In this contemporary era, fractional derivatives are widely used for the development of mathematical models to precisely describe the dynamics of real-world physical processes. In the field of fluid mechanics, analysis of thermal performance and flow behavior of non-Newtonian fluids is a topic of interest for a variety of researchers due to their significant applications in several industries, engineering operations, devices, and thermal equipment. The primary focus of this article is to investigate the effectiveness of jointly imposed time-controlled (ramped) boundary conditions in the electro-osmotic flow of a chemically reactive and radiative Walters' B fluid along with concentration and energy distributions. In Particular, the concept of using piece-wise time-dependent mass, motion, and energy conditions simultaneously for any non-Newtonian fluid is extensively explored in this work. The flow is developed due to the motion of the bounding vertical wall, which is suspended in a porous material subject to heat injection/absorption and uniform magnetic influences. Atangana-Baleanu derivative of order ψ is incorporated to establish the fractional form of ordinary modeled equations. Laplace transform method is adapted in light of some unit-less quantities to procure the exact solutions of the under observation model. Several graphical delineations are produced to comprehensively analyze the key characteristics of many physical and thermal parameters. To highlight the significance of operating surface conditions, solutions are compared for time-dependent and constant boundary conditions in every graph. Furthermore, the role of the fractional parameter, time-dependent conditions, and different other involved parameters in heat transfer, mass transfer, and flow rates is characterized by determining the expressions for Nusselt and Sherwood number and coefficient of skin friction. The numerical outcomes are organized in several tables to deeply scrutinize the noteworthy variations in the behavior of the aforementioned physical quantities. The graphical study reveals that the parameter Es accounting for electro-osmotic effects decelerates the flow of fluid. At the atomic level, such electro-osmotic flows are useful in the separation processes of the liquids. The fractional parameter ψ attenuates thicknesses of boundary layers for the evolution of time t but, it exhibits an opposite role for smaller values of t. It is also noted that the direct correspondence between velocity and time at the boundary for time duration t<1 plays a supportive part to effectively control the flow. The exercise tolerance level of cardiac patients is anticipated by following a ramped velocity based protocol. The fractional models are more effective than ordinary models for restricting the boundary shear stress. The occurrence of a chemical reaction leads to improving the mass transfer rate. Additionally, augmentation in heat transfer rate due to the ramped heating technique indicates the significance of this technique in cooling processes. The findings of this work are helpful for clear and comprehensive understanding of electro-osmotic flow of Walters' B fluid in a fractional framework together with chemically reacted mass transfer and thermally radiative heat transfer phenomena subject to wall ramping technique.
Citation: Asifa, Poom Kumam, Talha Anwar, Zahir Shah, Wiboonsak Watthayu. Analysis and modeling of fractional electro-osmotic ramped flow of chemically reactive and heat absorptive/generative Walters'B fluid with ramped heat and mass transfer rates[J]. AIMS Mathematics, 2021, 6(6): 5942-5976. doi: 10.3934/math.2021352
[1] | Ruishen Qian, Xiangling Zhu . Invertible weighted composition operators preserve frames on Dirichlet type spaces. AIMS Mathematics, 2020, 5(5): 4285-4296. doi: 10.3934/math.2020273 |
[2] | Liu Yang, Ruishen Qian . Volterra integral operator and essential norm on Dirichlet type spaces. AIMS Mathematics, 2021, 6(9): 10092-10104. doi: 10.3934/math.2021586 |
[3] | Dagmar Medková . Classical solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system. AIMS Mathematics, 2019, 4(6): 1540-1553. doi: 10.3934/math.2019.6.1540 |
[4] | Munirah Aljuaid, Mahmoud Ali Bakhit . Composition operators from harmonic H∞ space into harmonic Zygmund space. AIMS Mathematics, 2023, 8(10): 23087-23107. doi: 10.3934/math.20231175 |
[5] | Aydah Mohammed Ayed Al-Ahmadi . Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-. AIMS Mathematics, 2023, 8(7): 16240-16251. doi: 10.3934/math.2023831 |
[6] | Meichuan Lv, Wenming Li . Sharp bounds for multilinear Hardy operators on central Morrey spaces with power weights. AIMS Mathematics, 2025, 10(6): 14183-14195. doi: 10.3934/math.2025639 |
[7] | Li He . Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels. AIMS Mathematics, 2023, 8(2): 2708-2719. doi: 10.3934/math.2023142 |
[8] | Tingting Xu, Zaiyong Feng, Tianyang He, Xiaona Fan . Sharp estimates for the p-adic m-linear n-dimensional Hardy and Hilbert operators on p-adic weighted Morrey space. AIMS Mathematics, 2025, 10(6): 14012-14031. doi: 10.3934/math.2025630 |
[9] | Suixin He, Shuangping Tao . Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces. AIMS Mathematics, 2022, 7(1): 1000-1014. doi: 10.3934/math.2022060 |
[10] | Chun Wang . The coefficient multipliers on H2 and D2 with Hyers–Ulam stability. AIMS Mathematics, 2024, 9(5): 12550-12569. doi: 10.3934/math.2024614 |
In this contemporary era, fractional derivatives are widely used for the development of mathematical models to precisely describe the dynamics of real-world physical processes. In the field of fluid mechanics, analysis of thermal performance and flow behavior of non-Newtonian fluids is a topic of interest for a variety of researchers due to their significant applications in several industries, engineering operations, devices, and thermal equipment. The primary focus of this article is to investigate the effectiveness of jointly imposed time-controlled (ramped) boundary conditions in the electro-osmotic flow of a chemically reactive and radiative Walters' B fluid along with concentration and energy distributions. In Particular, the concept of using piece-wise time-dependent mass, motion, and energy conditions simultaneously for any non-Newtonian fluid is extensively explored in this work. The flow is developed due to the motion of the bounding vertical wall, which is suspended in a porous material subject to heat injection/absorption and uniform magnetic influences. Atangana-Baleanu derivative of order ψ is incorporated to establish the fractional form of ordinary modeled equations. Laplace transform method is adapted in light of some unit-less quantities to procure the exact solutions of the under observation model. Several graphical delineations are produced to comprehensively analyze the key characteristics of many physical and thermal parameters. To highlight the significance of operating surface conditions, solutions are compared for time-dependent and constant boundary conditions in every graph. Furthermore, the role of the fractional parameter, time-dependent conditions, and different other involved parameters in heat transfer, mass transfer, and flow rates is characterized by determining the expressions for Nusselt and Sherwood number and coefficient of skin friction. The numerical outcomes are organized in several tables to deeply scrutinize the noteworthy variations in the behavior of the aforementioned physical quantities. The graphical study reveals that the parameter Es accounting for electro-osmotic effects decelerates the flow of fluid. At the atomic level, such electro-osmotic flows are useful in the separation processes of the liquids. The fractional parameter ψ attenuates thicknesses of boundary layers for the evolution of time t but, it exhibits an opposite role for smaller values of t. It is also noted that the direct correspondence between velocity and time at the boundary for time duration t<1 plays a supportive part to effectively control the flow. The exercise tolerance level of cardiac patients is anticipated by following a ramped velocity based protocol. The fractional models are more effective than ordinary models for restricting the boundary shear stress. The occurrence of a chemical reaction leads to improving the mass transfer rate. Additionally, augmentation in heat transfer rate due to the ramped heating technique indicates the significance of this technique in cooling processes. The findings of this work are helpful for clear and comprehensive understanding of electro-osmotic flow of Walters' B fluid in a fractional framework together with chemically reacted mass transfer and thermally radiative heat transfer phenomena subject to wall ramping technique.
As usual, let D be the unit disk in the complex plane C, ∂D be the boundary of D, H(D) be the class of functions analytic in D and H∞ be the set of bounded analytic functions in D. Let 0<p<∞. The Hardy space Hp (see [5]) is the sets of f∈H(D) with
‖f‖pHp=sup0<r<112π∫2π0|f(reiθ)|pdθ<∞. |
Suppose that K:[0,∞)→[0,∞) is a right-continuous and nondecreasing function with K(0)=0. The Dirichlet Type spaces DK, consists of those functions f∈H(D), such that
‖f‖2DK=|f(0)|2+∫D|f′(z)|2K(1−|z|2)dA(z)<∞. |
The space DK has been extensively studied. Note that K(t)=t, it is Hardy spaces H2. When K(t)=tα, 0≤α<1, it give the classical weighted Dirichlet spaces Dα. For more information on DK, we refer to [3,7,8,9,10,14,15,16,19,23].
Let ϕ be a holomorphic self-map of D. The composition operator Cϕ on H(D) is defined by
Cϕ(f)=f∘ϕ, f∈H(D). |
It is an interesting problem to studying the properties related to composition operator acting on analytic function spaces. For example: Shapiro [17] introduced Nevanlinna counting functions studied the compactness of composition operator acting on Hardy spaces. Zorboska [23] studied the boundedness and compactness of composition operator on weighted Dirichlet spaces Dα. El-Fallah, Kellay, Shabankhah and Youssfi [7] studied composition operator acting on Dirichlet type spaces Dpα by level set and capacity. For general weighted function ω, Kellay and Lefèvre [9] using Nevanlinna type counting functions studied the boundedness and compactness of composition spaces on weighted Hilbert spaces Hω. After Kellay and Lefèvre's work, Pau and Pérez investigate more properties of composition operators on weighted Dirichlet spaces Dα in [14]. For more information on composition operator, we refer to [4,18].
We assume that H is a separable Hilbert space of analytic functions in the unit disc. Composition operator Cϕ is called power bounded on H if Cϕn is bounded on H for all n∈N.
Since power bounded composition operators is closely related to mean ergodic and some special properties (such as: stable orbits) of ϕ, it has attracted the attention of many scholars. Wolf [20,21] studied power bounded composition operators acting on weighted type spaces H∞υ. Bonet and Domański [1,2] proved that Cϕ is power bounded if and only if Cϕ is (uniformly) mean ergodic in real analytic manifold (or a connected domain of holomorphy in Cd). Keshavarzi and Khani-Robati [11] studied power bounded of composition operator acting on weighted Dirichlet spaces Dα. Keshavarzi [12] investigated the power bounded below of composition operator acting on weighted Dirichlet spaces Dα later. For more results related to power bounded composition operators acting on other function spaces, we refer to the paper cited and referin [1,2,11,12,20,21].
We always assume that K(0)=0, otherwise, DK is the Dirichlet space D. The following conditions play a crucial role in the study of weighted function K during the last few years (see [22]):
∫10φK(s)sds<∞ | (1.1) |
and
∫∞1φK(s)s2ds<∞, | (1.2) |
where
φK(s)=sup0≤t≤1K(st)/K(t),0<s<∞. |
Note that the weighted function K satisfies (1.1) and (1.2), it included many special case, such as K(t)=tp, 0<p<1, K(t)=loget and so on. Some special skills are needed in dealing with certain problems. Motivated by [11,12], using several estimates on the weight function K, we studying power bounded composition operators acting on DK. In this paper, the symbol a≈b means that a≲b≲a. We say that a≲b if there exists a constant C such that a≤Cb, where a,b>0.
We assume that H is a separable Hilbert space of analytic functions in the unit disc. Let R∈H(D) and {Rζ:ζ∈D} be an independent collection of reproducing kernels for H. Here Rζ(z)=R(ˉζz). The reproducing kernels mean that f(ζ)=⟨f,Rζ⟩ for any f∈H. Let RK,z be the reproducing kernels for DK. By [3], we see that if K satisfy (1.1) and (1.2), we have ‖RK,z‖DK≈1√K(1−|z|2). Before we go into further, we need the following lemma.
Lemma 1. Let K satisfies (1.1) and (1.2). Then
1+∞∑n=1tnK(1n+1)≈1(1−t)K(1−t) |
for all 0≤t<1.
Proof. Without loss of generality, we can assume 4/5<t<1. Since K is nondecreasing, we have
∞∑n=1tnK(1n+1)≈1(ln1t)K(ln1t)∫∞−lntγe−γK(ln1t)K(1γln1t)dγ≳1(1−t)K(1−t)∫∞ln2γe−γK(ln1t)K(1γln1t)dγ≳1(1−t)K(1−t)∫∞ln2γe−γdγ≈1(1−t)K(1−t). |
Conversely, make change of variables y=1x, an easy computation gives
∞∑n=1tnK(1n+1)≈∞∑n=1∫1n1n+1t1xx2K(x)dx≈∫10t1xx2K(x)dx≈∫∞1tyK(1y)dy. |
Let y=γ−lnt. We can deduce that
∞∑n=1tnK(1n+1)≈1(ln1t)∫∞−lntγe−γK(1γln1t)dγ=1(ln1t)K(ln1t)∫∞−lntγe−γK(ln1t)K(1γln1t)dγ≲1(1−t)K(1−t)∫∞−lntγe−γφK(γ)dγ. |
By [6], under conditions (1.1) and (1.2), there exists an enough small c>0 only depending on K such that
φK(s)≲sc, 0<s≤1 |
and
φK(s)≲s1−c, s≥1. |
Therefore,
∞∑n=1tnK(1n+1)≲1(1−t)K(1−t)∫∞−lntγe−γφK(γ)dγ≲1(1−t)K(1−t)(∫∞0e−γγ2−cdγ+∫∞0e−γγ1+cdγ)≈1(1−t)K(1−t)(Γ(3−c)+Γ(2+c)), |
where Γ(.) is the Gamma function. It follows that
1+∞∑n=1tnK(1n+1)≲1(1−t)K(1−t). |
The proof is completed.
Theorem 1. Let K satisfy (1.1) and (1.2). Suppose that ϕ is an analytic selt-map of unit disk which is not the identity or an elliptic automorphism. Then Cϕ is power bounded on DK if and only if ϕ has its Denjoy-Wolff point in D and for every 0<r<1, we have
supn∈N,a∈D∫D(a,r)Nϕn,K(z)dA(z)(1−|a|2)2K(1−|a|2)<∞,(A) |
where
D(a,r)={z:|a−z1−¯az|<r}, 0<r<1 |
and
Nϕn,K(z)=∑ϕn(zj)=wK(1−|zj(w)|2). |
Proof. Suppose that w∈D is the Denjoy-Wolff point of ϕ and (A) holds. Then limn→∞ϕn(0)=w. Hence, there is some 0<r<1 such that {ϕn(0)}n∈N⊆rD. Thus,
|f(ϕn(0))|2≲‖RK,ϕn(0)‖2DK≲‖RK,r‖2DK, f∈DK. |
From [24], we see that
1−|a|≈1−|z|≈|1−¯az|, z∈D(a,r).(B) |
Let {ai} be a r-lattice. By sub-mean properties of |f′|, combine with (B), we deduce
∫D|f′(z)|2Nϕn,KdA(z)≲∞∑i=1∫D(ai,r)|f′(z)|2Nϕn,K(z)dA(z)≲∞∑i=1∫D(ai,r)1(1−|ai|)2∫D(ai,l)|f′(w)|2dA(w)Nϕn,K(z)dA(z)≲∞∑i=1∫D(ai,l)|f′(w)|2(∫D(ai,r)Nϕn,K(z)dA(z)(1−|ai|)2K(1−|ai|2))K(1−|w|2)dA(w)≲∞∑i=1∫D(ai,l)|f′(w)|2K(1−|w|2)dA(w)<∞. |
Thus,
‖f∘ϕn‖2DK=|f(ϕn(0))|2+∫D|f′(z)|2Nϕn,KdA(z)<∞. |
On the other hand. Suppose that Cϕ is power bounded on DK. Hence, for any f∈DK and any n∈N, we have |f(ϕn(0))|≲1. Hence, by [3], it is easily to see that ‖RK,ϕn(0)‖DK≈1√K(1−|ϕn(0)|2)≲1. Note that
lim|z|→1‖RK,z‖DK≈lim|z|→11√K(1−|z|)=∞. |
Therefore, we deduce that ϕn(0)∈rD, where 0<r<1 and n∈N. Also note that if w∈¯D is the Denjoy-Wolff point of ϕ, we have limn→∞ϕn(0)=w. Thus, w∈D. Let
fa(z)=1−|a|2¯a√K(1−|a|2)(1−¯az). |
It is easily to verify that fa∈DK and f′a(z)=1−|a|2√K(1−|a|2)(1−¯az)2. Thus, combine with (B), we have
∫D(a,r)Nϕn,K(z)dA(z)(1−|a|2)2K(1−|a|2)≲∫D(a,r)(1−|a|2)2K(1−|a|2)|1−¯az|4Nϕn,K(z)dA(z)≤∫D(1−|a|2)2K(1−|a|2)|1−¯az|4Nϕn,K(z)dA(z)≲‖fa∘ϕn‖2DK<∞. |
Thus, (A) hold. The proof is completed.
Theorem 2. Let K satisfy (1.1) and (1.2). Suppose that ϕ is an analytic selt-map of unit disk which is not the identity or an elliptic automorphism with w as its Denjoy-wolff point. Then Cϕ is power bounded on DK if and only if
(1). w∈D.
(2). {ϕn} is a bounded sequence in DK.
(3). If n∈N and |a|≥1+|ϕn(0)|2, then Nϕn,K(a)K(1−|a|2)≲1.
Proof. Suppose that Cϕ is power bounded on DK. By Theorem 1, we see that w∈D. Note that z∈DK and ϕn=Cϕnz, we have (2) hold. Now, we are going to show (3) hold. Let |a|≥1+|ϕn(0)|2 and Δ(a)={z:|z−a|<12(1−|a|)}. Thus,
|ϕn(0)|<|z|, z∈Δ(a). |
If K satisfy (1.1) and (1.2). By [9], Nϕn,K has sub-mean properties. Thus,
Nϕn,K(a)K(1−|a|2)≲∫Δ(a)Nϕn,K(z)dA(z)(1−|a|2)2K(1−|a|2)≲∫Δ(a)(1−|a|2)2K(1−|a|2)|1−¯az|4Nϕn,K(z)dA(z)≲∫D(1−|a|2)2K(1−|a|2)|1−¯az|4Nϕn,K(z)dA(z)≲‖f∘ϕn‖2DK<∞. |
Conversely. Suppose that (1)–(3) holds. Let f∈DK. Note that z∈DK, z′=1 and 1+|ϕn(0)|2<1. By Lemma 1, we see that
‖R′K,1+|ϕn(0)|2‖2DK≈1(1−1+|ϕn(0)|2)K(1−1+|ϕn(0)|2)<∞. |
Thus,
∫D|f′(z)|2Nϕn,K(z)dA(z)=∫|z|≥1+|ϕn(0)|2|f′(z)|2Nϕn,K(z)dA(z)+∫|z|<1+|ϕn(0)|2|f′(z)|2Nϕn,K(z)dA(z)≲∫|z|≥1+|ϕn(0)|2|f′(z)|2K(1−|z|2)dA(z)+‖R′K,1+|ϕn(0)|2‖2DK∫|z|<1+|ϕn(0)|2Nϕn,K(z)dA(z)≲‖f‖2DK+‖R′K,1+|ϕn(0)|2‖2DK‖ϕn‖2DK<∞. |
The proof is completed.
Theorem 3. Let K satisfy (1.1) and (1.2). Suppose that ϕ is an analytic selt-map of D with Denjoy-Wolff point w and Cϕ is power bounded on DK. Then f∈Γc,K(ϕ) if and only if for any ϵ>0,
limn→∞∫Ωϵ(f)Nϕn,K(z)dA(z)(1−|z|2)2K(1−|z|2)=0,(C) |
where Γc,K(ϕ)={f∈DK: Cϕnf is convergent} and Ωϵ(f)={z:(1−|z|2)2K(1−|z|2)|f′(z)|2≥ϵ}.
Proof. Let f∈DK and (C) hold. For any δ>0, we choose 0<ϵ<δ and ϵ is small enough such that
∫Ωϵ(f)c|f′(z)|2K(1−|z|2)dA(z)<δ. |
By our assumption, we also know that for this ϵ, there is some N∈N such that for each n≥N, we have
∫Ωϵ(f)Nϕn,K(z)(1−|z|2)2K(1−|z|2)dA(z)<δ. |
Since
|f′(z)|≲‖f‖DK(1−|z|2)√K(1−|z|2), f∈DK. |
We obtain
∫Ωϵ(f)|f′(z)|2Nϕn,K(z)dA(z)≲‖f‖2DK∫Ωϵ(f)Nϕn,K(z)(1−|z|2)2K(1−|z|2)dA(z)<δ‖f‖2DK |
and
∫Ωϵ(f)c|f′(z)|2Nϕn,K(z)dA(z)=∫Ωϵ(f)c∩rD|f′(z)|2Nϕn,K(z)dA(z)+∫Ωϵ(f)c∖rD|f′(z)|2Nϕn,K(z)dA(z)≲ϵ∫Ωϵ(f)c∩rDNϕn,K(z)(1−|z|2)2K(1−|z|2)dA(z)+∫Ωϵ(f)c∖rD|f′(z)|2K(1−|z|2)dA(z)≲ϵ∫Ωϵ(f)c∩rDNϕn,K(z)(1−r2)2K(1−r2)dA(z)+∫Ωϵ(f)c|f′(z)|2K(1−|z|2)dA(z)<δ‖ϕn‖2DK(1−r2)2K(1−r2)+δ. |
Thus,
∫D|f′(z)|2Nϕn,K(z)dA(z)=∫Ωϵ(f)|f′(z)|2Nϕn,K(z)dA(z)+∫Ωϵ(f)c|f′(z)|2Nϕn,K(z)dA(z)≲(‖f‖2DK+‖ϕn‖2DK(1−r2)2K(1−r2)+1)δ. |
Conversely. Suppose that f∈DK and w is the Denjoy-Wolff point of ϕ. Thus, f∘ϕn→f(w) uniform convergent and f∈Γc,K(ϕ) if and only if
limn→∞∫D|f′(z)|2Nϕn,K(z)dA(z)=0. |
Suppose there exist ϵ>0 such that (C) dose not hold. There is a sequence {nk}⊆N and some η>0 such that for any k∈N, we have
∫Ωϵ(f)Nϕn,K(z)dA(z)(1−|z|2)2K(1−|z|2)>η. |
Hence,
∫D|f′(z)|2Nϕn,K(z)dA(z)≥∫Ωϵ(f)|f′(z)|2Nϕn,K(z)dA(z)≥ϵ∫Ωϵ(f)Nϕn,K(z)(1−|z|2)2K(1−|z|2)dA(z)>ηϵ. |
That is a contradiction. The proof is completed.
The composition operator Cϕ is called power bounded below if there exists some C>0 such that ‖Cϕnf‖H≥C‖f‖H, for all f∈H and n∈N.
In this section, we are going to show the equivalent characterizations of composition operator Cϕ power bounded below on DK. Before we get into prove, let us recall some notions.
(1) We say that {Gn}, a sequence of Borel subsets of D satisfies the reverse Carleson condition on DK if there exists some positive constant δ such that for each f∈DK,
δ∫Gn|f′(z)|2K(1−|z|2)dA(z)≥∫D|f′(z)|2K(1−|z|2)dA(z). |
(2) We say that {μn}, a sequence of Carleson measure on D satisfies the reverse Carleson condition, if there exists some positive constant δ and 0<r<1 such that
μn(D(a,r))>δ|D(a,r)| |
for each a∈D and n∈N.
Theorem 4. Let K satisfy (1.1) and (1.2). Suppose that ϕ is an analytic selt-map of D and Cϕ is power bounded on DK. Then the following are equivalent.
(1). Cϕ is power bounded below.
(2). There exists some δ>0 such that ‖Cϕnfa‖≥δ for all a∈D and n∈N.
(3). There exists some δ>0 and ϵ>0 such that for all a∈D and n∈N,
∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)>δ, |
where Gϵ(n)={z∈D:Nϕn,K(z)K(1−|z|2)≥ϵ}.
(4). There is some ϵ>0 such that the sequence of measures {χGϵ(n)dA} satisfies the reverse Carleson condition.
(5). The sequence of measures {Nϕn,K(z)K(1−|z|2)dA} satisfies the reverse Carleson condition.
(6). There is some ϵ>0 such that the sequence of Borel sets {Gϵ(n)} satisfies the reverse Carleson condition.
Proof. Suppose that w is the Denjoy-Wolff point of ϕ. By Theorem 2, w∈D. Without loss of generality, we use φw∘ϕ∘φw instead of ϕ.
(1)⇒(2). It is obvious.
(3)⇒(4). By [6], there exist a small c>0 such that K(t)tc is nondecreasing (0<t<1). Thus, the proof is similar to [18,page 5]. Let 0<r<1 and C>0 such that
∫D∖rDK(1−|z|2)dA(z)≥K(1−r2)(1−r2)c∫D∖rD(1−|z|2)cdA(z)>1−Cδ2. |
Making change of variable z=φa(w)=a−z1−¯az, we obtain
Cδ2≥∫rDK(1−|z|2)dA(z))=∫D(a,r)(1−|a|2)2|1−¯aw|4K(1−|φa(w)|2)dA(w)≥C∫D(a,r)(1−|a|2)2K(1−|a|2)|1−¯aw|4K(1−|w|2)dA(w)=C∫D(a,r)|f′a(w)|2K(1−|w|2)dA(w). |
Thus,
∫D(a,r)∩Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)=∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)−∫D(a,r)|f′a(z)|2K(1−|z|2)dA(z)≥δ−δ2=δ2. |
(2)⇒(3). Let r=supn∈N1+|ϕn(0)|2. We claim that: there exists some ϵ>0 and some δ>0 such that for all a∈D and n∈N,
∫rD|f′a(z)|2Nϕn,K(z)dA(z)>δ |
or
∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)>δ. |
Suppose that there are no ϵ,δ>0 such that the above inequalities hold. Thus, there exists sequences {ak}⊆D and {nk}⊆N such that
∫rD|f′ak(z)|2Nϕnk,K(z)dA(z)<1k |
or
∫Gϵ(n)|f′ak(z)|2K(1−|z|2)dA(z)<1k. |
Hence,
∫D|f′a(z)|2Nϕnk,K(z)dA(z)=∫rD|f′a(z)|2Nϕnk,K(z)dA(z)+∫G1k(nk)∖rD|f′a(z)|2Nϕnk,K(z)dA(z)+∫D∖(G1k(nk)∖rD)|f′a(z)|2Nϕnk,K(z)dA(z)≤1k+Lk+ηk→0, |
as k→∞. Where
L=sup|a|≥1+|ϕn(0)|2,n∈NNϕn,K(a)K(1−|a|2), η=supa∈D‖fa‖2DK. |
This contradict (2), so our claim hold. Let ϵ,δ>0 be as in above. Since f′a→0, uniformly on the compact subsets of D, as |a|→1, there exists some 0<s<1 such that for all |a|>s, we have
∫rD|f′a(z)|2Nϕnk,K(z)dA(z)≤‖f′a|rD‖2H∞‖ϕn‖2DK≤δ. |
That is, for |a|>s, we deduce that
∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)>δ. |
Similar to the proof of (3)⇒(4), there must be α,β>0 such that
|Gϵ(n)∩D(a,α)|>β|D(a,α)|, ∀|a|>s, ∀n∈N. |
Therefore,
∫Gϵ(n)∩D(a,α)K(1−|z|2)dA(z)≳β∫D(a,α)K(1−|z|2)dA(z), ∀|a|>s, ∀n∈N. |
Now if {ak} is a α-lattice for D, we have
∞∑k=1∫Gϵ(n)∩D(ak,α)K(1−|z|2)dA(z)≳β∞∑k=1∫D(ak,α)K(1−|z|2)dA(z), ∀|a|>s, ∀n∈N. |
Therefore,
∫Gϵ(n)K(1−|z|2)dA(z)≳1 ∀n∈N. |
For any |a|≤s, we obtain |f′a(z)|≳(1−s2)2K2(1−s2). Hence,
∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)≳(1−s2)2K2(1−s2)∫Gϵ(n)K(1−|z|2)dA(z)≳1. |
Therefore, (3) hold.
(5)⇒(2). Let a∈D. Then
∫D|f′a(z)|2Nϕnk,K(z)dA(z)≥∫D(a,r)|f′a(z)|2Nϕnk,K(z)dA(z)≳∫D(a,r)Nϕnk,K(z)K(1−|z|2)dA(z)≳1. |
(4)⇒(6). Note that Luecking using a long proof to show that G satisfies the reverse Carleson condition if and only if the measure χGdA(z) is a reverse Carleson measure. Simlar to the proof of [13], we omited here.
(6)⇒(1). Let f∈DK. Then
‖Cϕnf‖2DK=|f(0)|2+∫D|f′(z)|2Nϕn,K(z)dA(z)≥|f(0)|2+∫Gϵ(n)|f′a(z)|2Nϕn,K(z)dA(z)≥|f(0)|2+ϵ∫Gϵ(n)|f′a(z)|2K(1−|z|2)dA(z)≥|f(0)|2+ϵδ∫D|f′a(z)|2K(1−|z|2)dA(z)≳‖f‖2DK. |
Thus, it is easily to get our result. The proof is completed.
In this paper, we give some equivalent characterizations of power bounded and power bounded below composition operator Cϕ on Dirichlet Type spaces, which generalize the main results in [11,12].
The authors thank the referee for useful remarks and comments that led to the improvement of this paper. This work was supported by NNSF of China (No. 11801250, No.11871257), Overseas Scholarship Program for Elite Young and Middle-aged Teachers of Lingnan Normal University, Yanling Youqing Program of Lingnan Normal University, the Key Program of Lingnan Normal University (No. LZ1905), The Innovation and developing School Project of Guangdong Province (No. 2019KZDXM032) and Education Department of Shaanxi Provincial Government (No. 19JK0213).
We declare that we have no conflict of interest.
[1] | F. F. Reuss, Charge-induced flow, Proc. Imp. Soc. Nat. Moscow, 3 (1809), 327–344. |
[2] |
G. Wiedemann, Ueber die Bewegung von Flüssigkeiten im Kreise der geschlossenen galvanischen Säule, Ann. Phys., 163 (1852), 321–352. doi: 10.1002/andp.18521631102
![]() |
[3] | M. V. Smoluchowski, Elektrische endosmose und stromungsstrome, Handbuch del Elektrizitat und des Magnetismus, 2 (1921), 366. |
[4] |
D. H. Gray, Electrochemical hardening of clay soils, Geotechnique, 20 (1970), 81–93. doi: 10.1680/geot.1970.20.1.81
![]() |
[5] | A. Asadi, B. B. Huat, H. Nahazanan, H. A. Keykhah, Theory of electroosmosis in soil, Int. J. Electrochem. Sci., 8 (2013), 1016–1025. |
[6] |
V. Chokkalingam, B. Weidenhof, M. Krämer, W. F. Maier, S. Herminghaus, R. Seemann, Optimized droplet-based microfluidics scheme for sol–gel reactions, Lab Chip, 10 (2010), 1700–1705. doi: 10.1039/b926976b
![]() |
[7] |
A. Manz, C. S. Effenhauser, N. Burggraf, D. J. Harrison, K. Seiler, K. Fluri, Electroosmotic pumping and electrophoretic separations for miniaturized chemical analysis systems, J. Micromech. Microeng., 4 (1994), 257. doi: 10.1088/0960-1317/4/4/010
![]() |
[8] |
S. Deng, The parametric study of electroosmotically driven flow of power-law fluid in a cylindrical microcapillary at high zeta potential, Micromachines, 8 (2017), 344. doi: 10.3390/mi8120344
![]() |
[9] | S. Sarkar, P. M. Raj, S. Chakraborty, P. Dutta, Three-dimensional computational modeling of momentum, heat, and mass transfer in a laser surface alloying process, Numer. Heat Transfer A, 42 (2002), 307–326. |
[10] |
Y. Hu, C. Werner, D. Li, Electrokinetic transport through rough microchannels, Anal. Chem., 75 (2003), 5747–5758. doi: 10.1021/ac0347157
![]() |
[11] | G. H. Tang, X. F. Li, Y. L. He, W. Q. Tao, Electroosmotic flow of non-Newtonian fluid in microchannels, J. non-Newton. Fluid Mech., 157 (2009), 133–137. |
[12] |
Q. Liu, Y. Jian, L. Yang, Alternating current electroosmotic flow of the Jeffreys fluids through a slit microchannel, Phys. Fluids, 23 (2011), 102001. doi: 10.1063/1.3640082
![]() |
[13] |
C. Zhao, E. Zholkovskij, J. H. Masliyah, C. Yang, Analysis of electroosmotic flow of power-law fluids in a slit microchannel, J. Colloid Interf. Sci., 326 (2008), 503–510. doi: 10.1016/j.jcis.2008.06.028
![]() |
[14] | Q. S. Liu, Y. J. Jian, L. G. Yang, Time periodic electroosmotic flow of the generalized Maxwell fluids between two micro-parallel plates, J non-Newton. Fluid Mech., 166 (2011), 478–486. |
[15] |
C. Zhao, C. Yang, Joule heating induced heat transfer for electroosmotic flow of power-law fluids in a microcapillary, Int. J. Heat Mass Tran., 55 (2012), 2044–2051. doi: 10.1016/j.ijheatmasstransfer.2011.12.005
![]() |
[16] |
A. Bandopadhyay, D. Tripathi, S. Chakraborty, Electroosmosis-modulated peristaltic transport in microfluidic channels, Phys. Fluids, 28 (2016), 052002. doi: 10.1063/1.4947115
![]() |
[17] |
S. S. Hsieh, H. C. Lin, C. Y. Lin, Electroosmotic flow velocity measurements in a square microchannel, Colloid Polym. Sci., 284 (2006), 1275–1286. doi: 10.1007/s00396-006-1508-5
![]() |
[18] | S. Hadian, S. Movahed, N. Mokhtarian, Analytical study of temperature distribution of the electroosmotic flow in slit microchannels, World Appl. Sci. J., 17 (2012), 666–671. |
[19] |
M. Dejam, Derivation of dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a porous-walled microchannel, Chem. Eng. Sci., 204 (2019), 298–309. doi: 10.1016/j.ces.2019.04.027
![]() |
[20] |
J. C. Misra, A. Sinha, Electro-osmotic flow and heat transfer of a non-Newtonian fluid in a hydrophobic microchannel with Navier slip, J. Hydrodynam. Ser. B, 27 (2015), 647–657. doi: 10.1016/S1001-6058(15)60527-3
![]() |
[21] |
R. Ponalagusamy, R. Manchi, Particle-fluid two phase modeling of electro-magneto hydrodynamic pulsatile flow of Jeffrey fluid in a constricted tube under periodic body acceleration, Eur. J. Mech. B Fluid., 81 (2020), 76–92. doi: 10.1016/j.euromechflu.2020.01.007
![]() |
[22] |
M. Azari, A. Sadeghi, S. Chakraborty, Electroosmotic flow and heat transfer in a heterogeneous circular microchannel, Appl. Math. Model., 87 (2020), 640–654. doi: 10.1016/j.apm.2020.06.022
![]() |
[23] |
M. Dejam, Hydrodynamic dispersion due to a variety of flow velocity profiles in a porous-walled microfluidic channel, Int. J. Heat Mass Tran., 136 (2019), 87–98. doi: 10.1016/j.ijheatmasstransfer.2019.02.081
![]() |
[24] | A. J. Moghadam, Heat transfer in electrokinetic micro-pumps under the influence of various oscillatory excitations, Eur. J. Mech. B Fluid., 85 (2020), 158–168. |
[25] |
T. Alqahtani, S, Mellouli, A. Bamasag, F. Askri, P. E. Phelan, Thermal performance analysis of a metal hydride reactor encircled by a phase change material sandwich bed, Int. J. Hydrog. Energy, 45 (2020), 23076–23092. doi: 10.1016/j.ijhydene.2020.06.126
![]() |
[26] |
U. Khan, A. Zaib, D. Baleanu, M. Sheikholeslami, A. Wakif, Exploration of dual solutions for an enhanced cross liquid flow past a moving wedge under the significant impacts of activation energy and chemical reaction, Heliyon, 6 (2020), e04565. doi: 10.1016/j.heliyon.2020.e04565
![]() |
[27] |
H. R. Kataria, H. R. Patel, Effects of chemical reaction and heat generation/absorption on magnetohydrodynamic (MHD) casson fluid flow over an exponentially accelerated vertical plate embedded in porous medium with ramped wall temperature and ramped surface concentration, Propuls. Power Res., 8 (2019), 35–46. doi: 10.1016/j.jppr.2018.12.001
![]() |
[28] |
J. Zhao, Thermophoresis and Brownian motion effects on natural convection heat and mass transfer of fractional Oldroyd-B nanofluid, Int. J. Fluid Mech. Res., 47 (2020), 357–370. doi: 10.1615/InterJFluidMechRes.2020030598
![]() |
[29] |
P. K. Gaur, R. P. Sharma, A. K. Jha, Transient free convective radiative flow between vertical parallel plates heated/cooled asymmetrically with heat generation and slip condition, Int. J. Appl. Mech. Eng., 23 (2018), 365–384. doi: 10.2478/ijame-2018-0021
![]() |
[30] |
L. Wang, D. W. Sun, Recent developments in numerical modelling of heating and cooling processes in the food industry–a review, Trends Food Sci. Tech., 14 (2003), 408–423. doi: 10.1016/S0924-2244(03)00151-1
![]() |
[31] |
S. Islam, A. Khan, P. Kumam, H. Alrabaiah, Z. Shah, W. Khan, et al., Radiative mixed convection flow of Maxwell nanofluid over a stretching cylinder with Joule heating and heat source/sink effects, Sci. Rep., 10 (2020), 17823. doi: 10.1038/s41598-020-59925-0
![]() |
[32] |
A. Baslem, G. Sowmya, B. J. Gireesha, B. C. Prasannakumara, M. R. Gorji, N. M. Hoang, Analysis of thermal behavior of a porous fin fully wetted with nanofluids: convection and radiation, J. Mol. Liq., 307 (2020), 112920. doi: 10.1016/j.molliq.2020.112920
![]() |
[33] |
T. Hayat, M. W. A. Khan, M. I. Khan, A. Alsaedi, Nonlinear radiative heat flux and heat source/sink on entropy generation minimization rate, Physica B, 538 (2018), 95–103. doi: 10.1016/j.physb.2018.01.054
![]() |
[34] |
C. Sulochana, G. P. Ashwinkumar, N. Sandeep, Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alex. Eng. J., 57 (2018), 2573–2584. doi: 10.1016/j.aej.2017.08.006
![]() |
[35] | K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993. |
[36] |
S. Das, T. Das, S. Chakraborty, Analytical solutions for the rate of DNA hybridization in a microchannel in the presence of pressure-driven and electroosmotic flows, Sensors Actuat. B Chem., 114 (2006), 957–963. doi: 10.1016/j.snb.2005.08.012
![]() |
[37] | S. Das, S. Chakraborty, Transverse electrodes for improved DNA hybridization in microchannels, AIChE J., 53 (2007), 1086–1099. |
[38] |
D. Kumar, J. Singh, M. A. Qurashi, D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equ., 2019 (2019), 278. doi: 10.1186/s13662-019-2199-9
![]() |
[39] |
I. Ahmed, I. A. Baba, A. Yusuf, P. Kumam, W. Kumam, Analysis of Caputo fractional-order model for COVID-19 with lockdown, Adv. Differ. Equ., 2020 (2020), 1–14. doi: 10.1186/s13662-019-2438-0
![]() |
[40] |
S. Ullah, M. A. Khan, J. F. G. Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Contr. Appl. Meth., 40 (2019), 529–544. doi: 10.1002/oca.2493
![]() |
[41] |
B. Acay, E. Bas, T. Abdeljawad, Fractional economic models based on market equilibrium in the frame of different type kernels, Chaos Soliton. Fract., 130 (2020), 109438. doi: 10.1016/j.chaos.2019.109438
![]() |
[42] | A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. |
[43] | I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Amsterdam, The Netherlands: Elsevier, 1998. |
[44] | M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1–13. |
[45] | A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 4 (2016), 763–769. |
[46] | A. Gemant, XLV. On fractional differentials, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25 (1938), 540–549. |
[47] |
S. Aman, Q. A. Mdallal, I. Khan, Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium, J. King Saud Univ. Sci., 32 (2020), 450–458. doi: 10.1016/j.jksus.2018.07.007
![]() |
[48] |
A. Awan, M. D. Hisham, N. Raza, The effect of slip on electro-osmotic flow of a second-grade fluid between two plates with Caputo-Fabrizio time fractional derivatives, Can. J. Phys., 97 (2019), 509–516. doi: 10.1139/cjp-2018-0406
![]() |
[49] |
Y. Jiang, H. Qi, H. Xu, X. Jiang, Transient electroosmotic slip flow of fractional Oldroyd-B fluids, Microfluid. Nanofluid., 21 (2017), 7. doi: 10.1007/s10404-016-1843-x
![]() |
[50] |
M. I. Asjad, M. Aleem, A. Ahmadian, S. Salahshour, M. Ferrara, New trends of fractional modeling and heat and mass transfer investigation of (SWCNTs and MWCNTs)-CMC based nanofluids flow over inclined plate with generalized boundary conditions, Chin. J. Phys., 66 (2020), 497–516. doi: 10.1016/j.cjph.2020.05.026
![]() |
[51] |
C. Bardos, F. Golse, B. Perthame, The Rosseland approximation for the radiative transfer equations, Commun. Pure Appl. Math., 40 (1987), 691–721. doi: 10.1002/cpa.3160400603
![]() |
[52] |
L. M. Ottosen, A. J. Pedersen, I. R. Dalgaard, Salt-related problems in brick masonry and electrokinetic removal of salts, J. Building Appraisal, 3 (2007), 181–194. doi: 10.1057/palgrave.jba.2950074
![]() |
[53] |
S. Chakraborty, Towards a generalized representation of surface effects on pressure-driven liquid flow in microchannels, Appl. Phys. Lett., 90 (2007), 034108. doi: 10.1063/1.2433037
![]() |
[54] |
H. M. Park, W. M. Lee, Effect of viscoelasticity on the flow pattern and the volumetric flow rate in electroosmotic flows through a microchannel, Lab Chip, 8 (2008), 1163–1170. doi: 10.1039/b800185e
![]() |
[55] |
K. R. Rajagopal, M. Ruzicka, A. R. Srinivasa, On the Oberbeck-Boussinesq approximation, Math. Mod. Meth. Appl. Sci., 6 (1996), 1157–1167. doi: 10.1142/S0218202596000481
![]() |
[56] |
I. Khan, F. Ali, N. A. Shah, Interaction of magnetic field with heat and mass transfer in free convection flow of a Walters'-B fluid, Eur. Phys. J. Plus, 131 (2016), 77. doi: 10.1140/epjp/i2016-16077-7
![]() |
[57] |
F. Ali, M. Iftikhar, I. Khan, N. A. Sheikh, Aamina, K. S. Nisar, Time fractional analysis of electro-osmotic flow of Walters's-B fluid with time-dependent temperature and concentration, Alex. Eng. J., 59 (2020), 25–38. doi: 10.1016/j.aej.2019.11.020
![]() |
[58] |
F. Ali, M. Saqib, I. Khan, N. A. Sheikh, Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters'-B fluid model, Eur. Phys. J. Plus, 131 (2016), 377. doi: 10.1140/epjp/i2016-16377-x
![]() |