# Mixed Convection Squeezing Flow of Nanofluids in a Rotating Channel with Thermal Radiation

Authors: Wankui Bu; Hui Xu; Ilyas Khan; Sheikh Irfan Ullah Khan; Anwar Zeb
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present study, 3-dimensional squeezing movement in a circling conduit under the stimulus effective Prandtl number with the aid of thermal radiation is taken into account. Water and ethylene glycol are the base fluids along with gamma-alumina nanoparticles. The coupled nonlinear system of PDEs is transformed into a system of ODEs with the support of some appropriate resemblance alterations. Then, the explanation was obtained numerically by the Runge–Kutta–Fehlberg (RKF) method. The emerging parameters such as quotient of the electric magnetic field to viscous forces (M), Prandtl number (Pr), and Reynolds number (Re), along with physical parameters such as the Nusselt number and skin friction coefficient, will be integrated graphically. The Prandtl number is important for regulating the momentum and thermal boundary layers. As a result, the effect of the effective Prandtl number on the nanoboundary layer and laminar incompressible flow of γAl2O3−H2O and γAl2O3−C2H6O2 nanoparticles is considered. The impact of the radiation parameter (Rd) favors the temperature distribution. Furthermore, the thermal conductance enriches with the enhancement of solid volume fraction.


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## Body

## 1. Introduction

Mixed convection occurs when the free or natural convection (induced by buoyancy forces) and the forced convection (induced by external means) act together in a heat transfer phenomenon. An internal flow, such as a pipe flow, is surrounded by solid walls that can limit the development of its boundary layer. External flows, on the contrary, occur over bodies submerged in an unbounded fluid, allowing the flow boundary layer to freely develop in one direction. Flows over air foils, ship hulls, turbine blades, and other structures are examples. It carries great importance due to its application in nuclear reactors, electronic cooling, and solar energy systems. For heat management and control of such systems, natural convection alone is sometimes insufficient; hence, forced convection is required. Convection in a cavity is also of great importance which has two aspects under consideration, one which assists forced flow motion in the channel with natural convection in the cavity and the other that opposes it. It has a significant impact in the field of heat transmission and has a wide range of applications, including furnaces, ovens, and combustion chambers, as well as incinerators. However, the mixed convection is causing concern, particularly among electronic system manufacturers. In fact, as a result of reaching more and better hardware potential, a greater and more effective cooling system for boards and components was required. Sharif [1] studied laminar mixed convection in a shallow inclined cavity with the hot upper wall and cool lower wall. Mahmoodi [2, 3] investigated mixed convection flow of water-Al2O3 nanofluid in a rectangular cavity. He observed that due to the presence of nanoparticles in the base fluid, heat transfer rate increases for all ranges of the Richardson number under consideration. In recent times, Khan et al. [4–6] examined the MHD convection squeezing flow of nanofluids under the influence of mixed convection.The fluid particles must deform under the influence of squeezing flow based on the squeezing rate, for example, the flow of blood platelets in blood vessels with a diameter equal to or slightly bigger than the size of platelets. In such a case, when blood cells flow between those narrow gaps, they get squeezed and hence deformed. Chen and Skalak studied Stoke’s flow having spheroidal particles in a cylindrical tube in effect of both wide and narrow gaps [7]. Squeezing a sample between parallel plates has long been used to characterize the rheological behavior of soft, clear viscous materials, and small-amplitude oscillatory squeezing has recently been proposed as a method for determining the linear viscoelastic properties of molten polymers and suspensions. The biggest and most significant advantage of a squeeze flow rheometer over a rotational device is its simplicity as it has no air bearing and is significantly less expensive and simple to use. Squeezing flow is not an isometric flow; hence, devices that use this deformation for non-Newtonian fluids can only be used qualitatively unless a constitutive model is given. Rotating channel has great applications in geophysics, and astrophysics also has great importance in turbo machines, vortex-type MHD power generators, nuclear reactors using the liquid metal coolant, material processing, etc.Due to various applications in practical situations, the study of fluid flow on the surface of a rotating disc has attracted a lot of interest from researchers all over the world. Fluid flow over the surface of a spinning disc is frequently used for lubrication reasons in electric power generation systems, rotating machinery, corotating turbines, chemical processes, and computer storage in the fields of aerodynamics engineering and geothermal technology.MHD flow analysis is important because the magnetic field acting on the viscid current of an electrically transmitting liquid is used in a variety of industrial techniques, including the processing of magnetized materials, crude oil refining, MHD electric propagation, glass fabrication, paper production, and geophysical sciences, to name a few. The MHD flow issue going through an enhanced upright plate has a wide range of applications, including filtration, drying porous materials in textile manufacturing, and chemical impregnation of porous materials.A fluid which contains small nanometer-sized particles in a suspension called nanoparticles is termed as a nanofluid. These particles are usually made of metals, oxides, carbides, or carbon nanotubes. Commonly used nanofluids are water, ethylene glycol, and oil. Choi of the Argonne National Laboratory [8] was the first who proposed the term nanofluids in 1995 in the USA. The presence of these nanoparticles in nanofluids increases the thermal properties of the fluids. They have many applications such as from houses such as domestic refrigerator, chiller, heat exchanger in grinding, boiling, and machining fuel gas temperature reduction to large scale such as microelectronic fuel cells, hybrid powered engines, thermal management engine cooling, pharmaceutical processes, ultrasonic field display, and solar collectors (because of their optical properties). Routbort reviewed a project in 2008 that employed nanofluids for industrial cooling that could result in great energy savings and resulting emission reductions. Kim et al. [9, 10] at the Nuclear Science and Engineering Department of the Massachusetts Institute of Technology (MIT) performed a study to assess the feasibility of nanofluids in nuclear applications by improving the performance of any water-cooled nuclear system. Kandaswamy et al. [11] portrayed the impact of convection near its density maximum with partially active vertical walls. In addition to enhance the thermal conductivity, nanofluids also have many vast ranges of applications in various fields including energy, mechanical, and biomedical fields, reducing pollution, heating buildings, space and defense, transportation, etc. Modern researchers synthesize nanofluids by two famous ways that are two-step and one-step methods. Two-step method is a very renowned method in which nanoparticles, nanotubes, or nanofibers are first synthesized as a powder in a dry form, and then it will further be immersed in the fluid with the help of intensive magnetic force agitation, ultrasonic agitation, high-shear mixing, homogenizing, and ball milling. As it is a convenient and widely used method, it still lacks the ability to produce the nanoparticles of greater stability, and due to this loophole, the researchers’ interests dragged towards a more advanced method that is the one-step method; Eastman developed a one-step physical vapor condensation method to prepare Cu/ethylene glycol nanofluids. The one factor of this process is that it avoids the accumulation of nanoparticles, and hence, stability increased [12–19].Sketching in mind all the above-discussed detailed efforts and extensive work, we will further go through and extend the idea to three-dimensional mixed convection squeezing flow of nanofluids with a nonlinear term of temperature induced over here. The required gained model will be solved iteratively with the RKF (Runge–Kutta–Fehlberg) scheme. The rate of change of displacement and the temperature outcomes will be consequent by alumina and gamma-alumina nanoparticles with different base fluids such asH2O and C2H6O2 and also depicted graphically. In spite of these two outcomes, we will be focusing our devotion to evaluate wide-ranging consequences of embedding and physical parameter realistically.


## 2. Mathematical Formulation

Contemplate a time-dependent 3D squeezing flow of Newtonian base fluids, water and ethylene glycol, which are electrically conducting and incompressible, adjourned by the alumina and gamma-alumina nanoparticles in a rotating channel, in between two plates, set in parallel position. This position is in such an order that the bottom plate is stretched with the velocityUw=ax/1−bt at y=0 with the constant temperature Tw being maintained, while the higher plate is at the fluctuating distance ht=ν1−bt/a, where the temperature is Th. Vh=dh/dt represents the velocity of squeezing flow in the y-negative direction. The base fluids and the walls are rotating in counterclockwise in the constructive y-axis with the transverse component of velocity Ω⟶=ωj/1−bt. Bt=B0/1−bt signifies the variable transverse magnetic field executed perpendicularly. The suction of nanofluid occurred at the lower plate at y=0. The thermophysical possessions of nanofluids are taken from Khan et al. [3]. The geometry of the flow prototype is given in Figure 1.Figure 1
Geometry of the problem.The precise model of mixed convection squeezing flow of electrically conducting incompressible nanofluids, with a nonlinear temperature term, in the presence of applied magnetic field is distinct by the succeeding set of PDEs [4]:(1)∂u∂x+∂u∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y+2ω1−btw=−1ρnf∂p∂x+νnf∂2u∂x2+∂2u∂y2−σnfB02ρnf1−btu+gρβnfρnfT−T0,(3)∂v∂t+u∂v∂x+v∂v∂y=−1ρnf∂p∂y+νnf∂2v∂x2+∂2v∂y2,(4)∂u∂t+u∂u∂x+v∂u∂y−2ω1−btu=νnf∂2w∂x2+∂2w∂y2−σnfB02ρnf1−btw,(5)∂T∂t+u∂T∂x+v∂T∂y=−knfρCpnf∂2T∂x2+∂2T∂y2+16σ∗T∞33k∗ρcpnf∂2T∂y2.The apposite boundary surroundings for the prototype are(6)u=Uw=ax1−bt,v=Vw=−V01−bt,w=0,T=Tw,aty=0,(7)u=0,v=Vh=−a2να1−at1/2,w=0,T=Th,aty=ht.In equations (1) up to (7), u,v, and w are the rate of change components along x-, y-, and z-directions correspondingly, B0 is the magnetic field, p denotes the pressure, b is the distinctive limit of time reciprocal length t and bt<1, σnf is the effective electrically conducting parameter, magnitude of acceleration is g, T is the liquid’s temperature, νnf is the effective kinematic viscosity of the nanofluid, ρnf is the effective density of nanofluids, knf is the effective thermal conductivity of nanofluids, mean absorption coefficient is k∗,σ∗ is the SB (Stefan–Boltzmann) constant, and Cpnf is the heat capacity of nanofluids. For the present flow prototype, use the following effective models for dynamic viscosity, density, electrical conductivity, thermal conductivity, and heat capacitance [3].(8)ρnfρf=1−ϕ+ϕρsρf,ρCpnfρCpf=1−ϕ+ϕρCpsρCpf,where ϕ is termed as the solid size segment of the nanoparticles.The active viscosity of the nanoparticles is demarcated as(9)μnfμf=123ϕ2+7.3ϕ+1forγAl2O3−H2O,μnfμf=306ϕ2−0.19ϕ+1forγAl2O3−C2H6O2.The effective thermal conductivity of the nanoparticles is taken as(10)knfkf=4.97ϕ2−2.72ϕ+1,forγAl2O3−H2O,knfkf=28.905ϕ2−2.8273ϕ+1forγAl2O3−C2H6O2.The effective Prandtl number of the nanoparticles is given by(11)PrnfPrf=82.1ϕ2+3.9ϕ+1forγAl2O3−H2O,PrnfPrf=254.3ϕ2−3ϕ+1forγAl2O3−C2H6O2.Furthermore, the physical and the thermal properties of base fluids such asH2O and C2H6O2 along with alumina nanoparticles are given in Table 1.Table 1
Properties of nanoparticles and base fluids.
Modelρkgm−3Cpkg−1K−1kWm−1K−1σPrH2O997.141790.6130.0056.96C2H6O21116.623820.2495.5∗10−6204Al2O33970765403.5∗107—Here, the comparative alterations are defined in the directive to reduce the following boundary layer equation:(12)η=aνf1−bty,u=ax1−btf′η,ν=−aν1−bt1/2fη,θ=T−ThTw−Th.The ordinary differential equations in the nondimensionalized form are gained by dropping the governing boundary layer equations into the simplest form as follows:γAl2O3−H2O model:(13)fivη−B1f′ηf″η−fηf‴η+β23f″η+ηf‴η+2Rg′η−B2M2f″η+B3λθ′η=0,g″η−B1fηg′η−f′ηgη+βgη+η2g′η+2Rf′η−B2M2g′η=0,1+Rd4.97ϕ2+2.72ϕ+1θ″η+PrnfB4fηθ′η−β2ηθ′η=0.γAl2O3−C2H6O2 model:(14)fivη−C13f′ηf″η−fηf‴η+β23f″η+ηf‴η+2Rg′η−C2M2f″η+C3λθ′η=0,g″η−C1fηg′η−f′ηgη+βgη+η2g′η+2Rf′η−C2M2g′η=0,1+Rd28.906ϕ2+2.8272ϕ+1θ″η+PrnfC4fηθ′η−β2ηθ′η=0.The flow conditions at the boundaries are(15)fη=S,f′η=1,gη=0,θη=1,atη=0,fη=β2,f′η=0,gη=0,θη=0,atη=1,where(16)B1=1−ϕ+ϕρs/ρf123ϕ2+7.3ϕ+1,B2=σs+2σf+2ϕσs−σf/σs+2σf−ϕσs−σf123ϕ2+7.3ϕ+1,B3=1−ϕ+ϕρβs/ρβf123ϕ2+7.3ϕ+1,B4=1−ϕ+ϕρCps/ρCpf4.97ϕ2+2.72ϕ+1,C1=1−ϕ+ϕρs/ρf306ϕ2−0.19ϕ+1,C2=σs+2σf+2ϕσs−σf/σs+2σf−ϕσs−σf306ϕ2−0.19ϕ+1,C3=1−ϕ+ϕρβs/ρβf306ϕ2−0.19ϕ+1,C4=1−ϕ+ϕρCps/ρCpf28.906ϕ2+2.8272ϕ+1.The nondimensional limitations entrenched in the flow prototype are specified as(17)β=ba,M2=σβ02aρf,R=ωb,Re=xUwνf,Gr=gβfT0x3νf2,λ=GrRe2,Pr=μCpfkf,S=V0ah.The measures such as rate of heat transfer and skin friction coefficient are specified as(18)Cf=−2μnfρfUw02∂u∂yy=0.Using this, we have the skin friction coefficient as follows.ForγAl2O3−H2O,(19)Cfy=0=ReCf∗=f″0123ϕ2+7.3ϕ+11−ϕ+ϕρs/ρf=f″0B1.ForγAl2O3−C2H6O2,(20)Cfy=0=ReCf∗=f″0306ϕ2−0.19ϕ+11−ϕ+ϕρs/ρf=f″0C1.By using the comparison variables and active thermophysical properties, we attain the subsequent formula of the Nusselt number as(21)Nu=xqwkfTw−Th.Using this, we have the required form of the Nusselt number as follows.ForγAl2O3−H2O:At the lower plate,(22)Nuy=0=1−at1.5Nu∗=−4.97ϕ2+2.72ϕ+1θ′0.At the upper plate,(23)Nuy=ht=1−at1.5Nu∗=−4.97ϕ2+2.72ϕ+1θ′1.ForγAl2O3−C2H6O2:At the lower plate,(24)Nuy=0=1−at1.5Nu∗=−28.905ϕ2+2.8273ϕ+1θ′0.At the upper plate,(25)Nuy=ht=1−at1.5Nu∗=−28.905ϕ2+2.8273ϕ+1θ′1.


## 3. Solution of the Problem

For explanation persistence of the existing flow prototype, we utilized the iterative scheme of RK along with the shooting method. In order to summarize the flow mode, we take the subsequent conventions:(26)c1=f,c2=f′,c3=f″,c4=f‴,c5=g,c6=g′,c7=θ,c8=θ′.Initially, we rewrite theγAl2O3−H2O model as(27)fivη=B1f′ηf″η−fηf‴η+β23f″η+ηf‴η+2Rg′η+B2M2f″η+B3λθ′η,g″η=−B1fηg′η−f′ηgη+βgη+η2g′η+2Rf′η+B2M2g′η,θ″η=−1+Rd4.97ϕ2+2.72ϕ+1−1PrnfB4fηθ′η−β2ηθ′η.Using the conditions in (26), we have the subsequent structure of the first-order ODE:(28)c1′c2′c3′c4′c5′c6′c7′c8′=c2c3c4B1c2c3−c1c4+β23c3+ηc4+2Rc6+B2M2c3+B3λc8c6−B1c1c6−c2c5+βc5+η2c6+2Rc2+B2M2c6c8−1+Rd4.97ϕ2+2.72ϕ+1−1PrnfB4c1c8−β2ηc8.The associated boundary conditions are(29)c1c2c3c4c5c6c7c8=S1β¯20010.Equations (28) and (29) can be computed with the aid of Mathematica 10.0.In a similar fashion, we can solve theγAl2O3−C2H6O2 nanofluid model.


## 4. Results and Discussion

In this part, the results are assessed with the help of graphic aids and tabularized ideals. These results are evaluated by different flow structures.This subset comprises the variations in the rate of change fieldsfη,f′η and transverse component of velocity gη due to fluctuating suction limit S, quotient of V0 to bh, squeeze parameter β, quotient of α to b (where b is the characteristic parameter), rotation parameter R, quotient of ω and a, and magnetic parameter M. The numerical results are depicted for γAl2O3−H2O and γAl2O3−C2H6O2 nanoparticles. It is imperious to remark that defining the factor S>0 represents the movement of the upper plate towards the lower one, and the factor S<0 signifies the measure of the lower plate away from the upper one. Figure 2 represents the possessions of rotation parameter R on the velocity field fη for γAl2O3−H2O and γAl2O3−C2H6O2. Here, Figure 2(a) depicts the velocity field fη for the increasing value of rotation parameter R=1,4,8,12. For S>0, one can see that axial velocity fη decreases for the increasing value of rotation parameter R, while for S<0, the axial velocity increases for R=1,4,8,12.Figure 2
R varying for f with S (for suction and injection).
(a)(b)Figure3 represents the effects of rotation parameter R on the velocity field gradient f′η for γAl2O3−H2O and γAl2O3−C2H6O2. Figure 3(a) depicts the velocity field gradient f′η for the increasing value of rotation parameter R=1,4,12,16. For S>0, one can see that velocity field gradient f′η decreases for the increasing value of rotation parameter R, and for S<0, the velocity field gradient decreases.Figure 3
R varying for f with S (suction and injection).
(a)(b)Figure4 inspects the effects of rotation parameter R for the transverse component of the velocity for γAl2O3−H2O and γAl2O3−C2H6O2. In Figure 4(a), it is elucidated that, in the presence of γAl2O3−C2H6O2, the velocity field increases slowly (shown by dotted red lines), while in γAl2O3−H2O, it increases more rapidly for S>0, while in Figure 4(b), for S<0, the velocity field decreases, and the nanoparticles in water decrease more rapidly.Figure 4
R varying for gη with S (suction and injection).
(a)(b)Figures5–7 examine the effects of squeezing parameter β for axial velocity fη, velocity field gradient f′η, and the transverse component of velocity gη for γAl2O3−H2O and γAl2O3−C2H6O2. Here, the behavior of nanoparticles of given nanofluids is quite dominant for S>0 as well as for S<0 for β=0.1,0.3,0.5,0.7 to axial velocity fη and the transverse component of velocity gη. However, the impact of higher values of β on radial velocity is different than that of the remaining. For S>0, upon incremental values of β, the radial velocity increases, and an opposite behavior is observed in the case of injection.Figure 5
β varying for f with S (suction and injection).
(a)(b)Figure 6
β varying for f′ with S (suction and injection).
(a)(b)Figure 7
β varying for g with S (suction and injection).
(a)(b)Figures8–10 depict the effects of magnetic parameter M for γAl2O3−H2O and γAl2O3−C2H6O2 on velocity field gradient f′η, axial velocity fη, and the transverse component of the velocity gη. In Figure 8(a), one can easily examine that, for the suction case, the magnitude of the velocity gradient increases with increasing values of the Hartmann number, while in Figure 8(b), for S<0, an opposite behavior is encountered for incremental values of M. However, in Figure 9, for higher values of M, the axial velocity for both S>0 and S<0 decreases due to the impedance of Lorentz forces. In Figure 10(a), the impact of angular velocity looks like that of radial velocity in Figure 8(a), while in the case of injection, the magnitude of gη is decreased due to incremental values of the Hartmann number.Figure 8
M varying for f′ with S (suction and injection).
(a)(b)Figure 9
M varying for f with S (suction and injection).
(a)(b)Figure 10
M varying for g with S (suction and injection).
(a)(b)Figure11 views the upshots of λ quotient of the Grashof number to square root of the Reynolds number for the velocity field gradient f′η for γAl2O3−H2O and γAl2O3−C2H6O2. In Figure 11(a), it is noticed that, for S>0, the velocity field gradient first decreases rapidly, and then, at the area between 0.4 and 0.6, its behavior instantly becomes adverse, and then onward, it increases for λ=0,100,300. On the contrary, in Figure 11(b), it is noticed that, for S<0, the velocity field gradient first increases rapidly, and then, at the region between 0.4 and 0.6, its behavior instantly changes, and then onward, it decreases for λ=0,100,300.Figure 11
λ varying for f′ with S (suction and injection).
(a)(b)Figure12 checks the upshots of λ quotient of Gr to Re for the axial velocity fη for γAl2O3−H2O and γAl2O3−C2H6O2. Here, one can examine that, upon incremental values of λ, the velocity field decreases for S>0, while in Figure 12(b), one can examine that the velocity field fη increases for the increasing value of mixed convection parameter λ=0,100,300 for S<0.Figure 12
λ varying for f with S (suction and injection).
(a)(b)Figure13 inspects the effects of mixed convection parameter λ for the transverse component of the velocity gη for γAl2O3−H2O and γAl2O3−C2H6O2. In Figure 13(a), it is explicated that, in the occurrence of γAl2O3−H2O, the transverse component of velocity increases more rapidly, while in the case of γAl2O3−C2H6O2, this impact slows down, while in Figure 13(b), for S<0, the velocity field decreases, and the nanoparticles adjoined with water decrease more rapidly than the particles based on ethylene glycol which shows the behavior of being decreasing quite slowly for λ=0,100,300.Figure 13
λ varying for g with S (suction and injection).
(a)(b)Figures14–16 check the effects of solid volume fraction ϕ for radial velocity f′η, axial velocity, and the transverse component of the velocity gη for γAl2O3−H2O and γAl2O3−C2H6O2. In Figure 14(a), it is clarified that, in the presence of γAl2O3−C2H6O2, the velocity field initially decreases, and at the region from 0.4 to 0.6, a variation occurs, and the velocity field increases onward for S>0, while in Figure 14(b), for S<0, the velocity field decreases in the presence of γAl2O3−C2H6O2 more rapidly than that of γAl2O3−H2O for the increasing value of ϕ=0,0.04,0.08,0.12. However, in Figure 15, upon incremental values of solid size segment ϕ, the velocity fη decreases for S>0 and increases for S<0, while in Figure 16, the component of transverse velocity behaves opposite to that of axial velocity for both suction and injection cases.Figure 14
ϕ varying for f′ with S (suction and injection).
(a)(b)Figure 15
ϕ varying for f with S (suction and injection).
(a)(b)Figure 16
ϕ varying for g with S (suction and injection).
(a)(b)The impact of emerging parameters on the temperature profile along with the thermal radiation is reflected in Figures17 and 18.Figure 17
Rd varying for θ with S (suction and injection).
(a)(b)Figure 18
Pr varying for θ with S (suction and injection).
(a)(b)Figure17 depicts the impact of radiation parameter Rd on temperature profile θη. Here, it is observed that, due to remarkable effects of thermal radiation, the temperature distribution increases with incremental values of Rd for both S>0 and S<0.Figure18 represents the effects of Prandtl number Pr on the temperature field θη for γAl2O3−H2O and γAl2O3−C2H6O2. Here, it depicts the temperature field θη for increasing values of Prandtl number Pr=0.72,3.0,6.2. One can see that temperature field θη increases for increasing values of Prandtl number Pr=0.72,3.0,6.2 for S>0 as well as for S<0.Our flow model is of wide range for the nanofluids. In the following, the numerical results are discussed in detail with the help of upper plotted graphs and tables as well. Table2 reflects the validity of our flow model for ϕ=0andM=0.5. Here, one can easily notice that, with the existing literature, our flow model is in best agreement to the present study which proves the novelty and reality of our flow model. The upshots of skin friction coefficient and local Nusselt number for γAl2O3−H2O are portrayed in Table 3, while the ethylene glycol-based nanoparticles are portrayed in Table 4. To provoke the enhancement of thermal conductivity due to the inclusion of thermal radiation parameter Rd, one can observe that the rate of heat transfer increases quite rapidly in the case of γAl2O3−C2H6O2 nanoparticles as compared to γAl2O3−H2O.Table 2
Comparison with the existing literature forϕ=0andM=0.5.
βRSf″0f″1PresentKhan et al. [4]PresentKhan et al. [4]−1.02.00.5−10.53119500−10.53119507.703172267.70317240.0−7.58900683−7.58900694.823590914.82359091.0−4.51255172−4.51255171.809171881.80917192.0−1.28942091−1.2894209−1.35422925−1.35422923.02.083503442.0835034−4.67005029−4.6700503Table 3
Numerical values off″1 and θ1 for γAl2O3−H2O.
βRdRλMf″1θ′1S>0S<0S>0S<00.50.721115.93987−11.89738−1.23755−1.9150818.99335−19.04397−1.22456−1.894051.52.00256−26.23351−1.21168−1.873212−5.03229−33.46580−1.19891−1.8525610.534.5299−19.04411−0.421818−1.90666147.1583−19.04380−0.643296−1.878391.560.444−19.04357−0.884845−1.8584110.58.99335−19.01137−1.27791−1.878311.08.99326−19.01785−1.27791−1.878331.58.99311−19.02866−1.27793−1.87835Table 4
Numerical values off″1 and θ1 for γAl2O3−C2H6O2.
βRdRλMf″1θ′1S>0S<0S>0S<00.50.721115.94174−11.8980−1.03713−1.9861018.99516−19.0446−1.02218−1.954411.52.00432−26.2342−1.00737−1.923102−5.03057−33.4665−0.99272−1.8921810.58.99571−19.04482−0.96640−1.9714518.99450−19.04440−1.09289−1.933101.58.99368−19.04410−1.18506−1.9057010.58.99495−19.01197−1.09284−1.932991.08.99486−19.01846−1.09285−1.933011.58.99471−19.02927−1.09287−1.93305


## 5. Conclusion

An unsteady nonlinear flow prototype under the influence of thermal radiation comprising theγ-aluminum particle is presented for H2O and C2H6O2. The given structure is consolidated iteratively, and the subsequent outcomes are deduced:(1)
Forγ−AL2O3−H2O, the temperature distribution is higher than γ−AL2O3−C2H6O2(2)
In the presence of the effective Prandtl number, the skin friction coefficient enhanced in buoyancy opposing flow for both nanofluids(3)
Forγ−AL2O3−C2H6O2, the skin friction coefficient is higher than γ−AL2O3−H2O in buoyancy opposing flow(4)
The rate of heat transfer is decreased in the absence of the effective Prandtl model, while the opposite trend is observed in the presence of the effective Prandtl model(5)
It is inspected that the transverse component of velocity decreases for increasing values ofβ for S>0 and increases for S<0(6)
Due to incremental values ofM, the velocity fη decreased for S>0, and the transverse component of velocity increased(7)
The radiation parameterRd favors the temperature distribution along with the incremental values of the Prandtl number(8)
Mixed convection parameterλ opposes the axial velocity(9)
For increasing values ofRd, the rate of heat transfer increases quite rapidly for the γAl2O3−C2H6O2 nanoparticles(10)
Under certain limitations, the skin friction coefficient and rate of heat transfer at both walls show an excellent agreement with the prevailing one



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