The dynamical behaviour and thermal transportation feature of mixed convective Casson bi-phasic flows of water-based ternary Hybrid nanofluids with different shapes are examined numerically in a Darcy- Brinkman medium bounded by a vertical elongating slender concave-shaped surface. The mathematical framework of the present flow model is developed properly by adopting the single-phase approach, whose solid phase is selected to be metallic or metallic oxide nanoparticles. Besides, the influence of thermal radiation is taken into consideration in the presence of an internal variable heat generation. A set of feasible similarity transformations are applied for the conversion of the governing PDEs into a nonlinear differential structure of coupled ODEs. An advanced differential quadrature algorithm is employed herein to acquire accurate numerical solutions for momentum and energy equations. Results of the conducted parametric study are explained and revealed in graphs using bvp5c in MATLAB to solve the governing system. The solution with three mixture compositions is provided (Type-I and Type-II). Al2O3 (Platelet), GNT (Cylindrical), and CNTs (Spherical), Type-II mixture of copper (Cylindrical), silver (Platelet), and copper oxide (Spherical). In comparison to Type-I ternary combination Type-II ternary mixtures is lesser in terms of the temperature distribution. The skin friction coefficient is more in Type-1 compared to Type-2.
Citation: Kiran Sajjan, N. Ameer Ahammad, C. S. K. Raju, M. Karuna Prasad, Nehad Ali Shah, Thongchai Botmart. Study of nonlinear thermal convection of ternary nanofluid within Darcy-Brinkman porous structure with time dependent heat source/sink[J]. AIMS Mathematics, 2023, 8(2): 4237-4260. doi: 10.3934/math.2023211
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The dynamical behaviour and thermal transportation feature of mixed convective Casson bi-phasic flows of water-based ternary Hybrid nanofluids with different shapes are examined numerically in a Darcy- Brinkman medium bounded by a vertical elongating slender concave-shaped surface. The mathematical framework of the present flow model is developed properly by adopting the single-phase approach, whose solid phase is selected to be metallic or metallic oxide nanoparticles. Besides, the influence of thermal radiation is taken into consideration in the presence of an internal variable heat generation. A set of feasible similarity transformations are applied for the conversion of the governing PDEs into a nonlinear differential structure of coupled ODEs. An advanced differential quadrature algorithm is employed herein to acquire accurate numerical solutions for momentum and energy equations. Results of the conducted parametric study are explained and revealed in graphs using bvp5c in MATLAB to solve the governing system. The solution with three mixture compositions is provided (Type-I and Type-II). Al2O3 (Platelet), GNT (Cylindrical), and CNTs (Spherical), Type-II mixture of copper (Cylindrical), silver (Platelet), and copper oxide (Spherical). In comparison to Type-I ternary combination Type-II ternary mixtures is lesser in terms of the temperature distribution. The skin friction coefficient is more in Type-1 compared to Type-2.
The classical Hermite-Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation and it has many applications. This inequality may be regarded as a refinement of the concept of convexity. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable refinements and generalizations have been studied [1,2].
The importance of the study of set-valued analysis from a theoretical point of view as well as from their applications is well known. Many advances in set-valued analysis have been motivated by control theory and dynamical games. Optimal control theory and mathematical programming were an engine driving these domains since the dawn of the sixties. Interval analysis is a particular case and it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena.
Furthermore, a few significant inequalities like Hermite-Hadamard and Ostrowski type inequalities have been established for interval valued functions in recent years. In [3,4], Chalco-Cano et al. established Ostrowski type inequalities for interval valued functions by using Hukuhara derivatives for interval valued functions. In [5], Román-Flores et al. established Minkowski and Beckenbach's inequalities for interval valued functions. For other related results we refer to the readers [6].
In this paper, we establish Hermite-Hadamard type inequalities and He's inequality for interval-valued exponential type pre-invex functions in the Riemann-Liouville interval-valued fractional operator settings.
We begin with recalling some basic concepts and notions in the convex analysis.
Let the space of all intervals of ℜ is ℜc and Λ∈ℜc given by
Λ1=[Λ↔,↔Λ]={v∈ℜ|Λ↔<v<↔Λ},Λ↔,↔Λ∈ℜ. |
Various binary operations are given as follows [7]:
Scalar multiplication: τ∈ℜ,
τ.Λ1={[τΛ↔,τ↔Λ],if0≤τ,0,ifτ=0,[τ↔Λ,τΛ↔],ifτ≤0. |
Difference, addition, product and reciprocal for Λ1,Λ2∈ℜc are respectively given by
Λ1−Λ2=[Λ1↔,↔Λ1]−[Λ2↔,↔Λ2]=[Λ1↔−Λ2↔,↔Λ1−↔Λ2],Λ1+Λ2=[Λ1↔,↔Λ1]+[Λ2↔,↔Λ2]=[Λ1↔+Λ2↔,↔Λ1+↔Λ2],Λ1×Λ2=[min{Λ1↔Λ2↔,↔Λ1Λ2↔,Λ1↔↔Λ2,↔Λ1↔Λ2},max{Λ1↔Λ2↔,↔Λ1Λ2↔,Λ1↔↔Λ2,↔Λ1↔Λ2}]={uv|u∈Λ1,v∈Λ2},1Λ={1v1|0≠v1∈Λ}=[1Λ↔,1↔Λ],Λ1.1Λ2={u.1v|u∈Λ1,0≠v∈Λ2}=[Λ1↔.1↔Λ2,↔Λ1.1↔Λ2]. |
Let ℜΛ,ℜ+Λandℜ−Λ denote the collection of all closed intervals of ℜ, the collection of all positive intervals of ℜ and the collection of all negative intervals of ℜ respectively. In this paper, we examine a few algebraic properties of interval arithmetic.
Definition 2.1. [7] A mapping Ω is called an interval-valued function of υ on [a1,b1] if it assigns a nonempty interval to every v∈[a1,b1], that is
Ω(v)=[↔Ω(v),Ω↔(v)], | (2.1) |
where ↔Ω(υ)andΩ↔(υ) are both real valued functions.
Consider any finite ordered subset ∁ be the partition of [a1,b1], that is
∁:a1=a1,...,an=b1. |
The mesh of ∁ is
mesh(∁)=max{ai+1−ai;i=1,...,n}. |
The Riemann sum of Ω:[a1,b1]→ℜΛ can be defined by
˜S(Ω,∁,c)=Σni=1Ω(di)(ai+1−ai), |
where mesh(∁)<c.
Definition 2.2. [8] A mapping Ω:[a1,b1]→ℜΛ is called an interval-Riemann integrable on [a1,b1]if∃Λ∈ℜΛ such that for every δ>0 satisfying
d(˜S(Ω,∁,c),Λ)<δ, |
we have
Λ1=(IR)∫b1a1Ω(v)dv. | (2.2) |
Lemma 2.1. [9] Let Ω:[a1,b1]→ℜΛ be an interval-valued function as in (2.1), then it is interval-Riemann integrable if and only if
(IR)∫b1a1Ω(v)dv=[(R)∫b1a1↔Ω(v)dv,(R)∫b1a1Ω↔(v)dv]. |
In simple words, Ω is interval-Riemann integrable if and only if ↔Ω(v)andΩ↔(v) are both Riemann integrable functions.
Definition 2.3. [10] Let Ω∈L1[a1,b1], then the Riemann-Liouville fractional integrals of order m>0 with 0≤a1 are defined by
Ima+1Ω(v)=1Γ(m)∫va1(v−r)m−1Ω(r)dr,v>a1, | (2.3) |
Imb−1Ω(v)=1Γ(m)∫b1v(r−v)m−1Ω(r)dr,v<b1. | (2.4) |
Definition 2.4. [11,12] Let Ω:[a1,b1]→ℜΛ be an interval-valued, interval-Riemann integrable function as in (2.1), then the interval Riemann-Liouville fractional integrals of order m>0 with 0≤a1 are defined by
Ima+1Ω(v)=1Γ(m)(IR)∫va1(v−r)m−1Ω(r)dr,v>a1, | (2.5) |
Imb−1Ω(v)=1Γ(m)(IR)∫b1v(r−v)m−1Ω(r)dr,v<b1. | (2.6) |
Corollary 2.1. [12] Let Ω:[a1,b1]→ℜΛ be an interval-valued function as in (2.1) such that ↔Ω(v)andΩ↔(v) are Riemann integrable functions, then
Ima+1Ω(v)=[Ima+1Ω↔(v),Ima+1↔Ω(v)], |
Imb−1Ω(v)=[Imb−1Ω↔(v),Imb−1↔Ω(v)]. |
Definition 2.5. [13] A set Λ⊂ℜn with respect to a vector function η:ℜn×ℜn→ℜn is called an invex set if
b1+τη(a1,b1)∈Λ,∀a1,b1∈Λ,τ1∈[0,1]. |
Definition 2.6. [13] A function Ω on the invex set Λ with respect to a vector function η:Λ×Λ→ℜn is called pre-invex function if
Ω(b1+τη(a1,b1))≤(1−τ)Ω(b1)+τΩ(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.7) |
Lemma 2.2. [14,15] If Λ is open and η:Λ×Λ→ℜ, then ∀a1,b1∈Λ,τ,τ1,τ2∈[0,1], we have
η(b1,b1+τη(a1,b1))=−τη(a1,b1), | (2.8) |
η(a1,b1+τη(a1,b1))=(1−τ)η(a1,b1), | (2.9) |
η(b1+τ2η(a1,b1),b1+τ1η(a1,b1))=(τ2−τ1)η(a1,b1). | (2.10) |
In [16], Noor presented Hermite-Hadamard-inequality for pre-invex function, as follows:
Ω(2a1+η(b1,a1)2)≤1η(b1,a1)∫a1+η(b1,a1)a1Ω(v)dv≤Ω(a1)+Ω(b1)2. |
Definition 2.7. [15] Let us consider an interval-valued function Ω on the set Λ, then Ω is pre-invex interval valued function with respect to η on an invex set Λ⊂ℜn with respect to a vector function η:Λ×Λ→ℜn if
Ω(b1+τ1η(a1,b1))⊇(1−τ1)Ω(b1)+τ1Ω(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.11) |
Taking motivation from the exponential type convexity proposed in [17], we introduce the following notion:
Definition 2.8. A function Ω on the invex set Λ is called exponential-type pre-invex function with respect to a vector function η:Λ×Λ→ℜn if
Ω(b1+τ1η(a1,b1))≤(e(1−τ1)−1)Ω(b1)+(eτ1−1)Ω(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.12) |
It is important to note that a pre-invex function need not to be convex function. For example, the function f(x)=−|x| is not a convex function but it is a pre-invex function with respect to η, where
η(v,u)={u−v,ifu≤0,v≤0,v≥0,u≥0,v−u,otherwise. |
Theorem 2.1. Let Ω:[a1,b1]→ℜ be an exponential-type pre-invex function with respect to a vector function η:Λ×Λ→ℜn. If a1<b1 and Ω∈L[a1,b1], then we have
12(e12−1)Ω(a1+12η(b1,a1))≤1η(b1,a1)∫a1+η(b1,a1)a1Ω(v)dv≤(e−2)[Ω(a1)+Ω(b1)]. |
Proof. At first, from exponential-type-pre-invexity of Ω, we have
Ω(a1+12η(b1,a1))=Ω(12[b1+τ1η(a1,b1)]+12[a1+τ1η(b1,a1)])≤(e12−1)[Ω(b1+τ1η(a1,b1))+Ω(a1+τ1η(b1,a1))]. |
Integrating the above inequality with respect to τ1∈[0,1] yields
Ω(a1+12η(b1,a1))≤(e12−1)(∫10Ω(b1+τ1η(a1,b1))dτ1+∫10Ω(a1+τ1η(b1,a1))dτ1)=2(e12−1)η(b1,a1)∫a1+η(b1,a1)a1Ω(v)dv. |
Now, taking v=b1+τ1η(a1,b1) gives
1η(b1,a1)∫a1+η(b1,a1)a1Ω(v)dv=∫10Ω(b1+τ1η(a1,b1))dτ1≤∫10{(eτ1−1)Ω(a1)+(e(1−τ1)−1)Ω(b1)}dτ1=(e−2)[Ω(a1)+Ω(b1)]. |
This completes the proof.
By merging the concepts of pre-invexity and exponential type pre-invexity, we propose the following notion:
Definition 2.9. Let Λ⊂ℜn be an invex set with respect to a vector function η:Λ×Λ→ℜn. The interval valued function Ω on the set Λ is exponential-type pre-invex interval valued function with respect to η if
Ω(b1+τ1η(a1,b1))⊇(e(1−τ1)−1)Ω(b1)+(eτ1−1)Ω(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.13) |
Remark 2.1. In Definition 2.9, by taking h(τ1)=eτ1−1, where h:[0,1]⊂[a1,b1]→ℜ and h≠0, then we get h-pre-invex interval valued function with respect to η, that is
Ω(b1+τ1η(a1,b1))⊇h(1−τ1)Ω(b1)+h(τ1)Ω(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.14) |
Remark 2.2. Let Λ⊂ℜn be an invex set with respect to a vector function η:ℜn×ℜn→ℜn. The interval valued function Ω on the set Λ is exponential-type-pre-invex function with respect to η if and only if ↔Ω,Ω↔ are exponential-type pre-invex functions with respect to η, that is
↔Ω(b1+τ1η(a1,b1))≤(e(1−τ1)−1)↔Ω(b1)+(eτ1−1)↔Ω(a1),∀a1,b1∈Λ,τ1∈[0,1], | (2.15) |
Ω↔(b1+τ1η(a1,b1))≤(e(1−τ1)−1)Ω↔(b1)+(eτ1−1)Ω↔(a1),∀a1,b1∈Λ,τ1∈[0,1]. | (2.16) |
Remark 2.3. If ↔Ω(v)=Ω↔(v), then we get (2.12).
Remark 2.4. Since τ1≤eτ1−1 and 1−τ1≤e1−τ1−1 for all τ1∈[0,1], so every nonnegative pre-invex interval valued function with respect to η is also exponential-type pre-invex interval valued function with respect to η.
In this section, we establish fractional Hermite-Hadamard type inequality for interval-valued exponential type pre-invex. The family of Lebesgue measurable interval-valued functions is denoted by L([v1,v2],ℜ0).
Theorem 3.1. Let Λ⊂ℜ be an open invex set with respect to η:Λ×Λ→ℜ and a1,b1∈Λ with a1<a1+η(b1,a1). If Ω:[a1,a1+η(b1,a1)]→ℜ is an exponential type pre-invex interval-valued function such that Ω∈L[a1,a1+η(b1,a1)] and m>0, then we have (considering Lemma 2.2 holds)
1(e12−1)Ω(c1+12η(d1,c1))⊇Γ(m+1)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1)+Imc+1Ω(c1+η(d1,c1))]⊇mP(Ω(c1+η(d1,c1))+Ω(c1)), | (3.1) |
where
P=−1(m+1)(−1)m[(em+e)(−1)mΓ(m+1,1)+(−m−1)Γ(m+1,−1)+((−em−e)(−1)m+m+1)Γ(m+1)+2(−1)m]. | (3.2) |
Proof. Since Ω is an exponential type pre-invex interval-valued function, so
1(e12−1)Ω(a1+12η(b1,a1))⊇[Ω(a1)+Ω(b1)]. |
Taking a1=c1+(1−τ1)η(d1,c1) and b1=c1+(τ1)η(d1,c1) gives
1(e12−1)Ω(c1+(1−τ1)η(d1,c1)+12η(c1+(τ1)η(d1,c1),c1+(1−τ1)η(d1,c1)))⊇[Ω(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1))], |
implies
1(e12−1)Ω(c1+12η(d1,c1))⊇[Ω(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1))]. |
By multiplying by τm−11 on both sides and integrating over [0,1] with respect to τ1, we get
(IR)∫10τm−111(e12−1)Ω(c1+12η(d1,c1))dτ1⊇(IR)∫10τm−11[Ω(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1))]dτ1, |
(IR)∫10τm−111(e12−1)Ω(c1+12η(d1,c1))dτ1=[(R)∫10τm−111(e12−1)Ω↔(c1+12η(d1,c1))dτ1,(R)∫10τm−111(e12−1)↔Ω(c1+12η(d1,c1))dτ1], |
(IR)∫10τm−111(e12−1)Ω(c1+12η(d1,c1))dτ1=[1m(e12−1)Ω↔(c1+12η(d1,c1)),1m(e12−1)↔Ω(c1+12η(d1,c1))]=1m(e12−1)Ω(c1+12η(d1,c1)), | (3.3) |
(IR)∫10τm−11Ω(c1+(τ1)η(d1,c1))=[1ηm(d1,c1)(R)∫c1+(τ1)η(d1,c1)c(i−c)m−1Ω↔(i)di,1ηm(d1,c1)(R)∫c1+(τ1)η(d1,c1)c(i−c)m−1↔Ω(i)di], |
(IR)∫10τm−11Ω(c1+(τ1)η(d1,c1))=Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω↔(c1),Im(c1+η(d1,c1))−↔Ω(c1)]=Γ(m)ηm(d1,c1)Im(c1+η(d1,c1))−Ω(c1). | (3.4) |
Similarly
(IR)∫10τm−11Ω(c1+(1−τ1)η(d1,c1))=Γ(m)ηm(d1,c1)[Imc+1Ω↔(c1+η(d1,c1)),Imc+1↔Ω(c1+η(d1,c1))]=Γ(m)ηm(d1,c1)Imc+1Ω(c1+η(d1,c1)). | (3.5) |
From (3.3)–(3.5), we get
1m(e12−1)Ω(c1+12η(d1,c1))⊇Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1)+Imc+1Ω(c1+η(d1,c1))]. | (3.6) |
Now, from the interval valued exponential type pre-invexity of Ω, we have
Ω(c1+τ1η(d1,c1))=Ω(c1+η(d1,c1)+(1−τ1)η(c1,c1+η(d1,c1)))⊇(eτ1−1)Ω(c1+η(d1,c1))+(e(1−τ1)−1)Ω(c1). | (3.7) |
Similarly
Ω(c1+(1−τ1)η(d1,c1))=Ω(c1+η(d1,c1)+(τ1)η(c1,c1+η(d1,c1)))⊇(e(1−τ1)−1)Ω(c1+η(d1,c1))+(eτ1−1)Ω(c1). | (3.8) |
Thus, by adding (3.7) and (3.8), we get
Ω(c1+τ1η(d1,c1))+Ω(c1+(1−τ1)η(d1,c1))⊇[eτ1+e(1−τ1)−2](Ω(c1+η(d1,c1))+Ω(c1)). |
By multiplying by τm−11 on both sides and integrating over [0,1] with respect to τ1, we get
(IR)∫10τm−11Ω(c1+τ1η(d1,c1))dτ1+(IR)∫10τm−11Ω(c1+(1−τ1)η(d1,c1))dτ1⊇(IR)∫10τm−11[eτ1+e(1−τ1)−2](Ω(c1+η(d1,c1))+Ω(c1))dτ1. |
Now, from (3.2) we get
(IR)∫10τm−11[eτ1+e(1−τ1)−2](Ω(c1+η(d1,c1))+Ω(c1))dτ1=[(R)∫10τm−11[eτ1+e(1−τ1)−2](Ω↔(c1+η(d1,c1))+Ω↔(c1))dτ1,(R)∫10τm−11[eτ1+e(1−τ1)−2](↔Ω(c1+η(d1,c1))+↔Ω(c1))dτ1]=[P(Ω↔(c1+η(d1,c1))+Ω↔(c1)),P(↔Ω(c1+η(d1,c1))+↔Ω(c1))]=P(Ω(c1+η(d1,c1))+Ω(c1)). | (3.9) |
Also from (3.4), (3.5) and (3.9), we get
Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1)+Imc+1Ω(c1+η(d1,c1))]⊇P(Ω(c1+η(d1,c1))+Ω(c1)). | (3.10) |
Combining (3.6) and (3.10), we get
1(e12−1)Ω(c1+12η(d1,c1))⊇Γ(m+1)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1)+Imc+1Ω(c1+η(d1,c1))]⊇mP(Ω(c1+η(d1,c1))+Ω(c1)). |
Corollary 3.1. If ↔Ω(v)=Ω↔(v), then (3.1) leads to the following fractional inequality for exponential type pre-invex function:
1(e12−1)Ω(c1+12η(d1,c1))≤Γ(m+1)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1)+Imc+1Ω(c1+η(d1,c1))]≤mP(Ω(c1+η(d1,c1))+Ω(c1)). |
Theorem 3.2. Let Λ⊂ℜ be an open invex set with respect to η:Λ×Λ→ℜ and a1,b1∈Λ with a1<a1+η(b1,a1). If Ω,Ω1:[a1,a1+η(b1,a1)]→ℜ are exponential type pre-invex interval-valued functions such that Ω,Ω1∈L[a1,a1+η(b1,a1)] and m>0, then we have (considering Lemma 2.2 holds)
Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)+Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))]⊇P1Υ1(a1,a1+η(b1,a1))+2P2Υ2(a1,a1+η(b1,a1)), | (3.11) |
where
P1=e2Γ(m)−e2Γ(m,2)2m+2eΓ(m,1)+2Γ(m,−1)−2Γ(m)(−1)m+Γ(m)−Γ(m,−2)(−1)m⋅2m−2eΓ(m)+2m, | (3.12) |
P2=eΓ(m,1)+Γ(m,−1)(−1)m−Γ(m)(−1)m−eΓ(m)+em+1m, | (3.13) |
Υ1(a1,a1+η(b1,a1))=[Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+Ω(a1).Ω1(a1)], | (3.14) |
and
Υ2(a1,a1+η(b1,a1))=[Ω(a1+η(b1,a1)).Ω1(a1)+Ω(a1).Ω1(a1+η(b1,a1))]. | (3.15) |
Proof. Since ΩandΩ1 are exponential type pre-invex interval-valued functions, so we have
Ω(a1+τ1η(b1,a1))=Ω(a1+η(b1,a1)+(1−τ1)η(a1,a1+η(b1,a1)))⊇(eτ1−1)Ω(a1+η(b1,a1))+(e(1−τ1)−1)Ω(a1) |
and
Ω1(a1+τ1η(b1,a1))=Ω1(a1+η(b1,a1)+(1−τ1)η(a1,a1+η(b1,a1)))⊇(eτ1−1)Ω1(a1+η(b1,a1))+(e(1−τ1)−1)Ω1(a1). |
Since Ω,Ω1∈ℜ+Λ, so
Ω(a1+τ1η(b1,a1)).Ω1(a1+τ1η(b1,a1))⊇(eτ1−1)2Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+(e(1−τ1)−1)2Ω(a1).Ω1(a1)+(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1)+Ω(a1).Ω1(a1+η(b1,a1))]. | (3.16) |
Similarly, we have
Ω(a1+(1−τ1)η(b1,a1)).Ω1(a1+(1−τ1)η(b1,a1))⊇(e(1−τ1)−1)2Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+(eτ1−1)2Ω(a1).Ω1(a1)+(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1)+Ω(a1).Ω1(a1+η(b1,a1))]. | (3.17) |
Adding (3.16) and (3.17) yields
Ω(a1+τ1η(b1,a1)).Ω1(a1+τ1η(b1,a1))+Ω(a1+(1−τ1)η(b1,a1)).Ω1(a1+(1−τ1)η(b1,a1))⊇[(e(1−τ1)−1)2+(eτ1−1)2][Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+Ω(a1).Ω1(a1)]+2(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1)+Ω(a1).Ω1(a1+η(b1,a1))]. |
From (3.14) and (3.15), we have
Ω(a1+τ1η(b1,a1)).Ω1(a1+τ1η(b1,a1))+Ω(a1+(1−τ1)η(b1,a1)).Ω1(a1+(1−τ1)η(b1,a1))⊇[(e(1−τ1)−1)2+(eτ1−1)2]Υ1(a1,a1+η(b1,a1))+2(eτ1−1)(e(1−τ1)−1)Υ2(a1,a1+η(b1,a1)). |
Multiplying by τm−11 on both sides and integrating over [0,1] with respect to τ1 gives
(IR)∫10τm−11Ω(a1+τ1η(b1,a1)).Ω1(a1+τ1η(b1,a1))dτ1+(IR)∫10τm−11Ω(a1+(1−τ1)η(b1,a1)).Ω1(a1+(1−τ1)η(b1,a1))dτ1⊇(IR)∫10τm−11[(e(1−τ1)−1)2+(eτ1−1)2]Υ1(a1,a1+η(b1,a1))dτ1+2(IR)∫10τm−11(eτ1−1)(e(1−τ1)−1)Υ2(a1,a1+η(b1,a1))dτ1. |
So
(IR)∫10τm−11Ω(a1+τ1η(b1,a1)).Ω1(a1+τ1η(b1,a1))dτ1=Γ(m)ηm(d1,c1)Im(c1+η(d1,c1))−Ω(c1).Ω1(c1) |
and
(IR)∫10τm−11Ω(a1+(1−τ1)η(b1,a1)).Ω1(a1+(1−τ1)η(b1,a1))dτ1=Γ(m)ηm(d1,c1)Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1)). |
From (3.12) and (3.13), we get
(IR)∫10τm−11[(e(1−τ1)−1)2+(eτ1−1)2]Υ1(a1,a1+η(b1,a1))dτ1=P1Υ1(a1,a1+η(b1,a1)) |
and
(IR)∫10τm−11(eτ1−1)(e(1−τ1)−1)Υ2(a1,a1+η(b1,a1))dτ1=P2Υ2(a1,a1+η(b1,a1)). |
Thus,
Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)+Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))]⊇P1Υ1(a1,a1+η(b1,a1))+2P2Υ2(a1,a1+η(b1,a1)). |
Corollary 3.2. If ↔Ω(v)=Ω↔(v), then (3.11) leads to the following fractional inequality for exponential type pre-invex function:
Γ(m)ηm(d1,c1)[Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)+Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))]≤P1Υ1(a1,a1+η(b1,a1))+2P2Υ2(a1,a1+η(b1,a1)). |
Theorem 3.3. Let Λ⊂ℜ be an open invex set with respect to η:Λ×Λ→ℜ and a1,b1∈Λ with a1<a1+η(b1,a1). If Ω,Ω1:[a1,a1+η(b1,a1)]→ℜ are exponential type pre-invex interval-valued functions such that Ω,Ω1∈L[a1,a1+η(b1,a1)] and m>0, then from (3.12)–(3.15), we have (considering Lemma 2.2 holds)
Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))⊇(e12−1)2[mP1Υ2(a1,a1+η(b1,a1))+mP2Υ1(a1,a1+η(b1,a1))+Γ(m+1)ηm(d1,c1)[Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))+Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)]]. | (3.18) |
Proof. Since Ω is an exponential type pre-invex interval-valued function, so we have
Ω(a1+12η(b1,a1))⊇(e12−1)[Ω(a1)+Ω(b1)]. |
Taking a1=c1+(1−τ1)η(d1,c1) and b1=c1+(τ1)η(d1,c1) gives
Ω(c1+(1−τ1)η(d1,c1)+12η(c1+(τ1)η(d1,c1),c1+(1−τ1)η(d1,c1)))⊇(e12−1)[Ω(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1))], |
implies
Ω(c1+12η(d1,c1))⊇(e12−1)[Ω(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1))]. | (3.19) |
Similarly
Ω1(c1+12η(d1,c1))⊇(e12−1)[Ω1(c1+(1−τ1)η(d1,c1))+Ω1(c1+(τ1)η(d1,c1))]. | (3.20) |
Multiplying (3.19) and (3.20) gives
Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))⊇(e12−1)2[Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))+Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))]. | (3.21) |
Since Ω,Ω1∈ℜ+Λ, are exponential type pre-invex interval-valued functions for τ1∈[0,1], so we have
Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))⊇(e(1−τ1)−1)2Ω(a1+η(b1,a1)).Ω1(a1)+(eτ1−1)2Ω(a1).Ω1(a1+η(b1,a1))+(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+Ω(a1).Ω1(a1)]. | (3.22) |
Similarly
Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))⊇(eτ1−1)2Ω(a1+η(b1,a1)).Ω1(a1)+(e(1−τ1)−1)2Ω(a1).Ω1(a1+η(b1,a1))+(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+Ω(a1).Ω1(a1)]. | (3.23) |
Adding (3.22) and (3.23) yields
Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))⊇[(eτ1−1)2+(e(1−τ1)−1)2](Ω(a1+η(b1,a1)).Ω1(a1)+Ω(a1).Ω1(a1+η(b1,a1)))+2(eτ1−1)(e(1−τ1)−1)[Ω(a1+η(b1,a1)).Ω1(a1+η(b1,a1))+Ω(a1).Ω1(a1)]. |
Now from (3.21), we can write
Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))⊇(e12−1)2[[(eτ1−1)2+(e(1−τ1)−1)2]Υ2(a1,a1+η(b1,a1))+2(eτ1−1)(e(1−τ1)−1)Υ1(a1,a1+η(b1,a1))+Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))+Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))]. |
Multiplying by τm−11 on both sides and integrating over [0,1] with respect to τ1 yields
(IR)∫10τm−11Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))dτ1⊇(e12−1)2[(IR)∫10τm−11[(eτ1−1)2+(e(1−τ1)−1)2]Υ2(a1,a1+η(b1,a1))dτ1+2(IR)∫10τm−11(eτ1−1)(e(1−τ1)−1)Υ1(a1,a1+η(b1,a1))dτ1+(IR)∫10τm−11Ω(c1+(1−τ1)η(d1,c1)).Ω1(c1+(1−τ1)η(d1,c1))dτ1+(IR)∫10τm−11Ω(c1+(τ1)η(d1,c1)).Ω1(c1+(τ1)η(d1,c1))dτ1]. |
Thus from (3.12)–(3.15), we get
Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))⊇(e12−1)2[mP1Υ2(a1,a1+η(b1,a1))+mP2Υ1(a1,a1+η(b1,a1))+Γ(m+1)ηm(d1,c1)[Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))+Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)]]. |
Corollary 3.3. If ↔Ω(v)=Ω↔(v), then (3.18) leads to the following fractional inequality for exponential type pre-invex function:
Ω(c1+12η(d1,c1)).Ω1(c1+12η(d1,c1))≤(e12−1)2[mP1Υ2(a1,a1+η(b1,a1))+mP2Υ1(a1,a1+η(b1,a1))+Γ(m+1)ηm(d1,c1)[Imc+1Ω(c1+η(d1,c1)).Ω1(c1+η(d1,c1))+Im(c1+η(d1,c1))−Ω(c1).Ω1(c1)]]. |
In this section, we establish Hermite-hadamard type inequality in the setting of the He's fractional derivatives introduced in [18].
Definition 4.1. Let Ω be an L1 function defined on an interval [0,n1]. Then the k1-th He's fractional derivative of Ω(n1) is defined by
Ik1n1Ω(n1)=1Γ(i−k1)didni1∫n10(τ1−n)i−k1−1Ω(τ1)dτ1. |
The interval He's fractional derivative based on left and right end point functions can be defined by
Ik1n1Ω(n1)=1Γ(i−k1)didni1∫n10(τ1−n)i−k1−1Ω(τ1)dτ1=1Γ(i−k1)didni1∫n10(τ1−n)i−k1−1[Ω↔(τ1),↔Ω(τ1)]dτ1,n>n1, |
where
Ik1n1Ω↔(n1)=1Γ(i−k1)didni1∫n10(τ1−n)i−k1−1Ω↔(τ1)dτ1,n>n1 | (4.1) |
and
Ik1n1↔Ω(n1)=1Γ(i−k1)didni1∫n10(τ1−n)i−k1−1↔Ω(τ1)dτ1,n>n1. | (4.2) |
Theorem 4.1. Let Ω:[n1,n2]→ℜ be an exponential type pre-invex interval-valued function defined on [n1,n2]⊂Λ, where Λ is an open invex set with respect to η:Λ×Λ→ℜ and Ω:[n1,n2]⊂ℜ→ℜ+c is given by Ω(n)=[Ω↔(n),↔Ω(n)] for all n∈[n1,n2]. If Ω∈L1([n1,n2],ℜ), then
(−1)i−k1−1Ω(n12)⊇(e12−1)nk1ni−k12[(−1)i−k1−1Ik1(1−n)bΩ((1−n)b)+Ik1nbΩ(nb)]. | (4.3) |
Proof. Let Ω:[n1,n2]→ℜ be an exponential type pre-invex interval-valued function defined on [n1,n2], then
Ω(n1+12η(n2,n1))⊇(e12−1)[Ω(n2+τ1η(n1,n2))+Ω(n1+τ1η(n2,n1))] |
and
Ω↔(n1+12η(n2,n1))≤(e12−1)[Ω↔(n2+τ1η(n1,n2))+Ω↔(n1+τ1η(n2,n1))]. |
Taking n2=0,0≤n1 and multiplying by (τ1−n)i−k1−1Γ(i−k1), we get
(τ1−n)i−k1−1Γ(i−k1)Ω↔(n12)≤(e12−1)(τ1−n)i−k1−1Γ(i−k1)[Ω↔((1−τ1)n1)+Ω↔(τ1n1)]. |
Integrating with respect to τ1 over [0,n1] gives
Ω↔(n12)1Γ(i−k1)∫n10(τ1−n)i−k1−1dτ1≤(e12−1)Γ(i−k1)∫n10(τ1−n)i−k1−1Ω↔((1−τ1)n1)dτ1+(e12−1)Γ(i−k1)∫n10(τ1−n)i−k1−1Ω↔(τ1n1)dτ1, |
implies
Ω↔(n12)(−1)i−k1−1ni−k1Γ(i−k1)≤(e12−1)Γ(i−k1)∫n10(τ1−n)i−k1−1Ω↔((1−τ1)n1)dτ1+(e12−1)Γ(i−k1)∫n10(τ1−n)i−k1−1Ω↔(τ1n1)dτ1. |
Getting i-th derivative on both sides and using (4.1), we get
(−1)i−k1−1Ω↔(n12)≤(e12−1)nk1ni−k11[(−1)i−k1−1Ik1(1−n)bΩ↔((1−n)b)+Ik1nbΩ↔(nb)]. |
Similarly
(−1)i−k1−1↔Ω(n12)≤(e12−1)nk1ni−k11[(−1)i−k1−1Ik1(1−n)b↔Ω((1−n)b)+Ik1nb↔Ω(nb)]. |
Thus, we can write
(−1)i−k1−1[Ω↔(n12),↔Ω(n12)]⊇(e12−1)nk1ni−k11[(−1)i−k1−1Ik1(1−n)b[Ω↔((1−n)b),↔Ω((1−n)b)]+Ik1nb[Ω↔(nb),↔Ω(nb)]]. |
So,
(−1)i−k1−1Ω(n12)⊇(e12−1)nk1ni−k11[(−1)i−k1−1Ik1(1−n)bΩ((1−n)b)+Ik1nbΩ(nb)]. |
Corollary 4.1. If ↔Ω(v)=Ω↔(v), then (4.3) leads to the following fractional inequality for exponential type pre-invex function:
(−1)i−k1−1Ω(n12)≤(e12−1)nk1ni−k11[(−1)i−k1−1Ik1(1−n)bΩ((1−n)b)+Ik1nbΩ(nb)]. |
In this paper we studied the interval-valued exponential type pre-invex functions. We established He's and Hermite-Hadamard type inequalities for interval-valued exponential type pre-invex functions in the setting of Riemann-Liouville interval-valued fractional operator.
This work was sponsored in part by Henan Science and Technology Project of China (No:182102110292).
The author declares no conflict of interest.
[1] |
H. Yarmand, S. Gharehkhani, G. Ahmadi, S. F. S. Shirazi, S. Baradaran, E. Montazer, et al., Graphene nanoplatelets-silver hybrid nanofluids for enhanced heat transfer, Energy Convers. Management, 100 (2015), 419–428. https://doi.org/10.1016/j.enconman.2015.05.023 doi: 10.1016/j.enconman.2015.05.023
![]() |
[2] |
O. K. Koriko, K. S. Adegbie, I. L. Animasaun, M. A. Olotu, Numerical solutions of the partial differential equations for investigating the significance of partial slip due to lateral velocity and viscous dissipation: the case of blood-gold Carreau nanofluid and dusty fluid, Numer. Methods Part. Differ. Equ., 2021. https://doi.org/10.1002/num.22754 doi: 10.1002/num.22754
![]() |
[3] | Y. N. Jiang, X. M. Zhou, Y. Wang, Effects of nanoparticle shapes on heat and mass transfer of nanofluid thermocapillary convection around a gas bubble, Microgravity Sci. Technol., 32 (2020), 167–177. |
[4] | M. C. Arno, M. Inam, A. C. Weems, Z. Li, A. L. A. Binch, C. I. Platt, et al., Exploiting the role of nanoparticle shape in enhancing hydrogel adhesive and mechanical properties, Nat. commun., 11 (2020), 1–9. |
[5] |
L. Th. Benos, E. G. Karvelas, I. E. Sarris, A theoretical model for the manetohydrodynamic natural convection of a CNT-water nanofluid incorporating a renovated Hamilton-Crosser model, Int. J. Heat Mass Trans., 135 (2019), 548–560. https://doi.org/10.1016/j.ijheatmasstransfer.2019.01.148 doi: 10.1016/j.ijheatmasstransfer.2019.01.148
![]() |
[6] | H. Liu, I. L. Animasaun, N. A. Shah, O. K. Koriko, B. Mahanthesh, Further discussion on the significance of quartic autocatalysis on the dynamics of water conveying 47 nm alumina and 29 nm cupric nanoparticles, Arab. J. Sci. Eng., 45 (2020), 5977–6004. |
[7] |
A. S. Sabu, A. Wakif, S. Areekara, A. Mathew, N. A. Shah, 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 Trans., 129 (2021), 105711. https://doi.org/10.1016/j.icheatmasstransfer.2021.105711 doi: 10.1016/j.icheatmasstransfer.2021.105711
![]() |
[8] |
Q. Lou, B. Ali, S. U. Rehman, D. Habib, S. Abdal, N. A. Shah, et al., Micropolar dusty fluid: Coriolis force effects on dynamics of MHD rotating fluid when Lorentz force is significant, Mathematics, 10 (2022), 2630. https://doi.org/10.3390/math10152630 doi: 10.3390/math10152630
![]() |
[9] |
M. Z. Ashraf, S. U. Rehman, S. Farid, A. K. Hussein, B. Ali, N. A. Shah, et al., Insight into significance of bioconvection on MHD tangent hyperbolic nanofluid flow of irregular thickness across a slender elastic surface, Mathematics, 10 (2022), 2592. https://doi.org/10.3390/math10152592 doi: 10.3390/math10152592
![]() |
[10] |
K. Vajravelu, Convection heat transfer at a stretching sheet with suction or blowing, J. Math. Anal. Appl., 188 (1994), 1002–1011. https://doi.org/10.1006/jmaa.1994.1476 doi: 10.1006/jmaa.1994.1476
![]() |
[11] |
C-K. Chen, M-I. Char, Heat transfer of a continuous, stretching surface with suction or blowing, J. Math. Anal. Appl., 135 (1988), 568–580. https://doi.org/10.1016/0022-247X(88)90172-2 doi: 10.1016/0022-247X(88)90172-2
![]() |
[12] |
R. N. Kumar, R. J. P. Gowda, A. M. Abusorrah, Y. M. Mahrous, N. H. Abu-Hamdeh, A. Issakhov, et al., Impact of magnetic dipole on ferromagnetic hybrid nanofluid flow over a stretching cylinder, Phys. Scr., 96 (2021), 045215. https://doi.org/10.1088/1402-4896/abe324 doi: 10.1088/1402-4896/abe324
![]() |
[13] |
B. Ali, I. Siddique, I. Khan, B. Masood, S. Hussain, Magnetic dipole and thermal radiation effects on hybrid base micropolar CNTs flow over a stretching sheet: finite element method approach, Results Phys., 25 (2021), 104145. https://doi.org/10.1016/j.rinp.2021.104145 doi: 10.1016/j.rinp.2021.104145
![]() |
[14] |
G. C. Rana, S. K. Kango, Effect of rotation on thermal instability of compressible Walters' (Model B′) fluid in porous medium, J. Adv. Res. Appl. Math., 3 (2011), 44–57. https://doi.org/10.5373/jaram.815.030211 doi: 10.5373/jaram.815.030211
![]() |
[15] | A. V. Kuznetsov, D. A. Nield, Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model, Transp. Porous Media, 81 (2010), 409–422. |
[16] | L. J. Sheu, Linear stability of convection in a viscoelastic nanofluid layer, Eng. Int. J. Mech. Mechatron. Eng., 5 (2011), 1970–1976. |
[17] |
L. L. Lee, Boundary layer over a thin needle, Phys. Fluids, 10 (1967), 820–822. https://doi.org/10.1063/1.1762194 doi: 10.1063/1.1762194
![]() |
[18] |
J. L. S. Chen, Mixed convection flow about slender bodies of revolution, ASME J. Heat Mass Trans., 109 (1987), 1033–1036. https://doi.org/10.1115/1.3248177 doi: 10.1115/1.3248177
![]() |
[19] |
T. Fang, J. Zhang, Y. Zhong, Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput., 218 (2012), 7241–7252. https://doi.org/10.1016/j.amc.2011.12.094 doi: 10.1016/j.amc.2011.12.094
![]() |
[20] |
R. Jusoh, R. Nazar, I. Pop, Three-dimensional flow of a nanofluid over a permeable stretching/shrinking surface with velocity slip: A revised model, Phys. Fluids, 30 (2018), 033604. https://doi.org/10.1063/1.5021524 doi: 10.1063/1.5021524
![]() |
[21] |
A. Jamaludin, R. Nazar, I. Pop, Three-dimensional magnetohydrodynamic mixed convection flow of nanofluids over a nonlinearly permeable stretching/shrinking sheet with velocity and thermal slip, Appl. Sci., 8 (2018), 1128. https://doi.org/10.3390/app8071128 doi: 10.3390/app8071128
![]() |
[22] |
N. S. Khashi'ie, N. M. Arifin, R. Nazar, E. H. Hafidzuddin, N. Wahi, I. Pop, A stability analysis for magnetohydrodynamics stagnation point flow with zero nanoparticles flux condition and anisotropic slip, Energies, 12 (2019), 1268. https://doi.org/10.3390/en12071268 doi: 10.3390/en12071268
![]() |
[23] |
P. Rana, A. Kumar, G. Gupta, Impact of different arrangements of heated elliptical body, fins and differential heater in MHD convective transport phenomena of inclined cavity utilizing hybrid nanoliquid: Artificial neutral network prediction, Int. Commun. Heat Mass Trans., 132 (2022), 105900. https://doi.org/10.1016/j.icheatmasstransfer.2022.105900 doi: 10.1016/j.icheatmasstransfer.2022.105900
![]() |
[24] |
C. S. K. Raju, N. A. Ahammad, K. Sajjan, N. A. Shah, S-J. Yook, M. D. Kumar, Nonlinear movements of axisymmetric ternary hybrid nanofluids in a thermally radiated expanding or contracting permeable Darcy Walls with different shapes and densities: simple linear regression, Int. Commun. Heat Mass Trans., 135 (2022), 106110. https://doi.org/10.1016/j.icheatmasstransfer.2022.106110 doi: 10.1016/j.icheatmasstransfer.2022.106110
![]() |
[25] |
P. Rana, S. Gupta, G. Gupta, Unsteady nonlinear thermal convection flow of MWCNT-MgO/EG hybrid nanofluid in the stagnation-point region of a rotating sphere with quadratic thermal radiation: RSM for optimization, Int. Commun. Heat Mass Trans., 134 (2022), 106025. https://doi.org/10.1016/j.icheatmasstransfer.2022.106025 doi: 10.1016/j.icheatmasstransfer.2022.106025
![]() |
[26] |
N. A. Shah, A. Wakif, E. R. El-Zahar, T. Thumma, S.-J. Yook, Heat transfers thermodynamic activity of a second-grade ternary nanofluid flow over a vertical plate with Atangana-Baleanu time-fractional integral, Alexandria Eng. J., 61 (2022), 10045–10053. https://doi.org/10.1016/j.aej.2022.03.048 doi: 10.1016/j.aej.2022.03.048
![]() |
[27] |
R. Zhang, N. A. Ahammad, C. S. K. Raju, S. M. Upadhya, N. A. Shah, S-J. Yook, Quadratic and linear radiation impact on 3D convective hybrid nanofluid flow in a suspension of different temperature of waters: transpiration and fourier fluxes, Int. Commun. Heat Mass Trans., 138 (2022), 106418. https://doi.org/10.1016/j.icheatmasstransfer.2022.106418 doi: 10.1016/j.icheatmasstransfer.2022.106418
![]() |
[28] |
N. Acharya, Spectral quasi linearization simulation on the radiative nanofluid spraying over a permeable inclined spinning disk considering the existence of heat source/sink, Appl. Math. Comput., 411 (2021), 126547. https://doi.org/10.1016/j.amc.2021.126547 doi: 10.1016/j.amc.2021.126547
![]() |
[29] |
P. Rana, W. Al-Kouz, B. Mahanthesh, J. Mackolil, Heat transfer of TiO2-EG nanoliquid with active and passive control of nanoparticles subject to nonlinear Boussinesq approximation, Int. Commun. Heat Mass Trans., 126 (2021), 105443. https://doi.org/10.1016/j.icheatmasstransfer.2021.105443 doi: 10.1016/j.icheatmasstransfer.2021.105443
![]() |
[30] |
T. Elnaqeeb, I. L. Animasaun, N. A. Shah, Ternary-hybrid nanofluids: significance of suction and dual-stretching on three-dimensional flow of water conveying nanoparticles with various shapes and densities, Z. Naturforsch. A, 76 (2021). https://doi.org/10.1515/zna-2020-0317 doi: 10.1515/zna-2020-0317
![]() |
[31] |
N. A. Shah, A. Wakif, E. R. El-Zahar, S. Ahmad, S-J Yook, Numerical simulation of a thermally enhanced EMHD flow of a heterogeneous micropolar mixture comprising (60%)-ethylene glycol (EG), (40%)-water (W), and copper oxide nanomaterials (CuO), Case Study. Therm. Eng., 35 (2022), 102046. https://doi.org/10.1016/j.csite.2022.102046 doi: 10.1016/j.csite.2022.102046
![]() |
[32] | M. S. Upadhya, C. S. K. Raju, Implementation of boundary value problems in using MATLAB®, In: Micro and nanofluid convection with magnetic field effects for heat and mass transfer applications using MATLAB, Elsevier, 2022,169–238. https://doi.org/10.1016/B978-0-12-823140-1.00010-5 |
[33] | M. Khader, A. M. Megahed, Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity, Eur. Phys. J. Plus, 128 (2013), 100–108. |
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