Fractional simulations for thermal flow of hybrid nanofluid with aluminum oxide and titanium oxide nanoparticles with water and blood base fluids

Muhammad Ijaz Khan; Ibrahim B. Mansir; Ali Raza; Sami Ullah Khan; Samia Elattar; Hanaa Mohamed Said; Iskander Tlili; Khalid Abdulkhaliq M. Alharbi; Ahmed M. Galal

doi:10.1515/ntrev-2022-0156

Artikel Open Access

Fractional simulations for thermal flow of hybrid nanofluid with aluminum oxide and titanium oxide nanoparticles with water and blood base fluids

Muhammad Ijaz Khan , Ibrahim B. Mansir , Ali Raza , Sami Ullah Khan , Samia Elattar , Hanaa Mohamed Said , Iskander Tlili , Khalid Abdulkhaliq M. Alharbi und Ahmed M. Galal

Veröffentlicht/Copyright: 10. August 2022

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen

Aus der Zeitschrift Nanotechnology Reviews Band 11 Heft 1

Abstract

The fractional model has been developed for the thermal flow of hybrid nanofluid due to the inclined surface. The thermal investigation of the hybrid nanomaterial is predicted by utilizing the molybdenum disulphide nanoparticles and graphene oxide nanomaterials. The flow computations for mixed convection flow of nanoparticles and base fluids are performed due to vertical oscillating plate. The simulations for the formulated model have been done ρ-Laplace transform technique for Caputo fractional simulations. Definitions of Mittage–Leffler function and ρ-Laplace transform are also presented for the governing model. The application of updated definitions of ρ-Laplace transform for the Caputo fractional model is quite interesting unlike traditional Laplace transforms. The comparative investigation for both types of nanoparticles is performed for heat and mass transfer rates. It is observed that the heat enhancement rate due to water-based nanoparticles is relatively impressive compared to the kerosene oil-based nanomaterials.

Keywords: fractional derivatives; hybrid nanofluid; Caputo fractional model; Mittag–Leffler function

Nomenclature

v: velocity [m/s]
t: time [s]
U _o: constant velocity [m/s]
T: temperature [K]
C: concentration
Pr: Prandtl number
Sc: Schmidt number
Gr: heat Grashof number
Gm: mass Grashof number
CF: Caputo–Fabrizio fractional derivative
s: Laplace transformed variable
g: acceleration due to gravity [m/s²]
β: CF-fractional derivative operator
Nu: Nusselt number
C _f: skin friction
β ₁: second-grade fluid parameter

1 Introduction

Nowadays, researchers and engineers are showing great interest to examine nanofluids heat transportation problems. Although many traditional concepts for enhancing the thermal capacity are reported, the demand of the economic growth for industrial and engineering phenomena is affected due to low thermal requirements. The modern research in nanotechnology directed the researchers to report the thermal resources on the basis of nanoparticles. The uniform and homogenous suspension between water and nanomaterials induced the concept of nanofluids. The nanofluids are obtained from various materials like metals, graphite, carbide ceramics, oxide ceramics, carbide ceramics, and carbon nanotubes. A magnificently improved thermal nature of many base liquids on interacting nanoparticles is observed, which claim applications in the biofluids, thermal systems, extrusion processes, organic liquids, biomedical applications, solar systems, and many more. The basic continuation on nanofluid topic was addressed by Choi [1]. Shehzad et al. [2] identified the thermal transport of micropolar nanofluid with significant magneto-porosity effects. Turkyilmazoglu [3] observed the heat enhancement with the interaction of nanomaterials with implementation of Buongiorno nanofluid relations. Nadeem et al. [4] discussed the heating determination of nanofluid subject for viscoelastic base material. Abdelmalek et al. [5] highlighted the bioconvection prospective of micropolar nanofluid via accelerated space. Mahanthesh et al. [6] reported the nanomaterial applications for enhancing the heat transfer mechanism for the convective flow. Ramezanizadeh et al. [7] proposed a review detail for the nanomaterials research with different thermal conductivities via intelligence technique approach. Khan et al. [8] endorsed the thermal prediction for the heat transfer enhancement for the Reiner–Philippoff nanofluid with the effective role of higher order slip consequences. Reddy et al. [9] notified the thermal determination with the interaction of nanoparticles in the rotating disk. The lubrication problem for the third-grade nanofluid for the radiative flow has been observed by Abbasi et al. [10].

Among the nanomaterials, the hybrid nanofluid is the subclass with improved thermal features and enhanced characteristics. In comparison to simple nanofluids, hybrid materials offer more thermal stability and excellent thermal conductivities. Being the composite of metallic and nonmetallic nanoparticles with base fluids, various applications are observed for hybrid nanofluids. The enhancement of the thermal consequences of nanofluids is experimentally and theoretically observed by many researchers. Sabu et al. [11] reported the significant outcomes for the hybrid nanofluids via alumina nanoparticles when the role of slip is more prominent. Dawar et al. [12] conducted the thermal prospective research for nanoparticles with applications of solar energy. Hassan et al. [13] pronounced the thermal improvement of hybrid nanofluid with shear thinning and thickening base fluid via an experimental approach. The copper base hybrid nanofluid model for the channel flow due to cavity was inspected by Al-Farhany et al. [14]. Song et al. [15] addressed the multiwalled carbon nanotube thermal inspection with truncation of the boundary layer by confirming the results via the experimental model. Wahid et al. [16] addressed the aluminum oxide and copper nanoparticles mixture for the slip flow under the impact of thermal radiation. Madhukesh et al. [17] observed that enhancement in the heating phenomenon by involving the AA7072 and AA7075 nanoparticles for the curved surface numerically. Saleem et al. [18] discussed the role of activation energy and magnetic force impact for the hybrid nanofluid for the ciliary optimized transport. Rajesh et al. [19] evaluated the thermal prospective for the hybrid nanofluid in the vertical plate due to ramped temperature. Alghamdi et al. [20] discussed the heat transfer pattern with hybrid nanofluid for the blood flow.

Due to recent progress in the fractional calculus, some new concepts and definitions of derivatives and integrals are presented. On this end, different types of operators are classified for the integrals and derivatives. Among such operators, the concept of fractional calculus is also novel due to local and nonlocal Kernels. The knowledge about the local and nonlocal characteristics of kernels and operators are significantly reported via fractional calculus. The researchers provided many different and novel explanations about the definitions of fractional operators. Among this, the expressions for the Caputo and Fabrizio is very interesting. This definition of fractional derivatives presents the justification for the fractional operator of nonsingular kernel and about nonlocal phenomenon. Shah and Khan [21] implemented the fractional approach based on the Caputo–Fabrizio approach for the oscillating flow of the vertical plate due to second-grade material. Asjad et al. [22] used the Caputo–Fabrizio (CF) technique for the Newtonian heating flow of the viscoelastic fluid, and simulations were compared with the previous investigation. Raza et al. [23] focused on the utilization of the CF approach for the Burgers flow problem with time-dependent oscillations. The CF analytical determination for the second-grade fluid flow phenomenon has been observed by Siddique et al. [24]. Alshabanat et al. [25] used the modified concept of the CF approach for the electrical circuit problem. Anwar et al. [26] addressed the role of magnetic force along with the Newtonian heating effects toward the Oldroyd-B fluid flow via Caputo–Fabrizio approach. Khan et al. [27] performed a comparative fractional investigation for the slip flow of viscous fluid due to the radiative flow. Some more work on this topic can be addressed via refs. [28–33].

The concern of current research is to examine the hybrid nanofluid heating onset with the suspension of water and kerosene oil base liquids due to the oscillating plate. The thermal aspect of the hybrid nanofluid model is supported with the interaction of graphene oxide and molybdenum disulphide nanoparticles. The fractional simulations via Caputo fractional simulations with modified definitions of Laplace techniques are performed. The prime objectives of this continuation in the summarized form are as follows:

Inspect the comparative thermal applications of hybrid nanofluid for the water and kerosene oil base materials.
The thermal stability and efficiencies of both water and kerosene oil base fluids by interacting the graphene oxide and molybdenum disulphide nanoparticles have been theoretically evaluated.
The fractional approach based on the CF fractional simulations is considered.
The implementation of most modified definitions of ρ-Laplace transformations has been suggested with updated definitions. The integral tasks for the Caputo fractional approach are carried out by ρ-Laplace definitions.
The concept of Mittage–Leffler functions for the Caputo fractional approach in view of ρ-Laplace transformations is presented.
Based on the literature review, it is intended that different fractional studies regarding the thermal determination of nanofluid are available in the literature; however, the new fractional concept with ρ-Laplace transforms for such nanoparticles is not investigated in the existing literature.

2 Problem formulation

Consider a free convection viscoelastic second-grade nanofluid mixed with molybdenum disulfide and graphene-oxide nanoparticles with water and kerosene oil as base fluid via an exponentially vertical plate. The induced inclined plate is isothermally infinite whose mass and diffusion rate changes in the presence of heat absorption. Furthermore, the flowing fluid is considered electrically conducting and the value of the Reynolds number is very small compared to the transverse magnetic field. First, at t = 0, the static position of plate and nanoparticles is observed. Then, at t > 0⁺, the inclined plate starts to vibrate with constant velocity U o H ( t ) e − i ω t . The set of modeled equations for the hybrid nanofluid problem are as follows [1]:

(1) ρ nf ∂ v ( ζ , t ) ∂ t − β 1 * ∂ 3 v ( ζ , t ) ∂ t ∂ ζ 2 − μ nf ∂ 2 v ( ζ , t ) ∂ ζ 2 = g ( ρ β T ) nf ( T ( ζ , t ) − T ∞ ) + g ( ρ β C ) nf ( C ( ζ , t ) − C ∞ ) ,

(2) ( ρ C p ) n f ∂ T ( ζ , t ) ∂ t = κ n f ∂ 2 T ( ζ , t ) ∂ ζ 2 ,

(3) ∂ C ( ζ , t ) ∂ t = D n f ∂ 2 C ( ζ , t ) ∂ ζ 2 ,

subjected to the following conditions:

v ( ζ , 0 ) = 0 , T ( ζ , 0 ) = T ∞ , C ( ζ , 0 ) = C ∞ ; ζ ≥ 0 ,

v ( 0 , t ) − h ∂ v ( ζ , t ) ∂ ζ ζ = 0 = U 0 H ( t ) e − i ω t , T ( 0 , t ) = T w + ( T w − T ∞ ) t p , C ( 0 , t ) = C w + ( C w − C ∞ ) t p ,

v ( ζ , t ) → 0 , T ( ζ , t ) → T ∞ , C ( ζ , t ) → C ∞ ; ζ → ∞ , t > 0 .

where, ρ n f is the density, μ nf is the viscosity, σ n f is the electrical conductivity, ( ρ β T ) n f is the thermal expansion, ( ρ C p ) n f is the specific heat, and κ nf is the thermal conductivity. The thermal onset of these physical quantities is presented in Table 1.

Table 1

The thermal characteristics of blood, water single carbon-nanotubes, and multiwall carbon nanotubes

Material	Water	Kerosene oil	GO	MoS₂
ρ ( kg / m 3 )	997.1	78.3	1,800	5,060
C p ( J / kg K )	4,179	2,090	717	397.21
k ( W/m K )	0.613	0.145	5,000	904.4
β T × 10 5 ( K − 1 )	21	99	0.284	2.8424

Now introducing the nondimensional constraints as follows:

v ⁎ = v v o , ζ ⁎ = U o υ f ζ , t ⁎ = U o 2 υ f t , T ⁎ ( ζ , t ) = T − T ∞ T w − T ∞ , C ⁎ ( ζ , t ) = C − C ∞ C w − C ∞ ,

Pr = μ C p κ f , Gr = g ( υ β T ) f ( T w − T ∞ ) U o 3 , Gr = g ( υ β ϕ ) f ( C w − C ∞ ) U o 3 ,

M = v f σ f B o 2 ρ f U o 3 , β 1 = α 1 ⁎ U o 2 ρ f υ f Ω o , Sc = υ f D f .

The transformed set with dimensionless parameters is expressed as follows:

(4) ∂ v ( ζ , t ) ∂ t − λ 4 ∂ 2 v ( ζ , t ) ∂ ζ 2 − β 1 ∂ 3 v ( ζ , t ) ∂ t ∂ ζ 2 = λ 4 G r T ( ζ , t ) + λ 5 G m C ( ζ , t ) ,

(5) Pr ∂ T ( ζ , t ) ∂ t = Ω 3 ∂ 2 T ( ζ , t ) ∂ ζ 2 ,

(6) Sc ∂ C ( ζ , t ) ∂ t = Ω 4 ∂ 2 C ( ζ , t ) ∂ ζ 2 .

With consistent transformed conditions:

(7) v ( ζ , 0 ) = 0 , T ( ζ , 0 ) = 0 , C ( ζ , 0 ) = 0 , ζ ≥ 0 ,

(8) v ( 0 , t ) − h ∂ v ( ζ , t ) ∂ ζ ζ = 0 − H ( t ) e − i ω t = 0 , T ( 0 , t ) = t p , C ( 0 , t ) = t p ,

(9) v ( ζ , t ) → 0 , T ( ζ , t ) → 0 , C ( ζ , t ) → 0 ; ζ → ∞ , t > 0 ,

where

λ o = ( 1 − φ f ) + φ ρ s ρ f , λ 1 = μ n f 1 − φ , λ 2 = ( 1 − φ ) + φ ( ρ B T ) s ( ρ B T ) f ,

λ 3 = ( 1 − φ ) + φ ( ρ B ϕ ) s ( ρ B ϕ ) f , λ 4 = σ h n f σ f , λ 5 = λ 2 λ o , λ 6 = λ 3 λ o ,

Ω 1 = ( 1 − φ ) + φ ( ρ C p ) s ( ρ C p ) f , Ω 2 = κ nf κ f , Ω 3 = Ω 2 Ω 1 , Ω 4 = ( 1 − φ ) .

Performing the fractional definitions:

(10) D t β , ρ CF v ( ζ , t ) = λ 4 ∂ 2 v ( ζ , t ) ∂ ζ 2 + β 1 D t ζ , ρ CF ∂ 2 v ( ζ , t ) ∂ ζ 2 + λ 5 G r T ( ζ , t ) + λ 6 G m C ( ζ , t ) ,

(11) Pr CF D t β , ρ T ( β , t ) = Ω 3 ∂ 2 T ( ζ , t ) ∂ ζ 2 ,

(12) Sc CF D t β , ρ C ( ζ , t ) = Ω 4 ∂ 2 C ( ζ , t ) ∂ ζ 2 ,

where CF D t β , ρ is CF-time fractional operator [2] (Table 2).

Table 2

Thermophysical properties

Quantity	Mathematical representation
Effective density	ρ n f = ( 1 − φ ) ρ f + φ ρ s
Dynamic viscosity	μ n f = μ f ( 1 − φ ) 2.5
Electrical conductivity	σ n f = 1 + 3 σ s σ f − 1 φ σ s σ f + 2 − σ s σ f − 1 φ σ f
Thermal expansion	( ρ β T ) n f = ( 1 − φ ) ( ρ β T ) f + φ ( ρ β T ) s
Heat capacitance	( ρ C p ) n f = ( 1 − φ ) ( ρ C p ) f + φ ( ρ C p ) s
Thermal conductivity	k f = ( k CNTs − 2 φ ( k f − k CNTs ) + 2 k f ) ( k CNTs + 2 φ ( k f − k CNTs ) + 2 k f ) k f
Electrical conductivity	σ n f = σ f 1 + 3 σ s σ f − 1 φ σ s σ f + 2 − σ s σ f − 1 φ

3 Basic preliminaries

Definition 1

Let f ( t ) : [ 0 , ∞ ) → ℛ be a real-valued function and then ρ-Laplace transform of h(t) is defined by [3]

ℒ ρ { h ( t ) } ( q ) = ∫ ∞ 0 exp − q t p ρ h ( t ) t 1 − ρ d t .

Definition 2

Let h ( t ) : [ 0 , ∞ ) → ℛ be a real-valued function such that its ρ-Laplace transform exists [3]. Then

ℒ ρ { h ( t ) } ( q ) = ℒ { h ( ρ t ) ρ − 1 } ( q )

Here, ℒ { h ( t ) } is the ordinary Laplace of the defined function f(t).

Definition 3

(Convolution theorem) Let η and ζ be two piecewise continuous functions at each interval [ 0 , τ ] and of exponential order, then their ρ* convolution product of η and ζ is given by [3]

( η ⁎ ζ ) ( τ ) = ∫ τ 0 η ( τ ρ − t ρ ) − ρ ζ ( τ ) t 1 − ρ .

Then the commutatively of the ρ* convolution of two functions is expressed as follows:

( η ⁎ ζ ) ( τ ) = ( ζ ⁎ η ) ( τ ) .

Definition 4

Mittage–Leffler function

The Mittag–Leffler function is the generalized form of the exponential functions and plays a key role in the field of fractional order calculus. The mathematical expression for the Mittag–Leffler function for a single variable is given by

(13) E δ ( z ) = ∑ l = 0 ∞ z l Γ ( l δ + 1 ) , z ∈ ℂ , ℜ ( δ ) > 0 .

In the more general form, the Mittag–Leffler form is given by

(14) E δ , γ ( z ) = ∑ l = 0 ∞ z l Γ ( l δ + γ ) , z ∈ ℂ , ℜ ( δ ) > 0 .

From equations (13) and (14), the relation E δ , 1 ( z ) = E δ ( z ) is obvious.

Definition 5

The generalized CF-fractional operator is given, respectively, as follows [3]:

(15) ( ℘ β , ρ a C F h ) ( τ ) = 1 1 − γ ∫ 0 τ exp τ ρ − υ ρ ρ n − β − 1 h ( υ ) d υ υ 1 − ρ .

ρ-Laplace transform of some elementary functions:

(16) I . £ ρ { 1 } ( q ) = 1 q , q > 0 ,

(17) II . £ ρ { τ ρ } ( q ) = ρ p ρ Γ 1 + p ρ q 1 + p ρ , p ∈ ℜ , q > 0 ,

(18) III . £ ρ exp λ τ ρ ρ ( q ) = 1 q − λ , q > λ .

4 Solution of the problem

In this section, the solution of governing equations (concentration, temperature, and velocity profiles) will be evaluated by using ρ-Laplace transformation scheme.

4.1 Solution of the concentration profile

By employing the ρ-Laplace transform technique on the nonlocal concentration equation given in equation (12), we obtain

Sc s ρ ( 1 − β ) s ρ + β ℒ { C ( ζ , t ) } − ∑ n − 1 k = 0 s − k − 1 ( I k C ) ( 0 ) = Ω 4 ∂ 2 C ¯ ( ζ , s ) ∂ ζ 2 .

With

C ¯ ( 0 , s ) = ρ p / ρ Γ 1 + p ρ s 1 + p ρ , C ¯ ( ∞ , s ) = 0 .

Solving summation involved in the differential equation (12) and applying initial and transformed boundary conditions, equation (11) will take the form in ρ-Laplace transformed domain:

(19) C ¯ ( ζ , s ) = ρ p / ρ Γ 1 + p ρ s 1 + p ρ e − ζ S c Ω 4 a 1 s ρ s ρ + a 2 ,

where a 1 = 1 1 − β , a 2 = β a 2 , and the inverse of equation (19), which are tacked via Stehfest and Tzou’s techniques. The numerical values are presented in Table 3.

Table 3

Statistical analysis of concentration, temperature, and velocity profile by both numerical algorithms

ζ	C ( ζ , t )		T ( ζ , t )		v ( ζ , t )
ζ	Stehfest	Tzou’s	Stehfest	Tzou’s	Stehfest	Tzou’s
0.1	3.6173	3.5727	4.0490	4.0089	4.7115	4.6955
0.3	2.8129	2.7783	3.2589	3.2267	4.5181	4.5011
0.5	2.1865	2.1596	2.6202	2.5947	4.2010	4.1844
0.7	1.6987	1.6779	2.1053	2.0844	3.8289	3.8135
0.9	1.3191	1.3029	1.6891	1.6724	3.4428	3.4289
1.1	1.0238	1.0112	1.3534	1.3400	3.0663	3.0541
1.3	0.7941	0.7843	1.0827	1.0720	2.7126	2.7019
1.5	0.6155	0.6079	0.8646	0.8561	2.3878	2.3786
1.7	0.4767	0.4709	0.6890	0.6822	2.0943	2.0865
1.9	0.3689	0.3644	0.5477	0.5423	1.8320	1.8253

4.2 Solution of the temperature profile

Employing the ρ-Laplace transform technique on the nonlocal energy equation given in equation (11), we obtain (Figure 1).

Pr s ρ ( 1 − β ) s ρ + β ℒ { T ( ζ , t ) } − ∑ n − 1 k = 0 q − k − 1 ( I k T ) ( 0 ) = Ω 3 ∂ 2 T ¯ ( ζ , s ) ∂ ζ 2 .

Figure 1

Physical flow of problem.

With

T ¯ ( 0 , s ) = ρ p / ρ Γ 1 + p ρ s 1 + p ρ , T ¯ ( ∞ , s ) = 0 .

By employing such transformations, the expression for temperature profile is expressed as follows:

(20) T ¯ ( y , s ) = ρ p / ρ Γ 1 + p ρ s 1 + p ρ e − ζ P r Ω 3 a 1 s ρ s ρ + a 2 .

4.3 Solution of the velocity field

Applying the ρ-Laplace technique on equation (10) yield:

(21) λ 4 + β 1 a 1 s ρ s ρ + a 2 ∂ 2 v ¯ ( ζ , s ) ∂ ζ 2 − a 1 s ρ s ρ + a 2 v ¯ ( ζ , s ) = − λ 5 Gr T ¯ ( ζ , s ) − λ 8 Gm C ¯ ( ζ , s ) = 0 .

With transformed conditions:

v ¯ ( 0 , s ) − h ∂ v ¯ ( ζ , s ) ∂ ζ ζ = 0 − ρ 1 + ω ρ ρ s + ω = 0 , v ¯ ( ζ , s ) → 0 as ζ → ∞ ,

The solution is expressed as follows:

(22) v ¯ ( ζ , s ) = 1 1 + h g 1 ( s ) λ 4 + β 1 g 1 ( s ) λ 5 Gr ρ p / ρ Γ 1 + p ρ 1 + h Pr Ω 3 g 1 ( s ) s 1 + p ρ Pr Ω 3 g 1 ( s ) − g 1 ( s ) λ 4 + β 1 g 1 ( s ) + λ 6 Gm ρ p / ρ Γ 1 + p ρ 1 + h Sc Ω 4 g 1 ( s ) s 1 + p ρ Sc Ω 4 g 1 ( s ) − g 1 ( s ) λ 4 + β 1 g 1 ( s ) + ρ 1 + ω ρ ρ s + ω e − ζ g 1 ( s ) λ 4 + β 1 g 1 ( s ) − λ 5 Gr ρ p / ρ Γ 1 + p ρ e − ζ Pr Ω 3 g 1 ( s ) s 1 + p ρ Pr Ω 3 g 1 ( s ) − g 1 ( s ) λ 4 + β 1 g 1 ( s ) − λ 6 Gm ρ p / ρ Γ 1 + p ρ e − y Sc Ω 4 g 1 ( s ) s 1 + p ρ Sc Ω 4 g 1 ( s ) − g 1 ( s ) λ 4 + β 1 g 1 ( s ) ,

where

g 1 ( q ) = a 1 s ρ s ρ + a 2 .

The Laplace transform is expressed as follows:

w ( ξ , t ) = ln ( 2 ) t ∑ N n = 1 v n w ¯ ξ , n ln ( 2 ) t ,

where N is a positive integer, and

v n = ( − 1 ) n + N 2 ∑ r = q + 1 2 min q , N 2 r N 2 ( 2 r ) ! N 2 − r ! r ! ( r − 1 ) ! ( q − r ) ! ( 2 r − q ) ! ,

and

w ( ξ , t ) = e 4.7 t 1 2 w ¯ r , 4.7 t + Re ∑ N j = 1 ( − 1 ) k w ¯ r , 4.7 + k π i t .

5 Results and discussion

In this article, the molybdenum disulfide and graphene-oxide mixed nanofluid theoretical flow work is performed for fractional technique namely CF-fractional derivative. The numerical solution of governed equations is obtained by ρ-Laplace technique is worked out. The physical onset in view of modeled parameters is observed in this section. Figure 2(a) reports the thermal enhancement of kerosene oil and water base material when fractional parameter β gets varies. The thermal impact of both base materials declined for β. However, the change in thermal properties of kerosene oil is more progressive compared to water. Figure 2(a) explores the thermal observations for the temperature profile for kerosene oil and water liquid when the maximum value assigned to the Prandtl number. The results predict the temperature profile controls for the Prandtl number. The results for concentration profile due to enhancing β are determined in Figure 3(a). The declining convective transport of concentration profile for kerosene oil and water materials is noticed. To inspect the response of the fluid motion against the Schmidt number Sc Figure 3(b) has been plotted. A declination in the velocity field has been viewed against the higher values of Sc. This is due to the fact that Sc has direct contact with the viscosity of the fluid and opposite relation with diffusion, and because of this relations, declination of the field is observed for higher values. Figure 4(a) and (b) are plotted to inspect the behavior of fractional constraint and second-grade fluid parameters on momentum profile for both types of nanofluids. Figure 4(b) is plotted for the impact of the second-grade fluid parameter β ₁ on the velocity of different types of nanofluid. The figure shows the acceleration in the assumed nanofluid against β ₁. This is because of the inverse contact of β ₁ to the kinematic viscosity of the fluid; therefore, as the magnitude of β ₁ increases, the viscous forces in the based fluid decrease, which leads to a boost up the fluid motion. The change in the profile of velocity due to thermal Grashof number Gr and mass Grashof number Gm has been explored in Figure 5(a) and (b). The increasing effects on velocity due to Gr and Gm are preserved. This enhancing behavior is due to the presence of larger buoyancy forces.

Figure 2

(a and b) Effects of β, Pr for temperature profile.

Figure 3

(a and b) Effects of β, Sc on concentration profile.

$Figure 4 (a and b) Change in velocity for (a) fractional parameter β and (b) second-grade fluid parameter β 1 for velocity field with Pr = 5.6, Gr = 4.7, Gm = 4.2, Sc = 3.9. t = 1.3, φ = 0.03.$

Figure 4

(a and b) Change in velocity for (a) fractional parameter β and (b) second-grade fluid parameter β ₁ for velocity field with Pr = 5.6, Gr = 4.7, Gm = 4.2, Sc = 3.9. t = 1.3, φ = 0.03.

Figure 5

(a and b) Change in velocity for (a) hear Grashof number Gr and (b) mass Grashof number Gm for velocity field with β = 0.5, Pr = 5.6, Sc = 3.9. t = 1.3, φ = 0.03.

The variations in fluid behavior in the reaction of volume fraction of both fluids blood and water-based nanofluid have been plotted in Figure 6(a). Retardation in the fluid motion is noticed when the magnitude of ϕ reaches 0.04. The magnitude of the nanoparticles causes the mass and viscous forces in the fluid to increase, causing the fluid to move more slowly. In Figure 6(b), we have evaluated the influence of ρ on velocity field while keeping other parameters constant. From this figure, it can be concluded that the velocity field is the increasing function of order ρ. As the order ρ raises, the velocity field shows augmentation. The graphical comparison for suspension of water and kerosene oil base fluids with graphene oxide nanoparticles and water and kerosene oil base materials with the interaction of graphene oxide and molybdenum disulphide nanoparticles has been presented in Figure 7(a). The change in the velocity for water–graphene oxide suspension is more enlacing. However, the lower velocity profile is claimed for the combination of kerosene oil and molybdenum disulphide nanoparticles. The comparative analysis for the Laplace technique like Stehfest and Tzous schemes is presented in Figure 7(b). A fine accuracy of simulations for both techniques is observed. The numerical onset for concentration, temperature, and the velocity profile is tabulated by Stehfest and Tzous numerical techniques in Table 3. The numerical values of concentration, temperature, and velocity profiles declines for ζ Moreover, the obtained values from both techniques are almost identical. Furthermore, the numerical investigation rate of Nusselt number, Sherwood number, and skin friction are analyzed in Table 4. The increasing Nusselt number numerical values are results for β.

Figure 6

(a and b) Change in temperature for (a) φ and (b) ρ with β = 0.5, Pr = 5.6, Gr = 4.7, Gm = 4.2, Sc = 3.9, t = 1.3.

Figure 7

(a and b) Comparison of (a) different base fluid parameters and (b) comparison of numerical schemes.

Table 4

Numerical analysis for Sh, Nu, and C _f by increasing

β	Sh	Nu	C _f
0.1	1.5487	2.2309	1.5304
0.2	1.6409	2.3734	1.5746
0.3	1.7490	2.5356	1.6197
0.4	1.8789	2.7292	1.6657
0.5	2.0398	2.9718	1.7129
0.6	2.2477	3.2927	1.7607
0.7	2.5319	3.7469	1.8107
0.8	2.9537	4.4549	1.8639
0.9	3.6646	5.7370	1.9224

6 Conclusions

Caputo–Fabrizio fractional operator has been applied to fractionalize the viscoelastic second-grade nanofluid on an erect exponential type oscillating vertical flat plate. The solutions for generalized governing equations have been obtained by using ρ-Laplace transform technique, which is more general than the ordinary Laplace transform technique. For graphical analysis, the graphs of different parameters have been plotted through mathematical software and elaborated physically. The key findings of the present analysis are listed as follows:

The change in temperature profile is lower for fractional parameter and Prandtl number for both kerosene oil and water base liquids.
The lower concentration profile for kerosene oil and water base fluids is results for larger fractional parameter.
The increase in velocity has been observed for second-grade fluid parameter. However, the increasing effects are comparatively smaller for water base fluid.
With a larger increment of thermal Grashof number and mass Grashof number, the improvement in velocity is observed.
The change in the velocity for water–graphene oxide suspension is more impressive compared to kerosene oil and molybdenum disulphide nanoparticles.

Funding information: Dr. Hanaa Mohamed Said would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No. R-2022-102. Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R163), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4310392DSR21.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. ASME-Publ-Fed. 1995;231:99–106.Suche in Google Scholar

[2] Shehzad SA, Khan SU, Abbas Z, Rauf A. A revised Cattaneo-Christov micropolar viscoelastic nanofluid model with combined porosity and magnetic effects. Appl Math Mech. 2020;41:521–32.10.1007/s10483-020-2581-5Suche in Google Scholar

[3] Turkyilmazoglu M. On the transparent effects of Buongiorno nanofluid model on heat and mass transfer. Eur Phys J Plus. 2021;136:376.10.1140/epjp/s13360-021-01359-2Suche in Google Scholar

[4] Nadeem S, Fuzhang W, Alharbi FM, Sajid F, Abbas N, El-Shafay AS, Al-Mubaddel FS. Al-Mubaddel, Numerical computations for Buongiorno nano fluid model on the boundary layer flow of viscoelastic fluid towards a nonlinear stretching sheet. Alexandria Eng J. February 2022;61(2):1769–78.10.1016/j.aej.2021.11.013Suche in Google Scholar

[5] Abdelmalek Z, Khan SU, Awais M, Mustfa MS. Analysis of generalized micropolar nanofluid with swimming of microorganisms over an accelerated surface with activation energy. J Thermal Anal Calorimetry. 2021;144:1051–63.10.1007/s10973-020-09474-5Suche in Google Scholar

[6] Mahanthesh B, Srikantha N, Mackolil J. A study on heat transfer in three-dimensional nonlinear convective boundary layer flow of nanomaterial considering the aggregation of nanoparticles. Heat Transfer Asian Res. January 2022;51(1):891–908.10.1002/htj.22334Suche in Google Scholar

[7] Ramezanizadeh M, Nazari MA, Ahmadi MH, Lorenzini G, Pop I. A review on the applications of intelligence methods in predicting thermal conductivity of nanofluids. J Thermal Anal Calorimetry. 2019;138(1):827–43.10.1007/s10973-019-08154-3Suche in Google Scholar

[8] Khan SU, Al-Khaled K, Bhatti MM. Numerical experiment of Reiner-Philippoff nanofluid flow subject to the higher-order slip features, activation energy, and bioconvection. Partial Differ Equ Appl Math. December 2021;4:100126.10.1016/j.padiff.2021.100126Suche in Google Scholar

[9] Reddy PS, Sreedevi P, Rao KV. Impact of heat generation/absorption on heat and mass transfer of nanofluid over rotating disk filled with carbon nanotubes. Int J Num Methods Heat Fluid Flow. 2021;31(9):2962–85.10.1108/HFF-10-2020-0621Suche in Google Scholar

[10] Abbasi A, Farooq W, Muhammad T, Khan MI, Khan SU, Mabood F, Bibi S. Implications of the third-grade nanomaterials lubrication problem in terms of radiative heat flux: A Keller box analysis. Chem Phys Lett. 16 November 2021;783:139041.10.1016/j.cplett.2021.139041Suche in Google Scholar

[11] Sabu AS, Wakif A, Areekara S, Mathew A, Shah NA. Significance of nanoparticles’ shape and thermo-hydrodynamic slip constraints on MHD alumina-water nanoliquid flows over a rotating heated disk: The passive control approach. Int Commun Heat Mass Transfer. December 2021;129:105711.10.1016/j.icheatmasstransfer.2021.105711Suche in Google Scholar

[12] Dawar A, Wakif A, Thumma T, Shah NA. Towards a new MHD non-homogeneous convective nanofluid flow model for simulating a rotating inclined thin layer of sodium alginate-based Iron oxide exposed to incident solar energy. Int Commun Heat Mass Transfer. January 2022;130:105800.10.1016/j.icheatmasstransfer.2021.105800Suche in Google Scholar

[13] Hassan M, El‐Zahar ER, Khan SU, Rahimi‐Gorji M, Ahmad A. Boundary layer flow pattern of heat and mass for homogenous shear thinning hybrid-nanofluid: An experimental data base modeling. Num Methods Partial Differ Equ. March 2021;37(2):1234–49.10.1002/num.22575Suche in Google Scholar

[14] Al‐Farhany K, Alomari MA, Al‐Saadi A, Chamkha A, Öztop HF, Al‐Kouz W. MHD mixed convection of a Cu–water nanofluid flow through a channel with an open trapezoidal cavity and an elliptical obstacle. Heat Transfer Asian Res. 2022;51(2):1691–710.10.1002/htj.22370Suche in Google Scholar

[15] Song YQ, Hassan M, Khan SU, Khan MI, Qayyum S, Chu YM, Nadeem A. Thermal and boundary layer flow analysis for MWCNT-SiO2 hybrid nanoparticles: An experimental thermal model. Modern Phys Lett B. 2021;35(18):2150303.10.1142/S0217984921503036Suche in Google Scholar

[16] Wahid NS, Md Arifin N, Turkyilmazoglu M, Hafidzuddin ME, Abd Rahmin NA. MHD hybrid Cu-Al2O3/water nanofluid flow with thermal radiation and partial slip past a permeable stretching surface: analytical solution. J Nano Res. 2021;64:75–91.10.4028/www.scientific.net/JNanoR.64.75Suche in Google Scholar

[17] Madhukesh JK, Kumar RN, Gowda RP, Prasannakumara BC, Ramesh GK, Khan MI, et al. Numerical simulation of AA7072-AA7075/water-based hybrid nanofluid flow over a curved stretching sheet with Newtonian heating: A non-Fourier heat flux model approach. J Mol Liquids. August 2021;335(1):116103.10.1016/j.molliq.2021.116103Suche in Google Scholar

[18] Saleem N, Munawar S, Tripathi D. Entropy analysis in ciliary transport of radiated hybrid nanofluid in presence of electromagnetohydrodynamics and activation energy. Case Studies Thermal Eng. December 2021;28:101665.10.1016/j.csite.2021.101665Suche in Google Scholar

[19] Rajesh V, Sheremet MA, Öztop HF. Impact of hybrid nanofluids on MHD flow and heat transfer near a vertical plate with ramped wall temperature. Case Studies in Thermal Eng. December 2021;28:101557.10.1016/j.csite.2021.101557Suche in Google Scholar

[20] Alghamdi W, Alsubie A, Kumam P, Saeed A, Gul T. MHD hybrid nanofluid flow comprising the medication through a blood artery. Sci Rep. 2021;11:11621.10.1038/s41598-021-91183-6Suche in Google Scholar PubMed PubMed Central

[21] Shah NA, Khan I. Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives. Eur Phys J C. 2016;76(7):362.10.1140/epjc/s10052-016-4209-3Suche in Google Scholar

[22] Asjad MI, Shah NA, Aleem M, Khan I. Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison. Eur Phys J Plus. 2017;132(8):340.10.1140/epjp/i2017-11606-6Suche in Google Scholar

[23] Raza N, Awan AU, Haque E, Abdullah M, Rashidi MM. Unsteady flow of a Burgers’ fluid with Caputo fractional derivatives: A hybrid technique. Ain Shams Eng J. 2019;10(2):319–25.10.1016/j.asej.2018.01.006Suche in Google Scholar

[24] Siddique I, Tlili I, Bukhari SM, Mahsud Y. Heat transfer analysis in convective flows of fractional second grade fluids with Caputo-Fabrizio and Atangana-Baleanu derivative subject to Newtonian heating. Mech Time-Depend Mater. 2021;21:291–311.10.1007/s11043-019-09442-zSuche in Google Scholar

[25] Alshabanat A, Jleli M, Kumar S, Samet B. Generalization of caputo-fabrizio fractional derivative and applications to electrical circuits. Front Phys. 2020;8:64.10.3389/fphy.2020.00064Suche in Google Scholar

[26] Anwar T, Kumam P, Thounthong P, Muhammad S, Duraihem FZ. Generalized thermal investigation of unsteady MHD flow of Oldroyd-B fluid with slip effects and Newtonian heating; a Caputo-Fabrizio fractional model. Alexandria Eng J. March 2022;61(3):2188–202.10.1016/j.aej.2021.06.090Suche in Google Scholar

[27] Khan MI, Raza A, Naseem M, Al-Khaled K, Khan SU, Khan MI, et al. Comparative analysis for radiative slip flow of magnetized viscous fluid with mixed convection features: Atangana-Baleanu and Caputo-Fabrizio fractional simulations. Case Studies Thermal Eng. December 2021;28:101682.10.1016/j.csite.2021.101682Suche in Google Scholar

[28] Sene N, Fall AN. Homotopy perturbation ρ-laplace transform method and its application to the fractional diffusion equation and the fractional diffusion-reaction equation. Fractal Fract. 2019;3(2):14.10.3390/fractalfract3020014Suche in Google Scholar

[29] Bhangale N, Kachhia KB, Gómez-Aguilar J. A new iterative method with $$\rho $$ ρ-Laplace transform for solving fractional differential equations with Caputo generalized fractional derivative. Eng Computers. 2020;1–14.10.1007/s00366-020-01202-9Suche in Google Scholar

[30] Jarad F, Abdeljawad T. A modified Laplace transform for certain generalized fractional operators. Results Nonlinear Anal. 2018;1(2):88–98.Suche in Google Scholar

[31] Zakian V. Optimisation of numerical inversion of Laplace transforms. Electron Lett. 1970;6(21):677–9.10.1049/el:19700471Suche in Google Scholar

[32] Zakian V. Properties of IMN approximants. ed. Cambridge: Academic Press Ne3. p. 141–4Suche in Google Scholar

[33] Halsted D, Brown D. Zakian’s technique for inverting Laplace transforms. Chem Eng J. 1972;3:312–3.10.1016/0300-9467(72)85037-8Suche in Google Scholar

Received: 2022-01-09

Revised: 2022-03-15

Accepted: 2022-04-27

Published Online: 2022-08-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/ntrev-2022-0156

Schlagwörter für diesen Artikel

fractional derivatives; hybrid nanofluid; Caputo fractional model; Mittag–Leffler function

Creative Commons

BY 4.0