https://doi.org/10.1007/s12043-022-02337-8
Numerical simulations for optimised flow of second-grade
nanofluid due to rotating disk with nonlinear thermal radiation:
Chebyshev spectral collocation method analysis
SHAMI A M ALSALLAMI1, USMAN2, SAMI ULLAH KHAN3, ABUZAR GHAFFARI4, M IJAZ KHAN5,∗, M A EL-SHORBAGY6,7and M RIAZ KHAN8,9
1Department of Mathematical Sciences, College of Applied Science, Umm Al-Qura University, Makkah, Saudi Arabia
2Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, Department
of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083, People’s Republic of China
3Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan
4Department of Mathematics, University of Education, Lahore (Attock Campus 43600), Pakistan
5Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44000, Pakistan
6Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
7Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt
8LSEC and ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China
9School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
∗Corresponding author. E-mail: ijazfmg_khan@yahoo.com
MS received 6 October 2021; revised 4 December 2021; accepted 20 December 2021
Abstract. The optimised flow of nanofluids is quite essential to improve the thermal mechanism of various reacting materials. The entropy generation phenomenon is essential to avoid heat losses in thermal transport systems, heating processes and various engineering devices. In this theoretical analysis, the aspects of entropy generation is presented for time-independent second-grade nanomaterials for disk flow which shows rotating behaviours. The second-grade constitutive relations result in highly nonlinear differential equations. The effects of MHD, nonlinear radiation and chemical reaction are manifested in momentum, heat and concentration equations. Precise numerical treatment for a wide range of non-Newtonian fluid parameters was adopted to tackle the resulting similarity equations. The fluctuation against the heat transfer system, wall shear stress and mass changing phenomenon were also calculated and examined for various parametrical values. The interesting Chebyshev spectral collocation numerical simulations were performed to present the solution. This research finds that the entropy generation and Bejan number show the same trend for temperature and concentration difference parameters, whereas an opposite trend can be seen for the fluid and magnetic parameters. Also, entropy generation increases for diffusion parameter and Brinkman number, but Bejan number shows two trends.
Keywords. Radiative flow; viscoelastic nanofluid; Brownian motion and thermophoretic diffusion; Chebyshev spectral collocation method.
PACS Nos 65.40.gd; 88.20.td; 44
1. Introduction
With familiar thermal activities, the nanoparticles attracted the global attention of researchers in recent
times. The nanomaterials have noteworthy significance in the era of thermal engineering, medical and biomed- ical sciences, industrial processes, technological appli- cations and plasma physics. The most attractive appli-
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cations of nanofluids in recent times include controlling the cooling and heating of devices, solar performances, nuclear reactions, automated operation, chemotherapy, magnetic retention, astronomical and safety etc. With less than 100 nm size, the nanomaterials have extraor- dinary thermal efficiencies and conductivities which usually cannot be observed in base liquids. Because of their enhanced thermal efficiencies, the nanoparti- cles are assumed to be cheaper and effectual sources of energy. Choi and Eastman [1] presented the basic properties of nanofluids with the help of experiment in 1995. Buongiorno [2] successfully worked out the Brownian movement and thermophoretic mechanism of nanofluid in the measuring the stability of nano- materials. Hsiao [3] studied the thermal mechanism of magnetic nanoparticles subject to the slip effects.
Ahmedet al[4] studied the nanofluid flow confined by vertical cylinder by assessing the entropy generation.
The continuation of this study by Turkyilmazoglu [5]
dealt with thermal contribution of micropolar nanofluid associated with the heating and cooling devices. Rashidi et al [6] found the thermal prospective of nanofluid accounted by convectively heated surface. The enhance- ment of heat transfer using tangent hyperbolic nanofluid was numerically investigated by Ibrahim [7]. Waqaset al[8] addressed the thermal consequences of micropolar and Maxwell nanofluid model in porous space numeri- cally by employing the shooting scheme. Wakifet al[9]
endorsed the instability prospective in hybrid nanofluid with alumina and copper oxide nanoparticles over a rough surface. Reddyet al [10] performed a statistical measurement of dusty nanofluid in view of modified heat flux relations. The heat transfer mechanism by utilising the Walter’s B nanofluid objecting the role of microor- ganisms was studied by Khanet al [11]. Ali et al[12]
surveyed the natural convective flow of hybrid materi- als by using the relations of Caputo fractional derivative.
Chuet al[13] analytically attributed the effective con- tribution of buoyancy force to the magnetic nanofluid flow. Hassanet al[14] studied experimentally the heat transfer prospective by using the shear thinning hybrid nanoparticles.
The thermal optimisation associated with the dis- tinct thermal phenomenon enhances the efficiency and sustainability of various devices. On the basis of thermo- dynamics theories, a variety of heat enhancement mech- anisms and thermal extrusion systems are designed.
Based on the assumptions of first law of thermody- namic, the energy can be transmuted within distinct medium and systems without any loss. However, this theory fails to rationalise the irreversibilities or entropy generation. On the other hand, the second law of thermo- dynamics attributes the enrolment of available energy
and minimises the loss in energy and subsequently help to achieve improved thermal transportation pat- tern. The assessment of entropy generation (EG) is signified with the implementation of second theory of thermodynamics. The contribution of various thermal features like materials friction, porous space, viscous dissipation, radiative phenomenon, Joule heating, con- centration and temperature gradient play numerous roles in the entropy generation phenomenon. Bejan [15]
presented the pioneer work on enrolment of entropy gen- eration mechanism in boundary layer problems. Seyyedi et al[16] analysed the optimised flow of nanoparticles in an L-shaped enclosure with magnetic effects. Daset al[17] examined the contribution of entropy generation pattern in viscous material flow subject to the contri- bution of magnetic force and radiation mode. Riazet al[18] focussed on the contribution of entropy genera- tion in viscoelastic nanofluid flow in annular channel with flexible walls. Rashed [19] assessed the appli- cations of entropy generation in dissipative flow of nanofluid with heat generation and absorption conse- quences. The entropy generation in Casson nanofluid between stretching disks was studied analytically by Khan et al [20]. The entropy generation pattern in a porous channel occupied by nanoparticles was studied by Turkyilmazoglu [21]. Nayaket al [22] studied the entropy generation phenomenon in Crosser model by examining the shape factors and presented the dynamics of interfacial layers. Khanet al[23] studied the entropy generation in rotating flow of Williamson nanofluid confined by permeable walls of the microchannel. The stability and entropy generation features in radiative flow of the nanofluid was studied by Turkyilmazoglu [24,25]. Hayat et al [26] addressed the prospective of Jeffrey nanofluid with dominant aspects of entropy generation and thermal radiation. Some recent develop- ments regarding fluid flow using various assumptions and boundary conditions are listed in refs [27–37]. Ref- erences [38–43] signify some modern development in fluid flow in the presence of different mathematical tech- niques and flow assumptions.
Current investigation presents the numerical sim- ulations for the applications of entropy productivity phenomenon in viscoelastic nanofluid flow induced by rotating disk along with interesting features such as magnetic force, chemical reaction and porous space.
Additionally, the impact of thermal radiation in view of nonlinear relations is also inspected. Interesting numerical simulations based on the most fascinating Chebyshev spectral collocation scheme are performed as a novelty. The optimised features of flow pattern are graphically accessed. Various results in tabular forms are presented after studying various physical quantities.
Figure 1. The flow geometry.
2. Governing equations
Consider a time-independent 3D flow of viscous incom- pressible second-grade nanomaterial due to the rotating flow of a moving disk as displayed in figure1.
The velocity components u, v, w are chosen along the cylindrical coordinatesr, θ∗,z. The flow is assumed to be axisymmetric due to which the variation ofθ∗is ignored. The rotatory disk attained the uniform angu- lar velocityalongz-axis. The disk stretches radially with rates with the assistance of the velocityv = sr. The magnetic force with the magnetic strength B0 is applied perpendicular to the z-axis towards the circu- lating disk. The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negli- gible compared to the applied magnetic field. The heat and mass transfer characteristics in the presence of non- linear radiation and chemical reaction are studied. Tf
andCf denote the surface temperature and nanoparticle concentration whereas T∞ and C∞ denote the ambi- ent temperature and nanoparticle concentration. Based on the above assumptions, the governing equations are written as
∂u
∂r +u r +∂w
∂z =0, (1)
u∂u
∂r − v2
r +w∂u
∂z =νf∂2u
∂z2
+α1
ρf
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝ 3∂u
∂z
∂2u
∂r∂z +u ∂3u
∂r∂z2 +∂w
∂z
∂2u
∂z2 +w∂3u
∂z3 +∂v
∂r
∂2v
∂z2 +∂v
∂z
∂2v
∂r∂z
−v r
∂2v
∂z2 −1 r
∂u
∂z 2
+2∂u
∂r
∂2u
∂z2
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
−σfB02 ρf
u, (2)
u∂v
∂r +uv
r +w∂v
∂z
=νf∂2v
∂z2 + α1
ρf
⎛
⎜⎜
⎝ u ∂3v
∂r∂z2 +w∂3v
∂z3 −1 r
∂u
∂z
∂v
∂z +u
r
∂2v
∂z2 −2∂v
∂z
∂2u
∂r∂z
⎞
⎟⎟
⎠
−σfB02
ρf v, (3)
u∂T
∂r +w∂T
∂z
= 1 (ρcp)f
kf +16σ∗T3 3k∗
∂2T
∂z2 +τ
DB∂T
∂z
∂C
∂z + DT
T∞ ∂T
∂z 2
, (4)
u∂C
∂r +w∂C
∂z = DT
T∞ ∂2T
∂z2
+DB
∂2C
∂z2
−kr(C−C∞), (5) with boundary conditions
u=sr, v=r, w=0, T =Tf, C =Cf atz =0
u→0, v→0, T →T∞, C →C∞ atz → ∞
⎫⎪
⎬
⎪⎭. (6) Note thatνf signifies the kinematic viscosity,u, v, ware the velocity components,ρf denotes density,r, θ∗,zare the cylindrical coordinates,α1is the second-grade fluid material,σf is the electrical conductivity,is the angu- lar velocity, B0 is the magnetic field strength, T is the temperature,cp is the specific heat,C is the concentra- tion,kf is the thermal conductivity,DBis the Brownian motion coefficient,σ∗is the Stefan–Boltzmann coeffi- cient,kr is the chemical reaction coefficient,k∗ is the mean absorption coefficient,τ is the ratio of the heat capacities,DT is the thermophoresis coefficient,T∞is the ambient temperature,Cf is the surface nanoparticle concentration,C∞is the ambient concentration,sis the stretching rate andTf is the surface temperature. Note that the subscript f stands for the nanofluid.
3. Similarity transformation
Similarity transformation can be chosen as η=z
νf , u =rF(η), v =rG(η), w=
νfH(η)T =T∞+
Tf −T∞ θ(η), C =C∞+
Cf −C∞ ϕ(η),
⎫⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎭ (7)
Introducing (7) into (1)–(6), we may achieve the follow- ing ODEs:
H+2F =0, (8)
F−H F−F2+G2 +β
2F2+F F+FH+G2
−M F =0, (9) G−H G−2G F
+β
−3FG+2F G+H G
−M G =0, (10)
1+4
3Rd(1+(θw−1) θ)3
θ
−PrHθ+Pr
Nbθϕ+Ntθ2
=0, (11) ϕ−Sc Hϕ+ Nt
Nbθ−Scγ ϕ=0, (12) with
F(0)=ω, G(0)=1, H(0)=0, θ(0)=1, ϕ(0)=1,F(∞)→0, G(∞)→0,
θ(∞)→0ϕ(∞)→0.
⎫⎪
⎬
⎪⎭ (13) The dimensionless parameters are
β = α1
ρfνf , M = σfB02
ρf , ω= s
, Pr= νf
αf , Nt=τDT(Tf −T∞)
νf , θw=Tf
T∞, Rd=4σ∗T∞3 3kfk∗ , Nb = τDB(Cf −C∞)
νf , Sc= νf
DB, γ = kr .
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎭ (14) Here,ηindicates the dimensionless similarity variable, F,G,H, θ, ϕ denote the dimensionless radial veloc- ity, azimuthal velocity, axial velocity, dimensionless temperature, dimensionless nanoparticle concentration, respectively,βis the second-grade fluid parameter,Mis the magnetic parameter, Rdis the radiation parameter, θw is the surface heating or ratio parameter, Pr is the Prandtl number, Nb is the Brownian motion parameter, Nt is the thermophoretic parameter,Sc is the Schmidt number,γ is the reaction parameter andωis the rotation parameter.
4. Physical variables
4.1 Surface drag force
The shear surface force in radial and azimuthal direc- tions is
CFr = τr z
ρf (r)2, (15)
where τr z =
μf ∂u
∂z +α1
u ∂2u
∂r∂z +w∂2u
∂z2 +∂v
∂r
∂v
∂z
−v r
∂v
∂z +2∂u
∂r
∂u
∂z
z=0, (16)
CGθ∗ = τθ∗z
ρf (r)2. (17)
Here τθ∗z =
μf∂v
∂z +α1
u ∂2v
∂r∂z +w∂2v
∂z2 +u
r
∂v
∂z − ∂v
∂z
∂w
∂z
z=0
. (18)
The non-dimensional form is ReeCFr
=
F(η)+β
3F(η)F(η)+H F(η)
η=0, (19) ReeCθ∗z
=
G(η)+β
4F(η)G(η)+H G(η)
η=0. (20) 4.2 Local Nusselt number
Let us define N ur = r qw
kf(Tf −T∞), (21) where
qw = −kf
∂T
∂z
z=0
−
1+Rd
1+(θw−1) θ3
. (22)
The non-dimensional form leads to N ur
√Ree
= −
1+Rd
1+(θw−1) θ3
θ(η)
η=0. (23) 4.3 Local Sherwood number
The local Sherwood number is Shr = r qm
DB(Cf −C∞), (24) where
qm = −DB
∂C
∂z
z=0
. (25)
The non-dimensional form leads to
Shr
√Ree
= −ϕ(η)η=0, (26) where√
Ree=r2/νf denotes the Reynolds number, CFr is the skin friction in the radial direction,Shris the Sherwood number,τr zis the shear stress along the radial
direction,qm is the mass flux,CGθ∗ is the skin friction in the azimuthal direction,qwis the heat flux,τθ∗zis the shear stress along azimuthal direction and N ur is the Nusselt number,
5. Entropy modelling
The entropy generation is expressed as the sum of irre- versibilities, i.e.,
SG = kf
T∞2
1+4σ∗T∞3 3kfk∗
∂T
∂z 2
Thermal irreversibility
+ R∗DB
T∞ ∂T
∂z
∂C
∂z
Mass transfer irreversibility
+ σf
T∞B02(u2+v2)
Joule heating irreversibility
+2μf
T∞ ∂u
∂z 2
+ ∂v
∂z 2
+2α1
T∞
⎡
⎢⎢
⎢⎣ u
∂u
∂z
∂2u
∂r∂z + ∂v
∂z
∂2v
∂r∂z
+w ∂u
∂z
∂2u
∂r∂z +∂v
∂z
∂2v
∂r∂z
+∂v
∂z u
r
∂v
∂z −v r
∂u
∂z
+
2∂u
∂z ∂u
∂z 2
+∂w
∂z ∂v
∂z 2
⎤
⎥⎥
⎥⎦
Viscous dissipation irreversibility
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎭
. (27)
The non-dimensional form will lead to NG=δ1
1+4
3Rd
(1+(θw−1) θ)3θ2
+χθϕ +χδ2
δ1ϕ2+M Br
F2+G2 +2Br
F2+G2
+2βBr 3F
F2+G2 +G
F G−FG +H
F F+GG
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎭ ,
(28) where
δ1= Tf −T∞
T∞ , χ = R∗D
Cf −C∞
kf ,
δ2= Cf −C∞
C∞ , Br = μfr22 kf
Tf −T∞, NG = T∞SGνf
kf
Tf −T∞
, (29)
where R∗ denotes the gas constant, SG is the entropy generation, Br is the Brinkman number, δ1 is the dimensionless temperature difference,NGis the entropy generation rate, χ is the diffusion parameter andδ2 is the dimensionless concentration difference.
The expression for Bejan number is
Be = δ1
1+43Rd
(1+(θw−1) θ)3θ2
+χθϕ+χδδ21ϕ2
NG , (30)
whereBestands for Bejan number.
6. Chebyshev spectral collocation method (CSCM) Chebyshev spectral collocation method (CSCM) is utilised to obtain results numerically. To find the solution of differential equations (eqs (8)–(12)), Newton lineari- sation scheme is adopted for attaining the linear set of
system. For(i+1)th iterates, we replaced all dependent variablesF,G, θ andϕby the expressions.
Fi+1 =Fi +δFi
Gi+1=Gi +δGi
θi+1 =θi +δθi
ϕi+1 =ϕi +δϕi
⎫⎪
⎬
⎪⎭, (31) whereδFi, δGi, δθiandδϕirepresent a minimal change inFietc. Applying eq. (31) in (8)–(12) and dropping the quadratic and higher-order terms inFi, δGi, δθiandδϕi, we obtain
Hi =D−1(−2Fi) , (32) a1,iδF+a2,iδF+a3,iδF+a4,iδF = R1,i, (33) b1,iδG+b2,iδG+b3,iδG+b4,iδG= R2,i, (34) c1,iδθ+c2,iδθ+c3,iδθ = R3,i, (35) d1,iδϕ+d2,iδϕ+d3,iδϕ =R4,i, (36) where
a1,i =βHi, a2,i =1+βFi, a3,i = −Hi +4βFi,
a4,i = −2Fi +βFi−M, (37) b1,i =βHi, b2,i =1+2βFi, b3,i = −Hi −3βFi,
b4,i = −2Fi −M, (38) c1,i =1+4
3Rd((1+(θw−1) θi))3, (39) c2,i =8Rd(θw−1)3θ2θi+16Rd(θw−1)2θθi
+8Rd(θw−1) θi−PrHi +Pr
Nbϕi +2Ntθi
(40)
c3,i =8Rd(θw−1)3θiθi2+4Rd(θw−1)3θiθi2
+8Rd(θw−1)2θi2+8Rd(θw−1)2θiθi
+4Rd(θw−1) θi, (41) d1,i =1, d2,i =Sc Hi, d3,i = −Scγ, (42)
R1,i = −Fi+HiFi+Fi2−G2i
−β
2Fi2+FiFi2+HiFi+Gi2
+M Fi, (43) R2,i = −Gi+HiGi +2FiGi
−β
−3FiGi+2FiGi+HiGi
+M Gi, (44) R3,i = −
1+4
3Rd(1+(θw−1) θi)3
θi
−4Rd(θw−1)
(1+(θw−1) θi)2 θi2 +PrHiθi−Pr
Nbθiϕi+Ntθi2
, (45) R4,i = −Nt
Nbθi, (46)
with differentiation matrix inverseD−1. The flow conditions are:
δFi =ω−Fi, δGi =1−Gi, Hi =0, δθi =1−θi, δϕi =1−ϕi,atη=0 δFi →0−Fi, δGi →0−Gi, δθi →0−θi, δϕi →0−ϕθi,whenη→ ∞
$
. (47)
Now CSCM is adopted for capturing simulations for the above system. The spectral method can be used over the finite domain[−1,1], so that the domain [0,∞] is converted to finite domain[0,L]with the transformation ofηj = 2L −1.
7. Results and discussion
The upshots of the emerging parameters like second- grade fluid parameterβ, magnetic parameterM, rotation constant ω, Prandtl Pr, thermophoresis varaible Nt, Brownian motion Nb, surface heating parameter θw, radiation parameter Rd, Schmidt numberSc, chemical reactionγ, temperature difference parameterδ1, diffu- sion parameter χ, concentration difference parameter
Figure 2. Impact ofMandβ onF(η).
Figure 3. Impact ofM andβonG(η).
δ2 and Brinkman number Br on profiles of the radial velocity F(η), azimuthal velocity G(η), temperature θ (η), entropy generation NG(η), the concentration of
nanoparticleϕ (η)and Bejan numberBe(η)are graph- ically analysed. The Physical quantities such as surface drag forces (CFr,CGθ∗), heat transfer rate N ur and mass transfer rateShr are also calculated.
The impacts of magnetic parameter M and second- grade fluid parameterβon the non-dimensional distribu- tions of radialF(η)and azimuthalG(η)velocities and temperatureθ (η)are shown in figures2–4. Both veloc- ity profiles increase alongηfor various values ofβ and decrease for M. The temperature profile, on the other hand goes in the opposite direction. The second-grade fluid parameter causes a decrease in fluid viscosity as it rises, allowing fluid velocity components to increase and temperature components to drop. The larger estimation of magnetic impact find out the role of Lorentz force in
Figure 4. Impact ofMandβ onθ(η).
Figure 5. Impact ofRdandθwonθ(η).
an electrically conducting fluid if applied perpendicu- larly. The Lorentz force causes flow resistance, due to which the fluid velocity components decrease. When the flow is obstructed, more heat energy is generated, which helps fluid temperature to increase.
Figure 5 shows the impacts of θw and Rd on the dimensionless temperature distribution θ (η). As the radiation parameter increases, the mean absorption parameter declines, delivering more heat to the flow causing the fluid heating rate to decay. It should be observed thatθw =1 stands for linear radiation, while θw >1 stands for nonlinear radiation. The changes are assigned to enhance the heat transfer rate at the bound- ary of the ambient walls. The enhancement inθwleads to an extremely large thermal state of the temperature, which allows the temperature to escalate. Also, nonlin- ear radiation has a higher rate of temperature escalation than linear radiation.
The contribution ofNbandNt onθ (η)andϕ (η)pro- files are presented in figures6and7. The escalation in temperature is noticed for higher values of both parame- ters, whereas the concentration shows two trends. When
Figure 6. Impact ofNtandNbonθ(η).
Figure 7. Impact ofNt andNbonϕ(η).
the Brownian motion parameter enhances, the nanopar- ticles collide with one another, producing a resistive force that raises the temperature but lowers the con- centration. Also, thermophoresis enhances the number of nanoparticles due to which nanoparticles move from hot to cold surface. As a result, both temperature and concentration increase. The effects of the Schmidt num- berScand chemical reactionγ onϕ (η)are displayed in figure8. The concentration profile decays as both param- eters rise. Higher Schmidt number possesses a weaker mass diffusivity which lowers the nanoparticle concen- tration. The parameter Sc shows a greater decrease in the concentration profile.
The impacts of M andβ on the dimensionless pro- files of NG(η) and Be are captured in figures 9 and 10. For higher values of both parameters, an inverse trend between two profiles is detected. Large magnetic parameter results in greater resistance to the fluid parti- cles, which increases the collision of the molecules and causes escalation of disorder within the thermal system.
As a result,NG(η)increases whileBereduces. Higher estimations ofβlead to an escalation in entropy genera- tion and a reduction in the Bejan number. The escalation
Figure 8. Impact ofScandγ onϕ(η).
Figure 9. Impact ofM andβonNG(η).
in the NG(η) profile due to β is more extensive than due to M. Also, the reduction in the Be profile due to M is more noticeable than due to β. Figures 11 and 12 are sketched to discuss how Brinkman number and diffusion parameter affect the nature of entropy gener- ation and Bejan number. The NG(η) profile enhances for higher values of both parameters, but theBeprofile predicts two diverse behaviours. The specified change in diffusion constant result in a rise in diffusivity of the fluid particles; thus, more disorder is produced, which enhances entropy generation. Also, as the diffu- sion parameter improves, the irreversibility of heat and mass transfer becomes more prominent, resulting in a higher Bejan number. With higher Brinkman number, the viscous force increases the system’s disorder, help- ing an escalation in the entropy generation but the Bejan number decreases.
The physical determination ofδ1, δ2 on NG(η) and Beand dimensionless distributions are shown in figures 13 and14. Both profiles appear to increase for higher values of both parameters. As these parameters rise, heat and mass transfer irreversibility takes precedence over fluid friction irreversibility, resulting in an increase in
Figure 10. Impact ofMandβonBe(η).
Figure 11. Impact ofχandBronNG(η).
Figure 12. Impact ofχandBronBe(η).
both profiles. Also, the increase in both profiles due to δ1is larger than due toδ2.
The numerical values of the skin friction coefficients, local Nusselt number and local Sherwood number are calculated in tables 1–3 for various ranges of the involved parameters whenβ = 0.0 andβ = 0.5. The expressed numerical values from table 1 convey that friction wall coefficients decay for the variations ofM
Figure 13. Impact ofδ1andδ2onNG(η).
Figure 14. Impact ofδ1andδ2onBe(η).
Table 1. Numerical change in skin wall friction coefficients when Rd = 0.5, θw = 1.3, Pr = 6.2, Nb = γ = 0.3, Nt =Sc=0.2,δ1=δ2=0.5 andBr=χ=1.0.
√ReeCFr
√ReeCGθ∗
M ω β=0.0 β =0.5 β =0.0 β=0.5 0.5 0.1 0.2911 0.4878 −0.9119 −0.9584
0.6 0.2678 0.4677 −0.9567 −1.0141
0.8 0.2273 0.4350 −1.0438 −1.1216
1.0 0.1933 0.4100 −1.1271 −1.2235
0.5 0.05 0.3452 0.5271 −0.8715 −0.8276 0.15 0.2352 0.4417 −0.9512 −1.0953 0.20 0.1777 0.3888 −0.9896 −1.2377 0.25 0.1185 0.3290 −1.0270 −1.3851 0.30 0.0576 0.2625 −1.0635 −1.5370
and ω. Table 2 explains that the local Nusselt num- ber increases for the parameters Rd and θw, whereas it plummets for the parametersM,Pr,NtandNb. Also, table 3 signifies that the local Sherwood number gets enhanced for the variations of the parameters M, Sc, Nt,Nb andγ.
Table 2. Numerical values of the local Nusselt number when ω=δ1=δ2=0.5, γ =0.3,Sc=0.2,Br=χ =1.0.
N ur/√ Ree
M Rd θw Pr Nt Nb β =0.0 β=0.5 0.5 0.1 1.3 6.2 0.2 0.3 0.3278 0.3530
0.6 0.3213 0.3480
0.8 0.3093 0.3385
1.0 0.2984 0.3299
0.5 0.2 0.4262 0.4591
0.6 0.7697 0.8332
1.0 0.0453 1.1361
0.1 1.5 0.3723 0.4007
1.7 0.4269 0.4592
1.9 0.4914 0.5286
1.3 7.0 0.3059 0.3293
8.0 0.2785 0.2998
10.0 0.2267 0.2442
6.2 0.4 0.2255 0.2436
0.6 0.1612 0.1742
0.8 0.1200 0.1794
0.2 0.5 0.1646 0.1796 0.7 0.0793 0.0876 0.9 0.0369 0.0413 1.1 0.0168 0.0189
Table 3. Numerical values of the local Sherwood number whenω=Rd =0.5, θw =1.3, δ1=δ2=0.5,Br=χ= 1.0.
Shr/√ Ree
M Sc Nt Nb γ β =0.0 β=0.5
0.5 0.2 0.2 0.3 0.3 0.1940 0.1755
0.6 0.1990 0.1791
0.8 0.2086 0.1862
1.0 0.2174 0.1930
0.5 0.4 0.3295 0.3192
0.8 0.5436 0.5441
2.0 0.9739 0.9889
0.2 0.4 0.1428 0.1024
0.6 0.1550 0.1106
0.8 0.2183 0.1391
0.2 0.5 0.2719 0.2631
0.7 0.3029 0.2985
0.9 0.3183 0.3164
0.3 0.6 0.2990 0.2829
0.9 0.3882 0.3742 1.2 0.4659 0.4537 1.5 0.5350 0.5243
An entropy generation analysis is carried out in a time-independent flow of second-grade nanofluid over the rotatory stretchable disk. The nonlinear PDEs are used to model flow, heat and mass dynamics, which are then transmuted into ODEs with similarity variables.
The numerical solution allowed us to investigate the pri- mary physical characteristics flow features on boundary layer distributions. The following are the main findings of this investigation:
• The radial and azimuthal velocities follow opposing trends forM andβ.
• The temperature enhances for Rd, θw, Nt, Nb and M, whereas it falls forβ.
• Nanoparticle concentration increases for Nt, but plummets forNb, Scandγ.
• Entropy generation increases for M, β, χ, Br, δ1
andδ2.
• Bejan number increases forχ, δ1 andδ2, while it loweres forM, βandBr.
Acknowledgements
The authors would like to thank the Deanship of Scien- tific Research at Umm Al-Quara University for support- ing this work under Grant No. 22UQU4290491DSR02.
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