# A Nonsingular Fractional Derivative Approach for Heat and Mass Transfer Flow with Hybrid Nanoparticles

Authors: Muhammad Imran Asjad; Rabia Naz; Muhammad Danish Ikram; Azhar Iqbal; Fahd Jarad
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper deals with the study of MHD Brinkman type fluid flow containing hybrid titanium (TiO2) and silver (Ag) nanoparticles with nonlocal noninteger type Atangana-Baleanu (ABC) fractional differential operator. The problem is designed for the convective flow restrained in a microchannel. With the Mittag–Leffler kernel, the conventional governing equations are converted into dimensionless form and then generalised with noninteger order fractional operators. The solutions for temperature and velocity fields obtained via Laplace transform method and expressed in the series form. The effect of related parameters is dignified graphically with the help of Mathcad and presented in the graphical section. Finally, the results show that the AB fractional operator exhibited improved memory effect as compared to CF fractional operator. Furthermore, due to increasing the values volume fractional temperature can be enhanced and velocity decreases. In comparison between nanoparticles for different types of based fluid, velocity and temperature of water based (TiO2) and silver (Ag) is higher than other base fluids.


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## Body

## 1. Introduction

Fractional calculus is a field of mathematics that deals with the integration and derivation of real and complex numbers. Despite the fact that calculus is an old subject, in recent decades, it has regained popularity due to its numerous applications. Researchers have recently investigated mathematical models of heat transfer fluids with considerable applications, which play an essential role in manufacturing. These simulations are usually represented as integer-order partial differential equations. Its important to note that traditional PDEs cannot account for the dynamical behavior of physical flow parameters or retention effects. To cater and eliminate the above mentioned flaws researchers concentrated on fractional dynamic systems of heat transmission in simple and complex fluid models. Nanoparticles are tiny particles with a diameter of 1 to 100 nanometers. Nanoparticles, which are invisible to the naked eye, can have drastically varying physical and chemical characteristics than their bigger material counterparts. Only a few hundred atoms make up most nanoparticles. Nanofluids are often used as coolants in heat transfer equipment such as heat exchangers, electronic cooling systems (such as solid surfaces), and radiators due to their better cooling characteristics [1]. Nanofluids have unique features that make them potentially helpful in a variety of heat transfer applications, such as microelectronics, battery storage, pharmaceutical processes, hybrid-powered engines, engine cooling/vehicle thermal management, domestic refrigerator, chiller, superheater, and crusher [2]. Das et al. [3] conducted a research nanofuids and their application in research and technology. Heat transmission procedures play a significant role in general energy intake, that is why improvement of its effectiveness is a significant research area [4]. Nonrenewable assets such as natural gas, coal and petroleum are the core energy resources for international economy motion [5]. First of all, Caputo established the fractional derivative with Laplace transform technique. The (RL) fractional derivative issue was solved using this operator. In the literature, (RL) and (C) fractional operators were usually rejected in numerous fields of science therefore, a new fractional derivative (ABC) was introduced, which provided more accurate and promising results [6]. Caputo and Fabrizio introduced (CF) fractional derivative with a nonsingular kernel [7]. It has been effectively applied in many fields of science.Atangana andBaleanu [8] developed a new operator in 2016 which was better than Caputo and Fabrizio’s model in describing the physical behaviour of natural phenomena. Saqib et al. [9] described MHD movement of Casson fluid in a microchannel by (ABC) fractional operator. Atangana and Koca [10] discussed (ABC) fractional operator in nonlinear baggs and freedman model. Siddique et al. [11] explained unsteady free convection movements of an incompressible fluid over an infinite vertical plate via (ABC) and (CF) approaches. Imran et al. [12] provided a full description of fractional (ABC) and (CF) MHD viscous fluid convective flow. Mohammad et al. [13] used fractional derivatives to examine the outcomes of the uncertain fractional backward difference equations. (CF) and (ABC) fractional operators were used to discover the influences of carbon nanotubes on magnetohydrodynamic movement of methanol based nanofluid in [14]. Hammouch and Makkaoui [15] described the dynamical behavior of a novel 3D fractional-ordered chaotic system. Ikram et al. [16] analyzed the heat transfer of viscous nanofluid over an exponentially moving vertical plate via (ABC) approach. The thermal conductivity of Newtonian fluid with C and CF fractional operator was discussed in [17, 18].In [19] nonsingular kernel inside the Atangana-Baleanu fractional operator, the Painlev as well as BagleyTorvik calculations are answered according to the initial conditions. In [20] Imran et al. discussed the thermophoresis properties on simultaneous mass as well as heat transfer in MHD fluid over a porous medium by even heat flux with the help of (AB) fractional operator. In [21] paper emphases on conditions that a fresh high-order system for the arithmetic results of fractional instruction Volterra integro-DE’s with the help of AB operator. In [22] Khan discussed the novel idea of ABC fractional operator on human blood pour in Nanofluids. Sania and Yusuf [23] discussed the modeling and results of chickenpox infection with different fractional operators such as C, CF and ABC operators. Raza and Ullah [24] presented a comparison of C and CF fractional operators with heat transfer analysis of fractional Maxwell fluid. They used Tzous and Stehfests algorithms to find the solutions numerically. Imran et al. [25] studied the unsteady natural convection flow of Maxwell fluid with fractional derivative over an exponentially accelerated infinite vertical plate using Stehfests and Tzous algorithms. Chu et al. [26] explained a nonlocal fractional model of hybrid nanofluid with MHD free convection flow in microchannel. Imran et al. [27] used constant proportional Caputo (CPC) fractional operator to study the heat transfer flow of caly-water base nanofluids over an infinite vertical surface moving with constant velocity. Ikram et al. [28] calculated the heat transfer flow of hybrid nanofluid between parallel plates via (CPC) fractional approach and offered a comparison with CF fractional derivative. Imran et al. [29] discussed the heat and mass transfer flow for adhesive fluid between two upright plates through fractional approach. Dawar et al. [30] analyzed the heat transfer of 3D magnetohydrodynamic and sodium alginate based on iron oxide nanofluid moving through a spinning disk with homotopy analysis method. Vaidya et al. [31] calculated the influence of homogeneous and heterogeneous responses on the peristaltic flow of MHD Jeffrey fluid in a nonuniform vertical channel with slip conditions. Sabu et al. [32] presented the analysis of hydromagnetic alumina-water nanoliquid flow due to a rotating rigid disk numerically. They considered different nanoparticle shapes and the thermo-hydrodynamic slip conditions. The following references show a wide range of fractional models and their applications in applied sciences [33–36].Freshly, In [37] the fractional derivative of (BTF) along with hybrid nanofluid is explained with the help of (CF) fractional operator. We applied the LT technique to acquire the logical results for velocity as well as temperature fields and is indicated as M-function. After that a few figures were plotted to show the impression of parameters on velocity and temperature. The physical properties of hybrid nanofluids are presented in Table 1 and relation of the physical values of nano and hybrid nanofluids can be seen in Tables 2 and 3.Table 1
Nanoparticles and base fluid thermophysical characteristics [37].
MaterialBase fluidNanoparticlesNanoparticlesTiO2AgH2Oρ42510500997.1k8.95384290.613Cp68622354179σ1×10−123.6×1070.05βT×10−50.91.8921Pr——6.2Table 2
Expressions of nanofluid [37].
PropertiesNanofluidDensityρnf=ϕρs+1−ϕρfDynamic viscosityμnf=μf/1−ϕ2.5Thermal expansionρβTnf=ϕρβTs+1−ϕρβTfHeat capacitanceρCpnf=ϕρCps+1−ϕρCpfElectric conductivityσnf/σf=1+3σs/σf−1ϕ/σs/σf+2−σs/σf−1ϕThermal conductivityKnf/Kf=2kf+ks−2ϕks−kf/2kf+ks+ϕks−kfTable 3
Expressions of hybrid nanofluid [37].
PropertiesHybrid nanofluidDensityρhnf=ϕAgρAg+ϕTiO2ρTiO2+1−ϕhnfρfDynamic viscosityμhnf=μf/1−ϕAg+ϕTiO22.5Thermal expansionρβThnf=ϕAgρβTAg+ϕTiO2ρβTTiO2+1−ϕhnfρβTfHeat capacitanceρCphnf=ϕAgρCpAg+ϕTiO2ρCpTiO2+1−ϕhnfρCpfElectric conductivityσhnf/σf=1+3ϕAgσAg+ϕTiO2σTiO2/σf−ϕ/ϕAgσAg+ϕTiO2σTiO2/ϕσf+2−ϕAgσAg+ϕTiO2σTiO2/σf−ϕThermal conductivityKhnf/Kf=ϕAgkAg+ϕTiO2kTiO2/ϕhnf+2kf+2ϕAgkAg+ϕTiO2kTiO2−2ϕhnfkf/ϕAgkAg+ϕTiO2kTiO2/ϕhnf+2kf+ϕAgkAg+ϕTiO2kTiO2−ϕhnfkfThe significance of the present work is that a nonsingular fractional operator is applied in the heat and mass transfer phenomena containing nanoparticles of the type Ag andTiO2. As the nanoparticles used in the base fluid to improve the physical quantities of base fluid, this paper contains some new comparisons between fractional operators as well as different kinds of nanoparticles which can be useful in the heat transfer process where we can make a choice with different base fluids according to desired targets. These objectives have already been justified physically in this paper.


## 2. Mathematical Formulation

Suppose that magnetohydrodynamic natural convection movement of Brinkman fluid happens in microchannel of a comprehensive, electrically conductiveAg−TiO2−H2O hybrid nanofluid.The hypotheses forgoes as shown in Figure1:(a)
Width and length of microchannel isl and infinite, respectively(b)
Channel is beside horizontal line and normal to vertical line(c)
T0 is the temperature of framework, when t≤0(d)
Subsequentlyt=0+, temperature rises T0 into TW(e)
C0 is the concentration of framework, when t≤0(f)
Subsequentlyt=0+, concentration rises C0 into CW(g)
Motion is alongx-axis(h)
MF of strengthB0 works diagonally into the direction of flowFigure 1
Formation of microchannel and coordinate system.Flow of electrically conductiveAg−TiO2−H2O hybrid nanofluid undergoes electromotive energy, which generate current. The induced MF is overlooked for the assumption of reduced Reynolds number. The electromagnetic energy revolves on the power of electric instability [18].Velocity component of hybrid nanofluid is of the formvy,ti^,0,0. The electromagnetic force Fem is merged into the momentum equation of the natural convection flow of an incompressible Ag−TiO2−H2O hybrid nanofluid in the absence of pressure gradient and an inclined magnetic field is imposed.The governing equations are [37](1)ρhnfvty,t+βb∗vy,t=μhnfvyyy,t−σhnfB02vy,t+gρβThnfTy,t−T0+gρβchnfCy,t−C0,(2)ρCphnfTty,t=KhnfTyyy,t+Q0Ty,t−T0,(3)Cty,t=DCyyy,t,with the following initial and boundary conditions(4)vy,0=0,Ty,0=T0,Cy,0=C0,y≥0,t<0,(5)vl,t=0,Tl,t=Tw,Cl,t=Cw,t>0.By presenting the accompanying dimensionless variables,(6)τ1=νl2t,U=lνv,C=C−C0Cw−C0,ϖ=yl,φ=T−T0Tw−T0,into equations (1)–(5), we have the subsequent dimensionless problem:(7)C0Uτ1ϖ,τ1+βbUϖ,τ1=C1Uϖϖϖ,τ1−C2MUϖ,τ1+C3Grφϖ,τ1+C4GmCw,τ1,(8)C5Prφτ1ϖ,τ1=λhnfφϖϖϖ,τ1+Qφϖ,τ1,(9)Cτ1ϖ,τ1=1ScCϖϖϖ,τ1,where Sc is Schmidt number (dimensionless).Initial and boundary conditions which are dimensionless,(10)Uϖ,0=0,φϖ,0=0,Cϖ,0=0,ϖ≥0,τ1<0,Ul,τ1=0,φl,τ1=1,Cl,τ1=1,τ1>0,where(11)βb=l2βb∗ρhnfμ,M=l2σfB02μ,λhnf=khnfkf,Q=Q0l2kf,Pr=μCpkf,Gr=l3gβTfTw−T0ν2,Sc=νD,C0=1−ϕhnf+ϕAgρAg+ϕTiO2ρTiO2ρf,C1=11−ϕAg+ϕTiO22.5,C2=σhnfσf,C3=1−ϕhnf+ϕAgρβTAg+ϕTiO2ρβTTiO2ρβTf,C4=1−ϕhnf+ϕAgρβpAg+ϕTiO2ρβpTiO2ρβpf,C5=1−ϕhnf+ϕAgρCpAg+ϕTiO2ρCpTiO2ρCpf.


## 3. Result of the Problem

Fractional model obtained by equations (7)–(9) using the definition of ABC defined in [8].(12)Uϖϖϖ,τ1−D1ABCDτ1αUϖ,τ1−D2Uϖ,τ1+D3φϖ,τ1+D4Cϖ,τ1=0,(13)φϖϖϖ,τ1−D5ABCDτ1αφϖ,τ1+D6φϖ,τ1=0,(14)ScABCDτ1αCϖ,τ1=Cϖϖϖ,τ1,where as D1=C0/C1,D2=C2M+βb/C1,D3=C3Gr/C1,D4=C4Gm/C1,D5=C4Pr/λhnf,andD6=Q/λhnf.


### 3.1. Formulation of Concentration

Implementing LT on equation (14) and using initial and boundary conditions, we have(15)D2−Sc⋅qα1−αqα+αC¯ϖ,q=0,satisfying(16)C¯0,q=0,C¯1,q=1q.The general solution of equation (15) subject to equation (16)(17)C¯ϖ,q=SinhϖScqα/1−αqα+αqSinhScqα/1−αqα+α.Equation (17) can be written in suitable form by using formula [28].(18)SinhabSinha=∑n=0∞e−2n+1−ba−e−2n+1+ba.asC¯ϖ,q=1q∑i=0∞e−2i+1−ϖ⋅Scqα/1−αqα+α−e−2i+1+ϖ⋅Scqα/1−αqα+α,By converting into series form as in [28],(19)C¯ϖ,q=∑i=0∞∑i1=0∞∑i2=0∞−2i+1−ϖi1Sci2αi2Γi1/2+1i1!i2!1−αi1/2+i2qαi2+1Γi1/2+1−i2−∑i=0∞∑i3=0∞∑i4=0∞−2i+1−ϖi3Sci3αi4Γi3/2+1i3!i4!1−αi3/2+i4qαi4+1Γi3/2+1−i4,Taking inverse LT(20)Cϖ,τ1=∑i=0∞∑i1=0∞∑i2=0∞−2i+1−ϖi1Sci2αi2τ1αi2Γi1/2+1i1!i2!1−αi1/2+i2Γi1/2+1−i2−∑i=0∞∑i3=0∞∑i4=0∞−2i+1−ϖi3Sci3αi4τ1αi4Γi3/2+1i3!i4!1−αi3/2+i4Γi3/2+1−i4.


### 3.2. Formulation of Temperature Field

By implementing LT on equation (13) and using initial condition, we obtain(21)D2−D5qα1−αqα+α+D6φ¯ϖ,q=0,which satisfies(22)φ¯0,q=0,φ¯1,q=1q.The general solution of equation (21) subject to equation (22)(23)φ¯ϖ,q=1qsinhϖD5qα/1−αqα+α−D6sinhD5qα/1−αqα+α−D6,Suitably written in equivalent form(24)φ¯ϖ,q=1q∑ξ=0∞e−2ξ+1−ϖD5qα/1−αqα+α−D6−e−2ξ+1+ϖD5qα/1−αqα+α−D6.By applying LT to equation (24) after expressed in series form,(25)φ¯ϖ,q=1q+∑ξ=0∞∑ξ1=1∞∑ξ2=1∞∑ξ3=1∞ϖ−2ξ−1ξ1−1ξ1/2+ξ2+ξ3D5ξ2αξ3ξ1!ξ2!ξ3!D6ξ2−ξ1/21−αξ2+ξ3qαξ3+1Γξ1/2+1Γξ2+ξ3Γξ1/2+1−ξ2Γξ2−∑ξ=0∞∑ξ4=0∞∑ξ5=0∞∑ξ6=0∞−2ξ+1+ϖξ4−1ξ4/2+ξ5+ξ6D5ξ5αξ6ξ4!ξ5!ξ6!D6ξ5−ξ4/21−αξ5+ξ6qαξ6+1Γξ4/2+1Γξ5+ξ6Γξ4/2+1−ξ5Γξ5.Taking inverse LT on equation (25) and expressing in M-function defined in [7], we have(26)φϖ,τ1=1+∑ξ=0∞∑ξ1=1∞∑ξ2=1∞ϖ−2ξ−1ξ1−1ξ1/2+ξ2D5ξ2ξ1!ξ2!D6ξ2−ξ1/21−αξ2M32−ατ1α1−α|ξ1/2+1−ξ2,0,ξ2,0,1,αξ1/2+1,0,ξ2,1−∑ξ=0∞∑ξ4=0∞∑ξ5=0∞−2ξ+1+ϖξ4−1ξ4/2+ξ5D5ξ5ξ4!ξ5!D6ξ5−ξ4/21−αξ5M32−ατ1α1−α|ξ4/2+1−ξ5,0,ξ5,0,1,αξ4/2+1,0,ξ5,1.


### 3.3. Nusselt Number

We have computed the heat transfer rate in terms of Nusselt number through the following relation and presented in Table4.(27)Nu=−∂φ0,τ1∂ϖ.Table 4
Numerically analysis of Nusselt number for the effect of fractional parameter with comparison of (ABC) and (CF).
αNu (ABC)Nu (CF)Nu (ABC)Nu (CF)t = 0.5t = 0.5t = 2t = 20.12.0572.1112.0411.9360.22.0652.1692.0001.7980.32.0802.2251.9301.6420.42.1002.2771.8301.4690.52.1242.3171.6971.2880.62.1442.3301.5311.1110.72.1442.2881.3330.9560.82.0852.1381.1120.8380.91.8911.8380.8920.755


### 3.4. Formulation of Velocity Field

By utilizing the LT on equation (12) and using initial condition, we obtain(28)D2−D1qα1−αqα+α−D2U¯ϖ,q=−D3φ¯ϖ,q−D4c¯w,q,which satisfies(29)U¯0,q=0,U¯1,q=0.The general solution of equation (28) subject to equation (29) is(30)U¯ϖ,q=D3q∑ξ=0∞e−2ξD5qα/1−αqα+α−D6−e−2ξ+2D5qα/1−αqα+α−D6D7qα/1−αqα+α−D8×∑χ=0∞e−2χ+1−ϖD1qα/1−αqα+α+D2−e−2χ+1+ϖD1qα/1−αqα+α+D2−D3q∑ξ=0∞e−2ξ+1−ϖD5qα/1−αqα+α−D6−e−2ξ+1+ϖD5qα/1−αqα+α−D6D7qα/1−αqα+α−D8+D4q∑ı=0∞e−2ıScqα/1−αqα+α−e−2ı+2Scqα/1−αqα+α−D6D7qα/1−αqα+α−D8×∑χ=0∞e−2χ+1−ϖD1qα/1−αqα+α+D2−e−2χ+1+ϖD1qα/1−αqα+α+D2−D4q∑ı=0∞e−2ı+1−ϖScqα/1−αqα+α−e−2ξ+1+ϖScqα/1−αqα+αD7qα/1−αqα+α−D8.where D7=D5−D1,D8=D2+D6.Equation (30) is too much complicated, so that we cannot apply inverse LT. Therefore solutions obtained numerically.


## 3.1. Formulation of Concentration

Implementing LT on equation (14) and using initial and boundary conditions, we have(15)D2−Sc⋅qα1−αqα+αC¯ϖ,q=0,satisfying(16)C¯0,q=0,C¯1,q=1q.The general solution of equation (15) subject to equation (16)(17)C¯ϖ,q=SinhϖScqα/1−αqα+αqSinhScqα/1−αqα+α.Equation (17) can be written in suitable form by using formula [28].(18)SinhabSinha=∑n=0∞e−2n+1−ba−e−2n+1+ba.asC¯ϖ,q=1q∑i=0∞e−2i+1−ϖ⋅Scqα/1−αqα+α−e−2i+1+ϖ⋅Scqα/1−αqα+α,By converting into series form as in [28],(19)C¯ϖ,q=∑i=0∞∑i1=0∞∑i2=0∞−2i+1−ϖi1Sci2αi2Γi1/2+1i1!i2!1−αi1/2+i2qαi2+1Γi1/2+1−i2−∑i=0∞∑i3=0∞∑i4=0∞−2i+1−ϖi3Sci3αi4Γi3/2+1i3!i4!1−αi3/2+i4qαi4+1Γi3/2+1−i4,Taking inverse LT(20)Cϖ,τ1=∑i=0∞∑i1=0∞∑i2=0∞−2i+1−ϖi1Sci2αi2τ1αi2Γi1/2+1i1!i2!1−αi1/2+i2Γi1/2+1−i2−∑i=0∞∑i3=0∞∑i4=0∞−2i+1−ϖi3Sci3αi4τ1αi4Γi3/2+1i3!i4!1−αi3/2+i4Γi3/2+1−i4.


## 3.2. Formulation of Temperature Field

By implementing LT on equation (13) and using initial condition, we obtain(21)D2−D5qα1−αqα+α+D6φ¯ϖ,q=0,which satisfies(22)φ¯0,q=0,φ¯1,q=1q.The general solution of equation (21) subject to equation (22)(23)φ¯ϖ,q=1qsinhϖD5qα/1−αqα+α−D6sinhD5qα/1−αqα+α−D6,Suitably written in equivalent form(24)φ¯ϖ,q=1q∑ξ=0∞e−2ξ+1−ϖD5qα/1−αqα+α−D6−e−2ξ+1+ϖD5qα/1−αqα+α−D6.By applying LT to equation (24) after expressed in series form,(25)φ¯ϖ,q=1q+∑ξ=0∞∑ξ1=1∞∑ξ2=1∞∑ξ3=1∞ϖ−2ξ−1ξ1−1ξ1/2+ξ2+ξ3D5ξ2αξ3ξ1!ξ2!ξ3!D6ξ2−ξ1/21−αξ2+ξ3qαξ3+1Γξ1/2+1Γξ2+ξ3Γξ1/2+1−ξ2Γξ2−∑ξ=0∞∑ξ4=0∞∑ξ5=0∞∑ξ6=0∞−2ξ+1+ϖξ4−1ξ4/2+ξ5+ξ6D5ξ5αξ6ξ4!ξ5!ξ6!D6ξ5−ξ4/21−αξ5+ξ6qαξ6+1Γξ4/2+1Γξ5+ξ6Γξ4/2+1−ξ5Γξ5.Taking inverse LT on equation (25) and expressing in M-function defined in [7], we have(26)φϖ,τ1=1+∑ξ=0∞∑ξ1=1∞∑ξ2=1∞ϖ−2ξ−1ξ1−1ξ1/2+ξ2D5ξ2ξ1!ξ2!D6ξ2−ξ1/21−αξ2M32−ατ1α1−α|ξ1/2+1−ξ2,0,ξ2,0,1,αξ1/2+1,0,ξ2,1−∑ξ=0∞∑ξ4=0∞∑ξ5=0∞−2ξ+1+ϖξ4−1ξ4/2+ξ5D5ξ5ξ4!ξ5!D6ξ5−ξ4/21−αξ5M32−ατ1α1−α|ξ4/2+1−ξ5,0,ξ5,0,1,αξ4/2+1,0,ξ5,1.


## 3.3. Nusselt Number

We have computed the heat transfer rate in terms of Nusselt number through the following relation and presented in Table4.(27)Nu=−∂φ0,τ1∂ϖ.Table 4
Numerically analysis of Nusselt number for the effect of fractional parameter with comparison of (ABC) and (CF).
αNu (ABC)Nu (CF)Nu (ABC)Nu (CF)t = 0.5t = 0.5t = 2t = 20.12.0572.1112.0411.9360.22.0652.1692.0001.7980.32.0802.2251.9301.6420.42.1002.2771.8301.4690.52.1242.3171.6971.2880.62.1442.3301.5311.1110.72.1442.2881.3330.9560.82.0852.1381.1120.8380.91.8911.8380.8920.755


## 3.4. Formulation of Velocity Field

By utilizing the LT on equation (12) and using initial condition, we obtain(28)D2−D1qα1−αqα+α−D2U¯ϖ,q=−D3φ¯ϖ,q−D4c¯w,q,which satisfies(29)U¯0,q=0,U¯1,q=0.The general solution of equation (28) subject to equation (29) is(30)U¯ϖ,q=D3q∑ξ=0∞e−2ξD5qα/1−αqα+α−D6−e−2ξ+2D5qα/1−αqα+α−D6D7qα/1−αqα+α−D8×∑χ=0∞e−2χ+1−ϖD1qα/1−αqα+α+D2−e−2χ+1+ϖD1qα/1−αqα+α+D2−D3q∑ξ=0∞e−2ξ+1−ϖD5qα/1−αqα+α−D6−e−2ξ+1+ϖD5qα/1−αqα+α−D6D7qα/1−αqα+α−D8+D4q∑ı=0∞e−2ıScqα/1−αqα+α−e−2ı+2Scqα/1−αqα+α−D6D7qα/1−αqα+α−D8×∑χ=0∞e−2χ+1−ϖD1qα/1−αqα+α+D2−e−2χ+1+ϖD1qα/1−αqα+α+D2−D4q∑ı=0∞e−2ı+1−ϖScqα/1−αqα+α−e−2ξ+1+ϖScqα/1−αqα+αD7qα/1−αqα+α−D8.where D7=D5−D1,D8=D2+D6.Equation (30) is too much complicated, so that we cannot apply inverse LT. Therefore solutions obtained numerically.


## 4. Results and Discussion

The current study of MHD Brinkman type fluid flow is the focus of this article, which contain sliverAg and hybrid titanium TiO2 nanoparticles with nonlocal noninteger type fractional differential operators the problem is created for the convective flow restrained in a microchannel. With the help of LT temperature and velocity solutions are obtained and then converted to the series form. In order insight the physical significance of flow parameters some graphs are plotted.The fractional operator was used to measure the hybrid nanofluidTiO2−H2O. The (BTF) model with conflict along with affected by magnetic field was distinguished with the help of PDEs. Appraisal between the (ABC) fractional derivative was applied to determine the performance of hybrid nanofluid with the help of LT procedure. The results of the temperature and velocity fields are exposed as M-function. The parameters influences of α, βb, ϕhnf, Gr, Q and M on temperature as well as velocity fields are also calculated and shown graphically with their physical impact.Figures2-5 will compare the (ABC) derivative with the (CF) derivative, which is argued in [37], to analyze the impact of α on temperature and velocity fields. The velocity as well as temperature is decreasing functions of α as a result of reduction in momentum and heat boundary layers. It is due to that the fractional parameter controls the phenomena of fluid flow physically. The rate of heat transfer is also calculated numerically in the form of Nusselt number by comparing (ABC) and (CF) fractional operators as shown in Table 4. It is observed that (ABC) derivatives show better retention than (CF) derivatives. Figures 6 and 7 signify the appraisal of velocity and temperature fields for nanofluid particles Ag−H2O and TiO2−H2O. Meanwhile TiO2 is semiconductor and Ag is good conductor, consequently Ag−H2O had high temperature profile than TiO2−H2O. However the nanofluid concentration has more significance in the velocity field. Because of antiparticles adding in base fluid, the resultant nanofluid creates much denser which decrease the velocity. Ag−H2O decreases velocity more than TiO2−H2O relatively.Figure 2
Correlation between the velocities through CF [37] and present, when Pr=6.2, t = 3, M = 0.2, Gr = 20, Q = 0.5, ϕhnf=0.08 and β=0.8.Figure 3
Correlation between the temperatures through CF [37] and present, Pr=6.2, when t = 2, Q = 1.2 and ϕhnf=0.2.Figure 4
Correlation between the velocities through CF [37] and ABC, Pr=6.2, when t = 2, Q = 0.5, Gr = 23, M = 0.2, β=0.8 and ϕhnf=0.08.Figure 5
Correlation between the temperatures through CF [37] and ABC, Pr=6.2, when t = 3, Q = 0.5 and ϕhnf=0.04.Figure 6
Correlation between nanofluid for velocity, whenPr=6.2, t = 0.4, Gr = 40, Q = 0.5, M = 10, β=2 and α=0.2.Figure 7
Correlation among nanofluids for temperature, whent = 3, α=0.2, Q = 3, and Pr=6.2.Figures8 and 9 are designed to see the impact of Q which is very predictable. By increasing values of Q, both velocity and temperature increased. This is owing to heat producing by system through rising Q. Figures 10 and 11 display the influence of ϕhnf on the velocity and temperature fields. The velocity is a decreasing function of ϕhnf whereas temperature is increasing function. This is because of ϕhnf increases the viscosity and density of fluid from Table 2. The similar trend for ϕhnf was concluded by [34].Figure 8
Heat source Q Impact on velocity whenPr=6.2, ϕhnf=0.04, t = 0.2, β=0.5, Gr = 30, M = 3, and α=0.5.Figure 9
Effect of heat source Q on temperature whenPr=6.2, t = 0.4, ϕhnf=0.2 and α=0.5.Figure 10
Impact ofϕhnf on velocity when Pr=6.2, t = 0.1, M = 3, β=0.5, Q = 2, Gr = 30, and α=0.5.Figure 11
Effect ofϕhnf on temperature after Pr=6.2, t = 0.4, Q = 1.2 and α=0.5.Figures12 and 13 are made to show the evaluation of changed base fluids (engine oil, water, kerosene) for the temperature and velocity arenas. Because of the changed thermal conductivity of water and base fluids by hybrid nanoparticles has unusual temperature due to greater heat conductivity than further base fluids, it is observed that water-based hybrid nanoparticles has higher velocity as it moves away from the plate, whereas the velocity of engine oil-based fluid decrees near the plate. Figure 14 represent the effect of βb on the velocity field. After increasing βb the fluid velocity shrinks. This is due to higher values of βb make the drag forces stronger physically, so velocity decreases. The impression of M is verified in Figure 15. It guides that by rising M, velocity diminishes. Meanwhile values of M transmit to restive type forces, called Lorentz forces. Thus the velocity was declined. The effect of Gr on the velocity field represented in Figure 16. It is comprehended that by rising Gr, the velocity way quicker. As a result, Gr is linked to buoyancy forces, which elevate conventional convection and hence increase velocity.Figure 12
Comparison among base fluids (engine oil, water and kerosene) for velocity afterQ = 0.5, t = 4, Gr = 30, M = 4, β=0.2α=0.5 and ϕhnf=0.08.Figure 13
Comparison among unlike base fluids (engine oil, water and kerosene) for Temperature afterα=0.5, t = 0.4, Q = 3 and ϕhnf=0.2.Figure 14
The velocity effect of the BT fluidbeta, Pr = 6.2, Gr = 30, ϕhnf=0.08 when t = 3, Q = 0.5, and α=0.5.Figure 15
Effect of Magnetic field M on velocity after Pr = 6.2,t = 3, β=0.2, Q = 0.5, α=0.5,ϕhnf=0.08 and Gr = 30.Figure 16
The influence of Gron velocity whenβ=0,t=3,Pr=6.2, Q = 0.5, M = 4, ϕhnf=0.08, β=0.2 and α=0.5.


## 5. Conclusions

In present research work, the characteristics ofAg−TiO2−H2O hybrid nanofluid has been studied. In microchannel the flow of the hybrid nanofluid was calculated. The (MHD) effects and heat source Q were measure likewise. After converting into fractional model, (ABC) fractional operator used to solve the problem. The exact solutions were attained with the help of LT method and listed in the term of M-function, velocity fields.The significance of the present work is the nonsingular fractional operator is applied in the heat and mass transfer phenomena containing nanoparticles of the type Ag andTiO2. As the nanoparticles used in the base fluid to improve the physical quantities of base fluid. This paper contains some new comparisons between fractional operators as well as different kinds of nanoparticles which can be useful in the heat transfer process where we can make a choice with different base fluids according to desired targets. These objectives have already been justified physically in this study.Major outcomes of present paper are as per the following:(i)
The results show that the Atangana–Baleanu fractional operator exhibited improved memory effect as compared to Caputo–Fabrizio fractional operator.(ii)
Further, due to increasing the values volume fractional temperature can be enhanced and velocity decreases.(iii)
In comparison between nanoparticles for different types of based fluid, velocity and temperature of water basedTiO2 and silver (Ag) is higher than other base fluids.(iv)
Fractional parameter used to control the thermal and momentum boundary layer thickness.(v)
For larger values of concentration of nanoparticles, thermal as well as momentum boundary layer decrease.



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*Source: 9240772-2022-04-22.xml*