid; and py is the yield stress of the fluid. The velocity and the temperature are functions of y, t only.
Velocity and temperature profiles in free convection flow over a vertical plate.
Assuming that the flow in the laminar boundary layer is two-dimensional and steady, the heat transfer from the plate to the fluid is proportional to the local surface temperature T, using the Boussinesq approximation along with the assumption that the pressure is uniform across the boundary layer, and the equations for continuity, motion, and energy are given as(3)∂u∂x+∂v∂y=0,(4)ρnfu∂u∂x+v∂u∂y=1+1γμnf∂2u∂y2+gρβnfT−T∞,(5)ρcpnfu∂T∂x+v∂T∂y=knf∂2T∂y2−∂qr∂y.
Assuming no slip conditions, the appropriate boundary conditions are given as(6a)u=0,v=0,T=Tsaty=0,(6b)u=0,T=Tw,aty⟶∞,where the various physical and thermal properties in (3)–(5) are given as(7a)ρnf=ρf1−ϕ+ρsϕ,(7b)ρcpnf=ρcpf1−ϕ+ρcpsϕ,(7c)ρβnf=ρβf1−ϕ+ρβsϕ,(7d)μnf=μf1−ϕ2.5,(8)knf=kfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks,(9)∂qr∂y=−4σ3K∂T4∂y≅−16σTs33K∂2T∂y2usingRosseland’sapproximation,where m in the above Hamilton–Crosser model in (8) is the shape factor and its numerical values for different shapes are given in Table 1. It should be noted that the shape factor, m=3/λ, where λ is the sphericity (the ratio of the surface area of the sphere and the surface area of the real particles with equal volumes). Sphericity of sphere, platelet, cylinder, laminar, and brick are 1.000, 0.526, 0.625, 0.185, and 0.811, respectively. The Hamilton–Crosser model becomes a Maxwell–Garnett model, when the shape factor of the nanoparticle is 3 (m=3).
The values of different shapes of nanoparticles [41, 66].
S/N
Name
Shape
Shape factor (m)
Sphericity (ψ)
1
Sphere
3.0
1.000
2
Platelet
5.7
0.526
3
Cylinder
4.8
0.625
4
Lamina
16.2
0.185
5
Brick
3.7
0.811
Tables 2 and 3 present the physical and thermal properties of the base fluid and the nanoparticles, respectively. SWCNTs mean single-walled carbon nanotubes.
Going back to (3)–(5), if one introduces a stream function, ψx,y, such that (10)u=∂ψ∂y,v=−∂ψ∂x,and uses the following similarity and dimensionless variables:(11)η=ρf2gβfTw−T∞4μf2x1/4y,ψ=4μfρfρf2gβfTw−T∞x34μf21/4fη,θ=T−T∞Tw−T∞,Pr=μfcpkf,R=4σT∞33kK,one arrives at fully coupled third- and second-order ordinary differential equations:(12)1+1γf‴+1−ϕ2.51−ϕ+ϕρsρf3ff″−2f′2+1−ϕ+ϕρβsρβfθ=0,(13)1+43Rθ″+311−ϕ+ϕρCps/ρCpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksPrfθ′=0,and the boundary conditions as(14)f=0,f′=0,θ=1,whenη=0,f′=0,θ=0,whenη=∞.
Physical and thermal properties of the base fluid [41, 66–70].
Base fluid
ρ (kg/m^{3})
c_{
p} (J/kg K)
k (W/mK)
Pure water
997.1
4179
0.613
Ethylene glycol
1115
2430
0.253
Engine oil
884
1910
0.144
Kerosene
783
2010
0.145
Physical and thermal properties of the nanoparticles [41, 66–70].
Nanoparticles
ρ (kg/m^{3})
c_{
p} (J/kg K)
k (W/mK)
Copper (Cu)
8933
385
401
Aluminum oxide (Al_{2}O_{3})
3970
765
40
SWCNTs
2600
42.5
6600
Silver (Ag)
10500
235.0
429
Titanium dioxide (TiO_{2})
4250
686.2
8.9538
Copper(II) oxide (CuO)
783
540
18
It should be noted that, for a viscous fluid which does not have nanoparticles with negligible radiation, the nanoparticle volume fraction is zero, that is, ϕ=0,R=0, and γ⟶∞, then one recovers the earlier models [2–15] from (12) and (13), which are(15)f‴+3ff″−2f′2+θ=0,(16)θ″+3Prfθ′=0,and the boundary conditions remain the same as in (14).
3. Method of Solution: Differential Transform Method
The relatively new semi-analytical method, differential transformation method introduced by Zhou [48], has proven very effective in providing highly accurate solutions to differential equations, difference equation, differential-difference equations, fractional differential equation, pantograph equation, and integrodifferential equation. Therefore, this method is applied in the present study. The basic definitions and the operational properties of the method are as follows.
If ut is analytic in the domain T, then the function ut will be differentiated continuously with respect to time t:(17)dputdtp=φt,pforallt∈T.
If t=ti, then φt,p=φti,p, where p belongs to the set of nonnegative integers, denoted as the p-domain. We can therefore write (17) as(18)Up=φti,p=dputdtpt=ti,where Up is called the spectrum of ut at t=ti.
Express ut in Taylor’s series as(19)ut=∑p∞t−tipp!Up,where (19) is the inverse of Uk with symbol “D” denoting the differential transformation process and combining (18) and (19), we have(20)ut=∑p=0∞t−tipp!Up=D−1Up.
Table 4 contains the differential transform of some functions. Using the operational properties of the differential transformation method, the differential transformation of the governing differential (12) is given as(21)1+1γp+1p+2p+3Fp+3+1−ϕ2.51−ϕ+ϕρsρf3∑l=0pp−l+1p−l+2FlFp−l+2−2∑l=0pl+1p−l+1Fl+1Fp−l+1+1−ϕ+ϕρβsρβfΘp=0.
Operational properties of differential transformation method.
S/N
Function
Differential transform
1
ut±vt
Up±Vp
2
αut
αUp
3
dut/dt
p+1Up+1
4
utvt
∑r=0pVrUp−r
5
umt
∑r=0pUm−1rUp−r
6
dnut/dxn
p+1p+2⋯p+nUp+n
7
sinωt+α
ωp/p!sinπp/2!+α
8
cosωt+α
Zp=ωp/p!cosπp/2!+α
Equivalently, one can write the recursive relation for (21) in DTM domain as(22)Fp+3=1−ϕ2.51+1/γp+1p+2p+31−ϕ+ϕρsρf2∑l=0pl+1p−l+1Fl+1Fp−l+1−3∑l=0pp−l+1p−l+2FlFp−l+2−1−ϕ+ϕρβsρβfΘp.
For (13), the recursive relation in differential transform domain is given as(23)1+43Rp+1p+2Θp+2+3Pr11−ϕ+ϕρCps/ρCpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks×∑l=0pl+1Θl+1Fp−l=0,which can be written as(24)Θp+2=−3Pr1+4/3Rp+1p+211−ϕ+ϕρCps/ρCpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks×∑l=0pl+1Θl+1Fp−l.
Also, the recursive relation for the boundary conditions in (15) is(25)Fp=0⇒F0=0,p+1Fp+1=0⇒F1=0,θp=1⇒θ0=1,F2=a2,θ1=b,where a and b are unknown constants which will be found later.
From (25), the following boundary conditions in differential transform domain are established:(26)F0=0,F1=0,θ0=1,F2=a2,θ1=b.
Using p=0,1,2,3,4,5,6,7,… in the above recursive relations in (21), the following equations are developed:(27)F3=−1−ϕ2.561+1/γ1−ϕ+ϕρβsρβf,(28)F4=−1−ϕ2.5241+1/γ1−ϕ+ϕρβsρβfb,(29)F5=1−ϕ2.51201+1/γ1−ϕ+ϕρsρfa2,(30)F6=0,(31)F7=1−ϕ2.52101+1/γ21−ϕ+ϕρsρf−131−ϕ2.51−ϕ+ϕρβsρβfab+141−ϕ51−ϕ+ϕρβsρβf2−31−ϕ+ϕρsρf−71−ϕ2.5241−ϕ+ϕρβsρβfab+1−ϕ561−ϕ+ϕρβsρβf2+1−ϕ+ϕρβs/ρβfPr811−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksab,(32)F8=1−ϕ2.53361+1/γ21−ϕ+ϕρsρf1−ϕ2.5121−ϕ+ϕρsρfa3+1−ϕ561−ϕ+ϕρβsρβf2b−31−ϕ+ϕρsρf111−ϕ2.5201−ϕ+ϕρsρfa3+1−ϕ581−ϕ+ϕρβsρβf2b−1−ϕ+ϕρβs/ρβf2Pr1−ϕ2.54011−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksb,(33)F9=1−ϕ2.55401+1/γ21−ϕ+ϕρsρf−1−ϕ5241−ϕ+ϕρβsρβf1−ϕ+ϕρsρfa2+1−ϕ5361−ϕ+ϕρβsρβf2b2−31−ϕ+ϕρsρf−131−ϕ53601−ϕ+ϕρβsρβf1−ϕ+ϕρsρfa2+1−ϕ5481−ϕ+ϕρβsρβf2b2−1−ϕ+ϕρβs/ρβf2240Pr11−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks1−ϕ2.5b2,(34)F10=1−ϕ2.57201+1/γ21−ϕ+ϕρsρfa1−ϕ2.51521−ϕ+ϕρsρf−a1−ϕ2.531−ϕ+ϕρβsρβfb+1−ϕ2.541−ϕ+ϕρβsρβf2−31−ϕ+ϕρsρf−7a1−ϕ2.5241−ϕ+ϕρβsρβfb+1−ϕ561−ϕ+ϕρβsρβf2+1−ϕ+ϕρβs/ρβfPr811−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksab−1−ϕ51−ϕ+ϕρβs/ρβf721−ϕ+ϕρsρfa2b−31−ϕ+ϕρsρf11a1−ϕ2.510521−ϕ+ϕρsρf−131−ϕ2.51−ϕ+ϕρβsρβfab+141−ϕ51−ϕ+ϕρβsρβf2−31−ϕ+ϕρsρf−7241−ϕ2.51−ϕ+ϕρβsρβfab+1−ϕ561−ϕ+ϕρβsρβf2+1−ϕ+ϕρβs/ρβfPr811−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksab−1−ϕ5901−ϕ+ϕρβsρβfa2b+1−ϕ+ϕρβs/ρβfPr1411−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks−a2Pr411−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksb+1−ϕ2.51201−ϕ+ϕρsρfa2b.
In the same manner, the expressions for F [11], F [12], F [13], F [14], F [15] were found, which are too large expressions to be included in this paper.
Also, using p=0,1,2,3,… in the above recursive relations in (24), one arrives at(35)Θ2=0,(36)Θ3=0,(37)Θ4=−Pr81+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksab,(38)Θ5=Pr401+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ2.51+1/γ1−ϕ+ϕρβsρβfb,(39)Θ6=Pr1201+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ2.51+1/γ1−ϕ+ϕρβsρβfb2,(40)Θ7=−Pr141+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅−Pr411−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksa2b+1−ϕ2.51201+1/γ1−ϕ+ϕρsρfa2b,(41)Θ8=−Pr21281+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks21−ϕ2.51+1/γ1−ϕ+ϕρβsρβfab,(42)Θ9=−Pr241+4/3R11−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksPr3011−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ2.51+1/γ1−ϕ+ϕρβsρβfab2−Pr4811−ϕ+ϕρcps/ρcpfks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ51+1/γ1−ϕ+ϕρβsρβf2b+1−ϕ2.52101+1/γ21−ϕ+ϕρsρf−1−ϕ2.531+1/γ1−ϕ+ϕρβsρβfab+1−ϕ541+1/γ1−ϕ+ϕρβsρβf2−31−ϕ+ϕρsρf−71−ϕ2.5241+1/γ1−ϕ+ϕρβsρβfab+1−ϕ561+1/γ1−ϕ+ϕρβsρβf2+Pr811−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ+ϕρβsρβfabb,(43)Θ10=−Pr41+4/3R11−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks−aPr4Pr411−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksa2b+1−ϕ2.51201+1/γ1−ϕ+ϕρsρfa2b−31−ϕ5Pr3201+1/γ11−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks1−ϕ+ϕρβsρβf2b2−1−ϕ2.5a3Pr1201+1/γ1−ϕ+ϕρβsρβf11−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksb+1−ϕ2.53361+1/γ21−ϕ+ϕρβsρβf⋅a31−ϕ2.5121+1/γ1−ϕ+ϕρβsρβf+1−ϕ561+1/γ1−ϕ+ϕρβsρβf2b−31−ϕ+ϕρβsρβf⋅111−ϕ2.51201+1/γa31−ϕ+ϕρβsρβf+1−ϕ581+1/γ1−ϕ+ϕρβsρβf2b−Pr4011−ϕ+ϕρcps/ρcpf⋅ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ks⋅1−ϕ+ϕρβsρβf21−ϕ2.5bb.
Also, the expressions for Θ11, Θ12, Θ13, Θ14, Θ15, … are found in the same way, but they are too large expressions to be included in this paper.
Using the definition in (20), the solutions of (13) and (14) are given as(44)fη=F0+ηF1+η2F2+η3F3++η4F4+η5F5+η6F6+η7F7+η8F8+η9F9+η10F10+⋯,(45)θη=Θ0+ηΘ1+η2Θ2+η3Θ3++η4Θ4+η5Θ5+η6Θ6+η7Θ7+η8Θ8+η9Θ9+η10Θ10+⋯.
The solutions of (15) and (16) for a viscous fluid which does not have nanoparticles can be developed from (44) and (45) if nanoparticle volume fraction is set to zero, that is, ϕ=0.
4. The Basic Concept and the Procedure of Padé Approximant
The limitations of power series methods to a small domain have been overcomed by domain transformation and after-treatment techniques. These techniques increase the radius of convergence and also accelerate the rate of convergence of a given series. Among the so-called after-treatment techniques, Padé-approximant technique has been widely applied in developing accurate analytical solutions to nonlinear problems of large or unbounded domain problems of infinite boundary conditions [71]. The Padé-approximant technique manipulates a polynomial approximation into a rational function of polynomials. Such a manipulation gives more information about the mathematical behaviour of the solution. The basic procedures are as follows.
Suppose that a function fη is represented by a power series:(46)fη=∑i=0∞ciηi.
This above expression is the fundamental point of any analysis using Padé approximant. The notation ci,i=0,1,2,… is reserved for the given set of coefficient, and fη is the associated function. [L/M] Padé approximant is a rational function defined as(47)fη=∑i=0Laiηi∑i=0Mbiηi=a0+a1η+a2η2+⋯+aLηLb0+b1η+b2η2+⋯+bMηM,which has a Maclaurin expansion, agreeing with (46) (1) as far as possible. It is noticed that, in (47), there are L+1 numerator and M+1 denominator coefficients. So, there are L+1 independent number of numerator coefficients, making L+M+1 unknown coefficients in all. These numbers of coefficients of numerator and denominator suggest that L/M out of fit the power series in (46) through orders 1,η,η2,…,ηL+M.
In the notation of formal power series:(48)∑i=0∞ciηi=a0+a1η+a2η2+⋯+aLηLb0+b1η+b2η2+⋯+bMηM+ΟηL+M+1,which gives(49)b0+b1η+b2η2+⋯+bMηMc0+c1η+c2η2+⋯=a0+a1η+a2η2+⋯+aLηL+ΟηL+M+1.
Expanding the LHS and equating the coefficients of ηL+1,ηL+2,…,ηL+M, we get(50)bMcL−M+1+bM−1cL−M+2+bM−2cL−M+3+⋯+b2cL−1+b1cL+b0cL+1=0,bMcL−M+2+bM−1cL−M+3+bM−2cL−M+4+⋯+b2cL+b1cL+1+b0cL+2=0,bMcL−M+3+bM−1cL−M+4+bM−2cL−M+5+⋯+b2cL+1+b1cL+2+b0cL+3=0,⋮bMcL+bM−1cL+1+bM−2cL+2+⋯+b2cL+M−2+b1cL+M−1+b0cL+M=0,
If i<0,ci=0 for consistency. Since b0=1, (50) becomes a set of M linear equations for M unknown denominator coefficients:(51)cL−M+1cL−M+2cL−M+3⋯cLcL−M+2cL−M+3cL−M+4⋯cL+1cL−M+3cL−M+4cL−M+4⋯cL+2⋮⋮⋮⋮⋮cLcL+1cL+2⋯cL+M−1bMbM−1bM−2⋮b1=−cL+1cL+2cL+3⋮cL+M.
From the above (51), bi may be found. The numerator coefficients a0,a1,a2,…,aL follow immediately from (49) by equating the coefficient of 1,η,η2,…,ηL+M such that(52)a0=c0,a1=c1+b1c0,a2=c2+b1c1+b2c0,a3=c3+b1c2+b2c1+b3c0,a4=c4+b1c3+b2c2+b3c1+b4c0,a5=c5+b1c4+b2c3+b3c2+b4c1+b5c0,a6=c6+b1c5+b2c4+b3c3+b4c2+b5c1+b6c0,⋮aL=cL+∑i=1minL/MbicL−i.
Equations (51) and (52) normally determine the Padé numerator and denominator and are called Padé equations. The [L/M] Padé approximant is constructed which agrees with the equation in the power series through the order ηL+M. To obtain a diagonal Padé approximant of order [L/M], the symbolic calculus software Maple is used.
It should be noted as mentioned previously that Δ and Ω in the solutions are unknown constants. In order to compute their values for extended large domains solutions, the power series as presented in (44) and (45) are converted to rational functions using [18/18] Padé approximation through the software Maple, and then, the infinite boundary conditions, that is, η→∞,f′=0,θ=0 are applied. The resulting simultaneous equations are solved to obtain the values of Δ and Ω for the respective values of the physical and thermal properties of the nanofluids under considerations.
5. Flow and Heat Transfer Parameters
In addition to the determination of the velocity and temperature distributions, it is often desirable to compute other physically important quantities (such as shear stress, drag, heat transfer rate, and heat transfer coefficient) associated with the free convection flow and heat transfer problem. Consequently, two parameters, a flow parameter and a heat transfer parameter, are computed.
5.1. Fluid Flow Parameter
Skin friction coefficient is given by the following equation:(53)cf=τwρnfu2=μnf∂u/∂yy=0ρnfu2=μnf∂u/∂η⋅∂η/∂yy=0ρnfu2.
After the dimensionless exercise,(54)cfRex1/2=f″01−ϕ2.5,cfRex1/2τw4Grx31/4νμ=f″0f″01−ϕ2.5.
5.2. Heat Transfer Parameter
Heat transfer coefficient is given by the following equation:(55)hx=−knfTw−T∞∂T∂yy=0=−knfθ′01x14Grx1/4.
The local heat transfer coefficient at the surface of the vertical plate is obtained from (55). The local Nusselt number is(56)Nux=hxxknf=−xTw−T∞∂T∂yy=0=−θ′014Grx1/4,Nux=−θ′02Grx1/4=fPrGrx1/4,where ϕPr=−θ′0/2 is a function of Prandtl number. The dependence of ϕ on the Prandtl number is evidenced by (56).
It could also be shown that(57)NuxRex1/2=−knfkfθ′0=−ks+m−1kf−m−1ϕkf−ksks+m−1kf+ϕkf−ksθ′0,where Rex and Grx are the local Reynold and Grashof numbers, which are defined as(58)Rex=uxνnf,Grx=gβTw−T∞x3ν3.
6. Results and Discussion
The solutions of DTM with [18/18] Padé approximation is presented in this section. Tables 5–7 present various comparisons of results of the present study and the past works for viscous fluid, that is, when the volume fraction of the nanoparticles, thermal radiation, and Casson parameters are set to zero, that is, ϕ=R=β=0. It could be seen from the tables that there are excellent agreements between the past results and the present study. Moreover, the tables present the effects of Prandtl number on the flow and heat transfer processes.
Comparison of results of f″0 and θ′0 at different Prandtl numbers.
Kuo [51]
Na and Habib [12]
Present study
Pr
f″0
θ′0
f″0
θ′0
f″0
θ′0
0.72
0.6760
−0.5046
0.6760
−0.5046
0.6760
−0.5046
0.60
0.6947
−0.4721
0.6946
−0.4725
0.6947
−0.4721
0.50
0.7132
−0.4411
0.7131
−0.4420
0.7132
−0.4411
0.40
0.7356
−0.4053
0.7354
−0.4066
0.7356
−0.4053
0.30
0.7636
−0.3623
0.7633
−0.3641
0.7636
−0.3623
0.20
0.8015
−0.3078
0.8009
−0.3101
0.8015
−0.3078
0.10
0.8600
−0.2298
0.8590
−0.2326
0.8600
−0.2298
0.06
0.8974
−0.1834
0.8961
−0.1864
0.8974
−0.1834
0.04
0.9233
−0.1526
0.9221
−0.1556
0.9233
−0.1526
0.01
0.9845
−0.0832
0.9887
−0.0817
0.9885
−0.0832
1.00
0.6421
−0.5671
0.6421
−0.5671
0.6421
−0.5671
1.10
0.6323
−0.5862
0.6323
−0.5860
0.6323
−0.5862
1.20
0.6233
−0.6040
0.6234
−0.6036
0.6233
−0.6040
1.30
0.6151
−0.6208
0.6152
−0.6202
0.6151
−0.6208
1.40
0.6075
−0.6365
0.6076
−0.6358
0.6075
−0.6365
1.50
0.6005
−0.6515
0.6006
−0.6506
0.6005
−0.6515
1.60
0.5939
−0.6656
0.5940
−0.6646
0.5939
−0.6656
1.70
0.5877
−0.6792
0.5879
−0.6780
0.5877
−0.6792
1.80
0.5819
−0.6921
0.5821
−0.6908
0.5819
−0.6921
1.90
0.5764
−0.7045
0.5767
−0.7031
0.5764
−0.7045
2.00
0.5712
−0.7164
0.5715
−0.7149
0.5712
−0.7164
2.00
0.5712
−0.7164
0.5713
−0.7165
0.5712
−0.7164
3.00
0.5308
−0.8154
0.5312
−0.8145
0.5308
−0.8154
4.00
0.5029
−0.8914
0.5036
−0.8898
0.5029
−0.8914
5.00
0.4817
−0.9539
0.4827
−0.9517
0.4817
−0.9539
6.00
0.4648
−1.0073
0.4660
−1.0047
0.4648
−1.0073
7.00
0.4507
−1.0542
0.4522
−1.0512
0.4507
−1.0542
8.00
0.4387
−1.0961
0.4405
−1.0930
0.4387
−1.0961
9.00
0.4283
−1.1342
0.4304
−1.1309
0.4283
−1.1342
10.00
0.4191
−1.1692
0.4215
−1.1658
0.4191
−1.1692
Comparison of results of f″0 at different Prandtl numbers.
f″0
Pr
Kuiken [72]
Mosta et al. [17]
Present study
0.001
1.12313813
1.12313813
1.12313813
0.01
1.06338086
1.06338086
1.06338086
0.1
0.92408304
0.92408304
0.92408304
1
0.69321163
0.69321163
0.69321163
10
0.44711652
0.44711652
0.44711652
100
0.26452354
0.26452354
0.26452354
1000
0.15129020
0.15129020
0.15129020
10000
0.08554085
0.08554085
0.08554085
Comparison of results of −θ′0 at different Prandtl numbers.
−θ′0
Pr
Kuiken [72]
Mosta et al. [16]
Mosta et al. [17]
Present study
0.001
0.04680746
0.04680746
0.04680746
0.04680746
0.01
0.13576074
0.13576074
0.13576074
0.13576074
0.1
0.35005967
0.35005967
0.35005967
0.35005967
1
0.76986120
0.76986120
0.76986119
0.76986120
10
1.49709921
1.49709921
1.49709921
1.49709921
100
2.74688550
2.74688550
2.74688549
2.74688550
1000
4.93494763
4.93494763
4.93494756
4.93494763
10000
8.80444927
8.80444927
8.80444960
8.80444960
Although, the nonlinear partial differential equations in Mosta et al. [17] are the same in all aspects to the present problems under investigation, there are slight differences between the transformed nonlinear ordinary differential equations in (7) of Mosta et al. [17] and (13) and (14) developed in this present study (where the volume fraction of the nanoparticles, radiation, and Casson parameters are set to zero) due to the differences in the adopted similarity variables. It is shown that, using the DTM-Padé approximant as applied in this work to the transformed nonlinear ordinary differential equations in Mosta et al. [17], excellent agreements are recorded between the results of the present study and that of Mosta et al. [17] and Kuiken [72] as shown in Tables 6 and 7.
The variations in nanoparticle volume fraction with dynamic viscosity and thermal conductivity ratios of copper(II) oxide-water nanofluid are shown in Figures 2 and 3, respectively. Also, Figure 3 show the effects of nanoparticle shape on thermal conductivity ratio. It is depicted in the figure that the thermal conductivity of the nanofluid varies linearly and increases with increase in nanoparticle volume fraction. It is also observed that the suspensions of particles with high shape factor or low sphericity have higher thermal conductivity ratio of the nanofluid. Spherical-shaped nanoparticles have the lowest thermal conductivity ratio, and lamina-shaped nanoparticles have the highest thermal conductivity ratio.
Variation in nanofluid dynamic viscosity ratio with nanoparticle volume fraction.
Effects of nanoparticle shape on thermal conductivity ratio of the nanofluid.
The effects of the flow and heat transfer controlling parameters on the velocity and temperature distributions are shown in Figures 4–17 for different shapes, type, and volume fraction of the nanoparticles at Prandtl number of 0.01–1000.
(a) Effects of Casson parameter on the velocity profile of the Casson nanofluid. (b) Effects of Casson parameter on the temperature profile of the Casson nanofluid.
Effects of radiation parameter on the (a) velocity profile and (b) temperature profile of the Casson nanofluid.
(a) Effects of Prandtl number on the velocity profile when ϕ = 0.020. (b) Effects of Prandtl number on the temperature profile when ϕ = 0.020.
(a) Effects of Prandtl number on the velocity profile when ϕ = 0.040. (b) Effects of Prandtl number on the temperature profile when ϕ = 0.040.
(a) Effects of Prandtl number on the velocity profile when ϕ = 0.060. (b) Effects of Prandtl number on the temperature profile when ϕ = 0.060.
(a) Effects of Prandtl number on the velocity profile when ϕ = 0.080. (b) Effects of Prandtl number on the temperature profile when ϕ = 0.080.
(a) Effect of nanoparticle shape on the velocity distribution of the nanofluid. (b) Effects of nanoparticle shape on the temperature distribution of the nanofluid.
(a) Effects of Prandtl number on the velocity profile for spherical-shaped nanoparticle. (b) Effects of Prandtl number on the temperature profile for spherical-shaped nanoparticle.
(a) Effects of Prandtl number on the velocity profile for brick-shaped nanoparticle. (b) Effects of Prandtl number on the temperature profile for brick-shaped nanoparticle.
(a) Effects of Prandtl number on the velocity profile for cylindrical-shaped nanoparticle. (b) Effects of Prandtl number on the temperature profile for cylindrical-shaped nanoparticle.
(a) Effects of Prandtl number on the velocity profile for platelet-shaped nanoparticle. (b) Effects of Prandtl number on the temperature profile for platelet-shaped nanoparticle.
(a) Effects of Prandtl number on the velocity profile for lamina-shaped nanoparticle. (b) Effects of Prandtl number on the temperature profile for lamina-shaped nanoparticle.
(a) Effects of Prandtl number on the velocity profile for TiO_{2} nanoparticle. (b) Effects of Prandtl number on the temperature profile for TiO_{2} nanoparticle.
(a) Effects of Prandtl number on the velocity profile for CuO nanoparticle. (b) Effects of Prandtl number on the temperature profile for CuO nanoparticle.
6.1. Effect of Casson Parameter on Casson Nanofluid Velocity and Temperature Distributions
Figures 4(a) and 4(b) depict the effects of Casson parameters on the velocity and temperature profiles of the Casson nanofluid, respectively. It is obvious from the figure that the Casson parameter has influence on axial velocity. From Figure 4(a), the magnitude of velocity near the plate for Casson nanofluid parameter decreases for increasing value of the Casson parameter, while temperature increases for increase in Casson fluid parameter as shown in Figure 4(b). Physically, increasing values of the Casson parameter develop the viscous forces. These forces have tendencies to decline the thermal boundary layer.
6.2. Effect of Thermal Radiation Parameter on Casson Nanofluid Velocity and Temperature Distributions
It is depicted that both viscous and thermal boundary layers increase with the increase in radiation parameter, R. Figure 5(a) depicts the effect of thermal radiation parameter on the velocity profiles. From the figure, it is shown that increase in radiation parameter causes the velocity of the fluid to increase. This is because, as the radiation parameter is augmented, the absorption of radiated heat from the heated plate releases more heat energy to the fluid and the resulting temperature increases the buoyancy forces in the boundary layer which also increases the fluid motion and the momentum boundary layer thickness. This is expected because the considered radiation effect within the boundary layer increases the motion of the fluid which increases the surface frictions. The maximum velocity for all values of R is at the approximated value of η = 0.5. Therefore, it can be concluded that the inner viscous layer does not increase for variation of radiation parameter. Only the outer layer thickness has a great influence on thermal radiation, R. Although the velocity gradient at the surface increases with the increase in radiation parameter, a reverse case has been established in the literature when water is used as the fluid under the study of the flow of viscous fluid over a flat surface.
Using a constant value of the Prandtl number, the influence of radiation parameter on the temperature field is displayed in Figure 5(b). Increase in the radiation parameter contributes in general to increase in the temperature. This is because, as the thermal radiation increases, the absorption of radiated heat from the heated plate releases heat energy to the fluid. Consequently, the thermal boundary layer of fluid increases as the temperature near the boundary is enhanced. This shows that influence of radiation is more effective when high temperature is required for the desired thickness of the end product. It is observed that the effect of the radiation parameter is not significant as we move away from the boundary. Also, it is observed that as the temperature of the fluid increases for increasing thermal radiation, the temperature difference between the plate and the ambient fluid reduces which turns to decrease the heat transfer rate in flow region.
6.3. Effect of Nanoparticle Volume Fraction on Casson Nanofluid Velocity and Temperature Distributions for Different Values of Prandtl Number
Figures 6–9 show the effects of nanoparticle concentration/volume fraction and Prandtl number on velocity and temperature profiles of copper(II) oxide-water Casson nanofluid. It is indicated in the figures that as the volume fraction or concentration of the nanoparticle in the nanofluid increases, the velocity decreases. However, an opposite trend or behaviour in the temperature profile is observed; that is, the nanofluid temperature increases as the volume fraction of the nanoparticles in the base fluid increases. This is because the solid volume fraction has significant impacts on the thermal conductivity. The increased volume fraction of the nanoparticles in the base fluid results in higher thermal conductivity of the base fluid which increases the heat enhancement capacity of the base fluid. Also, one of the possible reasons for the enhancement on heat transfer of nanofluids can be explained by the high concentration of nanoparticles in the thermal boundary layer at the wall side through the migration of nanoparticles. It should also be stated that the thickness of the thermal boundary layer rises with increase in the values of nanoparticle volume fraction. This consequently reduces the velocity of the nanofluid as the shear stress and skin friction are increased. The figures also show the effects of Prandtl number (Pr) on the velocity and temperature profiles. It is indicated that the velocity of the nanofluid decreases as the Pr increases, but the temperature of the nanofluid increases as the Pr increases. This is because the nanofluid with higher Prandtl number has a relatively low thermal conductivity, which reduces conduction and thereby reduces the thermal boundary layer thickness and, as a consequence, increases the heat transfer rate at the surface. For the case of the fluid velocity that decreases with the increase in Pr, the reason is that fluid of higher Prandtl number means more viscous fluid, which increases the boundary layer thickness and thus reduces the shear stress and consequently, retards the flow of the nanofluid. Also, it can be seen that the velocity distribution for small value of Prandtl number consists of two distinct regions: a thin region near the wall of the plate where there are large velocity gradients due to viscous effects and a region where the velocity gradients are small compared with those near the wall. In the later region, the viscous effects are negligible and the flow of fluid in the region can be considered to be inviscid. Also, such region tends to create uniform accelerated flow at the surface of the plate.
The use of nanoparticles in the fluids exhibited better properties relating to the heat transfer of fluid than heat transfer enhancement through the use of suspended millimeter- or micrometer-sized particles which potentially cause some severe problems, such as abrasion, clogging, high pressure drop, and sedimentation of particles. The very low concentration applications and nanometer size properties of nanoparticles in the base fluid prevent the sedimentation in the flow that may clog the channel. It should be added that the theoretical prediction of enhanced thermal conductivity of the base fluid and prevention of clogging, abrasion, high pressure drop, and sedimentation through the addition of nanoparticles in the base fluid have been supported with experimental evidences in the literature.
6.4. Effect of Nanoparticle Shape on Casson Nanofluid Velocity and Temperature Distributions for Different Values of Prandtl Number
It is observed experimentally that the nanoparticle shape has significant impacts on the thermal conductivity. Therefore, the effects of nanoparticle shape at different values of Prandtl number on the velocity and temperature profiles of copper(II) oxide-water nanofluid are shown in Figures 10–15. It is indicated that the maximum decrease in velocity and maximum increase in temperature are caused by lamina, platelets, cylinder, bricks, and sphere, respectively. It is observed that lamina-shaped nanoparticle carries maximum velocity, whereas spherical-shaped nanoparticle has better enhancement on heat transfer than other nanoparticle shapes. In fact, it is in accordance with the physical expectation since it is well known that the lamina nanoparticle has greater shape factor than other nanoparticles of different shapes; therefore, the lamina nanoparticle comparatively gains maximum temperature than others. The decrease in velocity is highest in spherical nanoparticles as compared with other shapes. The enhancement observed at lower volume fractions for nonspherical particles is attributed to the percolation chain formation, which perturbs the boundary layer and thereby increases the local Nusselt number values.
It is evident from this study that proper choice of nanoparticles will be helpful in controlling velocity and heat transfer. It is also observed that irreversibility process can be reduced by using nanoparticles, especially the spherical particles. This can potentially result in higher enhancement in the thermal conductivity of a nanofluid containing elongated particles compared to the one containing spherical nanoparticle, as exhibited by the experimental data in the literature.
6.5. Effect of Type of Nanoparticle on Casson Nanofluid Velocity and Temperature Distribution for Different Values of Prandtl Number
The variations of the velocity and temperature profiles against η for various types of nanoparticles (TiO_{2}, CuO, Al_{2}O_{3}, and SWCNTs) are shown in Figures 16–19. Using a common base fluid for all the nanoparticle types, it is observed that the maximum decrease in velocity and maximum increase in temperature are caused by TiO_{2}, CuO, Al_{2}O_{3}, and SWCNTs, respectively. It is observed that SWCNT nanoparticle carries maximumdecrease in velocity but has better enhancement on heat transfer than other nanoparticle shapes. In accordance with the physical expectation well, the SWCNT nanoparticle has higher thermal conductivity than other types of nanoparticles; therefore, the SWCNT nanoparticle comparatively gains maximum temperature than others. The increased thermal conductivity of the base fluid due to the use of nanoparticle of higher thermal conductivity increases the heat enhancement capacity of the base fluid.
(a) Effects of Prandtl number on the velocity profile for Al_{2}O_{3} nanoparticle. (b) Effects of Prandtl number on the temperature profile for Al_{2}O_{3} nanoparticle.
(a) Effects of Prandtl number on the velocity profile for SWCNT nanoparticle. (b) Effects of Prandtl number on the temperature profile for SWCNT nanoparticle.
Also, it is observed that the velocity decrease is maximum in SWCNT nanoparticles when compared with other types of nanoparticles. This is because the solid thermal conductivity has significant impacts on the momentum boundary layer of the nanofluid. The thickness of the momentum boundary layer increases with the increase in thermal conductivity. It is observed that the thickness of the thermal boundary layer enhances in the presence of higher thermal conductivity nanoparticle. Therefore, the sensitivity of the boundary layer thickness to the type of nanoparticle is correlated with the value of the thermal conductivity of the nanoparticle used, which consequently leads to enhancement of thermal conductivity of the nanofluid.
7. Conclusion
In this work, the influences of thermal radiation and nanoparticles on free convection flow and heat transfer of Casson nanofluids over a vertical plate have been analyzed. The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are transformed to system of nonlinear ordinary differential equation through similarity variables. The systems of fully coupled nonlinear ordinary differential equations have been solved using differential transformation method with Padé-approximant technique. The accuracies of the developed analytical solutions were verified with the results generated by some other methods as presented in the past works. The developed analytical solutions were used to investigate the effects of Casson parameter, thermal radiation parameter, Prandtl number, nanoparticle size, and nanoparticle shapes on the flow and heat transfer behaviour of various Casson nanofluids. From the parametric studies, the following observations were established:
The magnitude of velocity near the plate for the Casson nanofluid parameter decreases for increasing value of the Casson parameter, while temperature increases for increase in Casson fluid parameter.
Both the velocity and temperature of the nanofluid as well viscous and thermal boundary layers increase with increase in the radiation parameter.
The velocity of the nanofluid decreases as the Prandtl number increases, but the temperature of the nanofluid increases as the Prandtl number increases.
The velocity of the nanofluid decreases as the volume fraction or concentration of the nanoparticle in the base fluid increases. However, an opposite trend or behaviour in the temperature profile was observed which showed that as the nanofluid temperature increases, the volume fraction of the nanoparticles in the base fluid increases.
The lamina-shaped nanoparticle carries maximum velocity, whereas spherical-shaped nanoparticle has better enhancement on heat transfer than other nanoparticle shapes. The maximum decrease in velocity and maximum increase in temperature are caused by lamina-shaped nanoparticle and followed by platelet-, cylinder-, brick-, and sphere-shaped nanoparticles, respectively.
Using a common base fluid to all the nanoparticle types considered in this work, it was observed that SWCNT nanoparticle carries maximum decrease in velocity but has better enhancement on heat transfer than other nanoparticle shapes. Also, it was observed that the maximum decrease in velocity and maximum increase in temperature are caused by TiO_{2} and followed by CuO, Al_{2}O_{3}, and SWCNT nanoparticles, in that order.
The present study reveals and exposes the predominant factors affecting the boundary layer of free convection flow and heat transfer of Casson nanofluids. Moreover, the high level of accuracy and versatility of differential transformation method with Padé-approximate technique have been demonstrated. It is hoped that the present study will enhance the understanding as it provides physical insights into the free convection boundary layer problems of Casson nanofluid under various parameters.
AbbreviationsNomenclaturec_{
p}:
Specific heat capacity
k:
Thermal conductivity
K:
The absorption coefficient
m:
Shape factor
p:
Pressure
p_{
y}:
Yield stress of the fluid
Pr:
Prandtl number
u:
Velocity component in the x-direction
v:
Velocity component in the z-direction
y:
Axis perpendicular to plates
x:
Axis along the horizontal direction
y:
Axis along the vertical direction
Symbolsβ:
Volumetric extension coefficients
ρ:
Density of the fluid
μ:
Dynamic viscosity
η:
Similarity variable
γ:
Casson parameter
λ:
Sphericity
ϕ:
Volume fraction or concentration of the nanofluid
θ:
Dimensionless temperature
τ:
Shear stress
τ_{0}:
Casson yield stress
μ:
Dynamic viscosity
σ˙:
Shear rate
e_{
ij}:
The (i, j)th component of the deformation rate
π:
Product of the component of deformation rate with itself
π_{c}:
Critical value of this product based on the non-Newtonian model
μ_{B}:
Plastic dynamic viscosity of the non-Newtonian fluid
Subscriptf:
Fluid
s:
Solid
n_{
f}:
Nanofluid.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author expresses sincere appreciation to the University of Lagos, Nigeria, for providing material supports and good environment for this work.
SchmidtE.BeckmannW.Das temperatur-und geschwindigkeitsfeld vor einer wärme abgebenden senkrecher platte bei natürelicher conventionOstrachS.An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force1953Denton, TX, USAUniversity of North TexasNACA ReportSparrowE. M.GreggJ. L.Laminar free convection from a vertical plate with uniform surface heat fluxLefevreE. J.Laminar free convection from a vertical plane surface168Proceedings of the 9th International Congress on Applied Mechanics1956Brussels, BelgiumSparrowE. M.GreggJ. L.Similar solutions for free convection from a nonisothermal vertical plateStewartsonK.JonesL. T.The heated vertical plate at high Prandtl numberKuikenH. K.An asymptotic solution for large Prandtl number free convectionKuikenH. K.Free convection at low Prandtl numbersEshghyS.Free-convection layers at large Prandtl numberRoyS.High Prandtl number free convection for uniform surface heat fluxKuikenH. K.RotemZ.Asymptotic solution for the plume at very large and small Prandtl numbersNaT. Y.HabibI. S.Solution of the natural convection problem by parameter differentiationMerkinJ. H.A note on the similarity solutions for free convection on a vertical plateMerkinJ. H.PopI.Conjugate free convection on a vertical surfaceAliF. M.NazarR.ArifinN. M.Numerical investigation of free convective boundary layer in a viscous fluidMotsaS. S.ShateyiS.MakukulaZ.Homotopy analysis of free convection boundary layer flow with heat and mass transferMotsaS. S.MakukulaZ. G.ShateyiS.Spectral local linearisation approach for natural convection boundary layer flowGhotbiA. R.BararniaH.DomairryG.BarariA.Investigation of a powerful analytical method into natural convection boundary layer flowMosayebidorchehS.MosayebidorchehT.Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivitySheikholeslamiM.CVFEM for magnetic nanofluid convective heat transfer in a porous curved enclosureSheikholeslamiM.GanjiD. D.SheikholeslamiM.SheikholeslamiM.RanaP.SoleimaniS.Numerical study of MHD Natural convection liquid metal flow and heat transfer in a wavy enclosure using CVFEMMosayebidorchehS.Rahimi-GorjiM.GanjiD. D.MoayebidorchehT.PourmehranO.BiglarianM.Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties using DTM-FDMMosayebidorchehS.MakindeO. D.GanjiD. D.ChermahiniM. A.DTM-FDM hybrid approach to unsteady MHD Couette flow and heat transfer of dusty fluid with variable propertiesMosayebidorchehS.FarzinpoorM.GanjiD. D.Transient thermal analysis of longitudinal fins with internal heat generation considering temperature-dependent properties and different fin profilesMosayebidorchehS.MoayebidorchehT.RashidiM. M.Analytical solution of the steady state condensation film on the inclined rotating disk by a new hybrid methodMosayebidorchehS.VataniM. M.GanjiD. D.MoayebidorchehT.Investigation of the viscoelastic flow and species diffusion in a porous channel with high permeabilityMosayebidorchehS.Analytical investigation of the micropolar flow through a porous channel with changing wallsHatamiM.MosayebidorchehS.JingD.Thermal performance evaluation of alumina-water nanofluid in an inclined direct absorption solar collector (IDASC) using numerical methodMosayebidorchehS.Solution of the boundary layer equation of the power-law pseudoplastic fluid using differential transform methodMosayebidorchehS.HatamiM.GanjiD. D.MosayebidorchehT.MirmohammadsadeghiS. M.Investigation of transient MHD Couette flow and heat transfer of dusty fluid with temperature-dependent propertiesSheikholeslamiM.ShehzadS. A.CVFEM for influence of external magnetic source on Fe_{3}O_{4}-H_{2}O nanofluid behavior in a permeable cavity considering shape effectSheikholeslamiM.ShehzadS. A.Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBMSheikholeslamiM.SeyednezhadM.Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEMSheikholeslamiM.RokniH. B.Simulation of nanofluid heat transfer in presence of magnetic field: a reviewSheikholeslamiM.HayatT.AlsaediA.On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylindersSheikholeslamiM.RokniH. B.Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic fieldSheikholeslamiM.SadoughiM. K.Simulation of CuO-water nanofluid heat transfer enhancement in presence of melting surfaceSheikholeslamiM.SadoughiM.Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticlesSheikholeslamiM.BhattiM. M.Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticlesSheikholeslamiM.BhattiM. M.Active method for nanofluid heat transfer enhancement by means of EHDSheikholeslamiM.RokniH. B.Nanofluid two phase model analysis in existence of induced magnetic fieldSheikholeslamiM.ShehzadS. A.Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary conditionSheikholeslamiM.HayatT.AlsaediA.Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using lattice Boltzmann methodSheikholeslamiM.HayatT.AlsaediA.Numerical study for external magnetic source influence on water based nanofluid convective heat transferSheikholeslamiM.HayatT.AlsaediA.AbelmanS.Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosityZhouJ. K.Lien-TsaiY.Cha’o-KuangC.The solution of the Blasius equation by the differential transformation methodKuoB. L.Thermal boundary-layer problems in a semi-infinite flat plate by the differential transformation methodKuoB. L.Application of the differential transformation method to the solutions of the free convection problemRashidiM. M.LaraqiN.SadriS. M.A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-PadeCassonN.DashR. K.MehtaK. N.JayaramanG.Casson fluid flow in a pipe filled with a homogeneous porous mediumAnderssonH. I.DandapatB. S.Flow of a power-law fluid over a stretching sheetSajidM.AhmadI.HayatT.AyubM.Unsteadyflow and heat transfer of a second grade fluid over a stretching sheetEldabeN. T. M.SalwaM. G. E.Heat transfer of MHD non-Newtonian Casson fluid flow between two rotating cylinderNadeemS.HaqR. L.AkbarN. S.KhanZ. H.MHD three-dimensional Casson fluid flow past a porous linearly stretching sheetRaptisA.PerdikisC.Viscoelastic flow by the presence of radiationSeddeekM. A.Effects of radiation and variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flowMaboodF.ImtiazM.AlsaediA.HayatT.Unsteady convective boundary layer flow of Maxwell fluid with nonlinear thermal radiation: a numerical studyHayatT.MuhammadT.AlsaediA.AlhuthaliM. S.Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiationFarooqM.KhanM. I.WaqasM.HayatT.AlsaediA.KhanM. I.MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effectsShehzadS. A.AbdullahZ.AlsaediA.AbbasiF. M.HayatT.Thermally radiative three-dimensional flow of Jeffrey nanofluid with internal heat generation and magnetic fieldCassonN.A flow equation for the pigment oil suspension of the printing ink typeAkbarN. S.ButtA. W.Ferro-magnetic effects for peristaltic flow of Cu-water nanofluid for different shapes of nano-size particlesUl HaqR.NadeemS.KhanZ. H.NoorN. F. M.Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubesTalleyL. D.PickardG. L.EmeryW. J.SwiftJ. H.Pastoriza-GallegoM.LugoL.LegidoJ.PiñeiroM.Thermal conductivity and viscosity measurements of ethylene glycol-based Al_{2}O_{3} nanofluidsAberoumandS.JafarimoghaddamA.Experimental study on synthesis, stability, thermal conductivity and viscosity of Cu–engine oil nanofluidBakerG. A.Graves-MorrisP.KuikenH. K.On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small