Open AccessArticle

Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model

Ming Shen

¹,

Yihong Liu

²,

Qingan Yin

^2,*

Hongmei Zhang

¹ and

Hui Chen

School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China

School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China

Author to whom correspondence should be addressed.

Fractal Fract. 2024, 8(8), 491; https://doi.org/10.3390/fractalfract8080491

Submission received: 1 July 2024 / Revised: 18 August 2024 / Accepted: 18 August 2024 / Published: 21 August 2024

(This article belongs to the Section Mathematical Physics)

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Browse Figures

Figure 1
Schematic diagram of the physical model. "> Figure 2
Grid independence test for varying step sizes. "> Figure 3
Comparisons between the exact solutions and numerical solutions. "> Figure 4
Impact of β on temperature (a) and concentration (b) when Pr = 5, Bi = Le = 1, Nt = 0.5 and Nb = 0.7. "> Figure 5
Impact of Pr on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Bi = Le = 1, Nt = 0.5 and Nb = 0.7. "> Figure 6
Impact of Bi on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Pr = 5, Le = 1, Nt = 0.5 and Nb = 0.7. "> Figure 7
Impact of Nb on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Pr = 5, Bi = Le = 1 and Nt = 0.5. "> Figure 8
(a,b) Impact of Nb, Bi and β on NuRex1/2. "> Figure 9
(a,b) Impact of Nb, Bi and β on ShRex1/2. ">

Versions Notes

Abstract

This paper introduces fractional Brownian motion into the study of Maxwell nanofluids over a stretching surface. Nonlinear coupled spatial fractional-order energy and mass equations are established and solved numerically by the finite difference method with Newton’s iterative technique. The quantities of physical interest are graphically presented and discussed in detail. It is found that the modified model with fractional Brownian motion is more capable of explaining the thermal conductivity enhancement. The results indicate that a reduction in the fractional parameter leads to thinner thermal and concentration boundary layers, accompanied by higher local Nusselt and Sherwood numbers. Consequently, the introduction of a fractional Brownian model not only enriches our comprehension of the thermal conductivity enhancement phenomenon but also amplifies the efficacy of heat and mass transfer within Maxwell nanofluids. This achievement demonstrates practical application potential in optimizing the efficiency of fluid heating and cooling processes, underscoring its importance in the realm of thermal management and energy conservation.

Keywords:

fractional Brownian motion; Maxwell nanofluids; Riemann–Liouville fractional derivative; improved Buongiorno model

1. Introduction

Fluid heating and cooling technologies are essential for advancing industrial and engineering systems, with their role in enhancing energy efficiency and promoting environmental sustainability becoming increasingly prominent [1]. In this field, there is a pronounced emphasis on elevating the thermal conductivity of fluids, with a crucial focus on exploring the application of nanofluids [2,3,4,5] to optimize their performance. Numerous experiments with nanofluids have demonstrated exceptionally high thermal conductivity compared to base liquids without nanoparticles [6]. Taking into account various physical mechanisms, many theories and mathematical models have been proposed to elucidate this phenomenon [7,8,9,10].

In addition to the improved effective thermal conductivity, the enhancement of convective heat transfer by nanofluids also relies on nanoparticle migration as an additional strengthening physical mechanism [11]. Buongiorno [12] reported seven factors and affirmed that Brownian diffusion and thermophoresis were dominant slip mechanisms for the abnormal convective heat transfer enhancement of nanofluids. Tiwari and Das [13] developed a model to analyze the behavior of nanofluids, taking into account the solid volume fraction.

Yang et al. [14] explored the enhancement of convective heat transfer in nanofluids, determining that particle migration led to a non-uniform distribution of thermal conductivity in the flow field, thereby improving heat conduction. These relevant findings are further supported in reference [15]. Makinde and Aziz [16] investigated the boundary layer flow of nanofluids induced by the linear stretching of a thin sheet, confirming that the enhanced Brownian motion and thermophoresis effects result in a thicker thermal boundary layer and increased local temperature. Muhammad Awais et al. [17] studied the heating/absorption effects in the boundary layer stretching flow of Maxwell nanofluids, finding that an increase in the intensity of the Brownian motion process leads to more effective movement of nanoparticles, significantly enhancing the temperature distribution and the thermal conductivity.

While there has been some progress in the field of nanofluid heat transfer mechanisms [18,19,20,21], unsolved questions and challenges still persist, making it a continuously active area of research. The non-Newtonian nature of nanofluids and the complex interactions between nanoparticles and the base fluid add to the complexity of the research. Fractional calculus has emerged as an alternative tool for modeling challenges encountered in various fields of science and engineering [22,23,24,25,26,27]. Recently, it has been discovered that fractional derivatives offer remarkable adaptability in describing the constitutive relations for the flow and heat transfer of non-Newtonian fluids and nanofluids. Numerous studies have concentrated on modeling these constitutive relationships using fractional calculus operators [28,29].

Zhang et al. [30] employed a time-fractional model to study water-based nanofluids, revealing that fractional-order models generate temperature distributions with smaller values on solid walls and within nanofluids. Asifa et al. [31] applied the Prabhakar fractional model to assess MgO-SiO₂–kerosene oil mixed nanofluids, finding higher heat transfer rates and lower surface friction coefficients. Khan et al. [32] investigated Maxwell nanofluids using Caputo–Fabrizio time-fractional derivatives, observing increased thermal conductivity in grease with nanoparticles. Asjad et al. [33] studied fractional-order MHD Oldroyd-B mixed nanofluids, confirming the impact of fractional parameters on controlling thermal and momentum boundary layers.

Pan et al. [34] established a fractional heat transfer equation using spatial fractional Caputo derivatives to describe the anomalous diffusion of nanoparticles in nanofluids based on the Das model, determining that the longitudinal flight of nanoparticles enhances convective heat transfer in nanofluids. Shen et al. [35] explored the effects of nanoparticle shape on heat conduction in Maxwell viscoelastic nanofluids using the fractional Cattaneo heat flux model, highlighting the superior enhancement of spherical nanoparticles. Additionally, Shen et al. [36] described anomalous heat transport in Sisko nanofluids, establishing an improved Buongiorno model with Caputo time-fractional derivatives for better explanation of abnormal thermal conductivity enhancement. Furthermore, Zhang et al. [37] introduced a novel time and spatial fractional heat conduction model to investigate the heat and mass transfer of Maxwell nanofluid.

Existing studies indicate that the migration of nanoparticles, serving as an additional enhancing physical mechanism, positively influences the heat transfer improvement in nanofluids [38]. Current research predominantly revolves around introducing the impact of Brownian motion into heat and mass transfer equations, with the equations used to describe Brownian motion typically being of integer order. Actually, the migration of nanoparticles exhibits characteristics of memory effects, and its influence persists on a longtime scale, thereby giving rise to the phenomenon of slow and long tails [39]. This memory effect can be described by introducing fractional calculus. In comparison to traditional integer-order Brownian motion, fractional Brownian motion more flexibly captures non-Markovian and non-Gaussian characteristics [40].

However, within the literature we have reviewed, there is currently no research on the impact of fractional Brownian motion on the heat and mass transfer of non-Newtonian nanofluids. An effort is made in this paper to address this issue by incorporating fractional Brownian motion into the Buongiorno model for investigating heat transport of Maxwell nanofluids over a stretching flat surface. The novelty of the current research lies in the application of spatial fractional-order Riemann–Liouville derivatives to redefine the Brownian motion constitutive equation. The aim of this study is to investigate how fractional Brownian motion affects the thermal and mass transfer dynamics of Maxwell nanofluids, which may contribute insights and guidance for research in the field of thermal management and the application of nanofluids.

2. Mathematical Formulation

2.1. Conservation Equations for Nanofluids with Fractional Calculus

The energy equation for Maxwell nanofluids can be written as [12]

{(ρ c)}_{f} (V \cdot \nabla T) = - \nabla \cdot q + h_{p} \nabla \cdot j_{p},

(1)

in which T is the temperature, V stands for the velocity vector, q is the energy flux,

j_{p}

is the diffusion mass flux for the nanoparticles, (ρc)_f is the heat capacity of the fluid and h_p is the specific enthalpy of the nanoparticle material.

In Buongiorno’s model,

j_{p}

is given by [12]

j_{p} = j_{p, B} + j_{p, T} = - ρ_{p} D_{B} \nabla ψ - ρ_{p} D_{T} \frac{\nabla T}{T_{\infty}},

(2)

where

ψ

is the nanoparticle concentration, ρ_p is the mass density of the nanoparticle and

D_{B}

and

D_{T}

represent the Brownian diffusion coefficient and thermophoresis diffusion coefficient, respectively. However, due to the presence of coupled effects and nonlinear properties across multiple time scales in Maxwell nanofluids, the traditional Brownian motion model fails to capture their complex behavior effectively.

Based on the continuous time random walk model, the complex behavior of nanoparticles can be described using fractional calculus when the asymptotic long-time limit of the waiting time probability density function has a heavy-tailed distribution, or when the random walks satisfy a long-tailed density of jump lengths (Lévy flights) [34]. In this context, we introduce the Riemann–Liouville fractional-order derivative into the Brownian diffusion mass flux model to describe the abnormal kinematic characteristics of nanoparticles within the base fluid, which is expressed as

j_{p, B} = - ρ_{p} D_{B} σ^{β - 1} \nabla^{β} ψ,

(3)

where σ is introduced to keep the dimension of constitutive equation balance with meter dimension, and β is the spatial fractional derivative parameter. When

ψ

ψ

(x, y)

,

\nabla^{β} ψ

is defined as

\nabla^{β} ψ = (\frac{\partial^{β} ψ}{\partial x^{β}}, \frac{\partial^{β} ψ}{\partial y^{β}}),

(4)

where the symbols ∂^β

ψ

/∂y^β are the Riemann–Liouville fractional derivative of order β (0 ≤ β < 1) given by

\frac{\partial^{β} ψ}{\partial y^{β}} = \frac{1}{Γ (n - β)} \frac{\partial^{n}}{\partial y^{n}} \int_{a}^{y} {(y - ξ)}^{- β + n - 1} ψ (x, ξ) d ξ,

(5)

so the fractional diffusion mass flux for the nanoparticles can be given by

j_{p} = j_{p, B} + j_{p, T} = - ρ_{p} D_{B} σ^{β - 1} \nabla^{β} ψ - ρ_{p} D_{T} \frac{\nabla T}{T_{\infty}} .

(6)

The heat flux is written as

q = - k_{f} \nabla T + h_{p} j_{p},

(7)

where

k_{f}

is the nanofluid thermal conductivity. Therefore, the fractional energy equation for Maxwell nanofluids in the presence of thermal radiation can be written as

{(ρ c)}_{f} (V \cdot \nabla T) = k_{f} Δ T + {(ρ c)}_{p} \nabla T \cdot (D_{B} σ^{β - 1} \nabla^{β} ψ + ρ_{p} D_{T} \frac{\nabla T}{T_{\infty}}),

(8)

where

\nabla h_{p}

is set equal to

c_{p} \nabla T

In addition, the fractional continuity equation for Maxwell nanoparticles in the absence of chemical reactions reads [12]

V \cdot \nabla ψ = - \frac{1}{ρ_{p}} \nabla \cdot j_{p},

(9)

with j_p given by Equation (7).

2.2. Problem Statement

By utilizing the convective boundary condition to elucidate energy transfer, we derive the governing equations for the flow and heat mass transfer of a two-dimensional Maxwell nanofluid over a stretching surface with Riemann–Liouville fractional Brownian diffusion. The physical depiction of the model is shown in Figure 1, where the x-axis represents the axis of fluid flow, and the y-axis is perpendicular to the x-axis. It is assumed that U_w(x) is the velocity of the stretching surface. Under the boundary layer approximation conditions, the governing equations are given as follows [12,17,36]:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(10)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = ν_{f} \frac{\partial^{2} u}{\partial y^{2}} - λ_{0} (u^{2} \frac{\partial^{2} u}{\partial x^{2}} + v^{2} \frac{\partial^{2} u}{\partial y^{2}} + 2 u v \frac{\partial^{2} u}{\partial x \partial y}),

(11)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α_{f} \frac{\partial^{2} T}{\partial y^{2}} + τ \{σ^{β - 1} D_{B} \frac{\partial T}{\partial y} \frac{\partial^{β} ψ}{\partial y^{β}} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial y})}^{2}\},

(12)

u \frac{\partial ψ}{\partial x} + v \frac{\partial ψ}{\partial y} = \frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}} + σ^{β - 1} D_{B} \frac{\partial^{β + 1} ψ}{\partial y^{β + 1}},

(13)

where ν_f is the kinematic viscosity, λ₀ is relaxation time,

{α_{f} = k}_{f}

/(ρc)_f is the thermal diffusivity of the nanofluid and τ = (ρc)_p/(ρc)_f is the ratio of effective heat capacity of the nanoparticle material to the heat capacity of the fluid. The initial and boundary conditions for the present model are given by

u (x, 0) = U_{w} (x) = \frac{U_{0}}{σ R e} x, v (x, 0) = 0, - k_{f} \frac{\partial T (x, 0)}{\partial y} = H_{f} (T_{w} - T (x, 0)), ψ (x, 0) = ψ_{w};

u (x, y) = 0, T (x, y) = T_{\infty}, ψ (x, y) = ψ_{\infty} a s y \to \infty;

where H_f is the convective heat transfer coefficient.

The dimensionless variables are introduced as

u^{*} = \frac{u {R e}^{\frac{3}{2}}}{U_{0}}, x^{*} = \frac{x {R e}^{\frac{1}{2}}}{σ}, ϕ = \frac{ψ}{ψ_{w} - ψ_{\infty}}, θ = \frac{T - T_{\infty}}{T_{w} - T_{\infty}},

(14)

v^{*} = \frac{v R e}{U_{0}}, y^{*} = \frac{y}{σ}, λ_{0}^{*} = \frac{U_{0} λ_{0}}{σ R e} .

(15)

The governing equations can be written as

\frac{\partial u^{*}}{\partial x^{*}} + \frac{\partial v^{*}}{\partial y^{*}} = 0,

(16)

u^{*} \frac{\partial u^{*}}{\partial x^{*}} + v^{*} \frac{\partial u^{*}}{\partial y^{*}} = \frac{\partial^{2} u^{*}}{\partial {y^{*}}^{2}} - λ_{0}^{*} ({u^{*}}^{2} \frac{\partial^{2} u^{*}}{\partial {x^{*}}^{2}} + {v^{*}}^{2} \frac{\partial^{2} u^{*}}{\partial {y^{*}}^{2}} + 2 u^{*} v^{*} \frac{\partial^{2} u^{*}}{\partial x^{*} \partial y^{*}}),

(17)

u^{*} \frac{\partial θ}{\partial x^{*}} + v^{*} \frac{\partial θ}{\partial y^{*}} = \frac{1}{P r} \frac{\partial^{2} θ}{\partial {y^{*}}^{2}} + \frac{N b}{P r} \frac{\partial θ}{\partial y^{*}} \frac{\partial^{β} ϕ}{\partial {y^{*}}^{β}} + \frac{N t}{P r} {(\frac{\partial θ}{\partial y^{*}})}^{2},

(18)

u^{*} \frac{\partial ϕ}{\partial x^{*}} + v^{*} \frac{\partial ϕ}{\partial y^{*}} = \frac{1}{P r L e} \frac{N t}{N b} \frac{\partial^{2} θ}{\partial {y^{*}}^{2}} + \frac{1}{P r L e} \frac{\partial^{β + 1} ϕ}{\partial {y^{*}}^{β + 1}},

(19)

in which Re is the generalized Reynolds numbers, Pr is the Prandtl number, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter and Le is the Lewis number, which are stated as follows:

N t = \frac{τ D_{T} (T_{w} - T_{\infty})}{T_{\infty} α_{f}}, N b = \frac{τ D_{B} (φ_{w} - φ_{\infty})}{α_{f}},

R e = \frac{σ U_{0}}{ν_{f}}, P r = \frac{ν_{f}}{α_{f}}, L e = \frac{α_{f}}{D_{B}} .

The corresponding initial and boundary conditions become

u^{*} (x^{*}, 0) = x^{*}, v^{*} (x^{*}, 0) = 0, \frac{\partial θ}{\partial y^{*}} = - \frac{σ H_{f}}{k_{f}} (1 - θ (x^{*}, 0)), ϕ (x^{*}, 0) = h;

(20)

u^{*} (x^{*}, y^{*}) = 0, θ (x^{*}, y^{*}) = 0, ϕ (x^{*}, y^{*}) = h - 1 a s y^{*} \to \infty;

(21)

The similarity transformations are given as

u^{*} = x^{*} f^{'} (y^{*}), v^{*} = - f (y^{*}), ϕ = ϕ (y^{*}), θ = θ (y^{*}) .

(22)

Omitting the dimensionless mart

*

for brevity, substituting (22) into (17), (18) and (19) yields

f^{‴} + f f^{″} - {f^{'}}^{2} - λ_{0} f^{2} f^{‴} + 2 λ_{0} f f^{'} f^{″} = 0,

(23)

f θ^{'} + \frac{1}{P r} θ^{″} + \frac{N t}{P r} {θ^{'}}^{2} + \frac{N b}{P r} θ^{'} \frac{d^{β} ϕ}{d y^{β}} = 0,

(24)

f φ^{'} + \frac{1}{P r L e} \frac{N t}{N b} θ^{″} + \frac{1}{P r L e} \frac{d^{β + 1} ϕ}{d y^{β + 1}} = 0 .

(25)

with

f (0) = 0, f^{'} (0) = 1; f^{'} \to 0 a s y \to \infty

(26)

{- B i (1 - θ (0)) = θ}^{'} (0); θ \to 0 a s y \to \infty,

(27)

ϕ (0) = h; ϕ \to h - 1 a s y \to \infty,

(28)

where h = ψ_w/(ψ_w − ψ_∞), Bi = σH_f/k_f is the thermal Biot number.

The local Nusselt number Nu and the Sherwood number Sh can be characterized as

N u = - \frac{x}{T_{w} - T_{\infty}} {(\frac{\partial T}{\partial y})}_{y = 0}, S h = - \frac{x}{ϕ_{w} - ϕ_{\infty}} {(\frac{\partial φ}{\partial y})}_{y = 0} .

(29)

In dimensionless form, the Equation (29) can be characterized as

N u {R e}_{x}^{1 / 2} = - θ^{'} (0), S h {R e}_{x}^{1 / 2} = - ϕ^{'} (0),

(30)

where the local Reynolds number Re_x =

\frac{U_{0} σ}{{x^{*}}^{2} ν_{f}}

3. Numerical Technique

3.1. Solution Procedure

The shooting method [41] is used to solve the integer-order differential equation for velocity (23) with boundary conditions (26) numerically. In energy and concentration Equations (24) and (25) with Riemann–Liouville fractional derivatives, we utilize a combination of finite differences and the Newton method to solve the coupled system of nonlinear fractional differential equations.

Denote y_j = jΔy (j = 0, 1, 2, …, N), where Δy = Y_max/N is space step. Based on the shifted Grünwald formulae, the space fractional derivative of order 0 ≤ β < 1 is approximated by [36]:

\frac{d^{β} ϕ}{d y^{β}} \approx \frac{1}{{(∆ y)}^{β}} \sum_{l = 0}^{j + 1} w_{l}^{β} ϕ (y_{j - l + 1}),

(31)

where the symbol w_l^β refers to the Grünwald weight coefficient [37] with w₀^β= 1 and w_l^β = (1 − (β + 1)/l) w_l₋₁^β. The iterative equations and corresponding Jacobian matrix are presented in detail in Appendix A.

The conditions for the convergence of iteration are that the absolute value of the difference between successive nodes is less than 10⁻⁶, and the iteration results tend to be stable. To demonstrate the stability of the numerical solution, we employed a series of meshes with varying densities, observing that as the grid spacing decreases, the numerical results converge to a consistent value, thereby validating the stability of the numerical approach, as shown in Figure 2.

In order to validate the effectiveness of our numerical methods, we compare the results of

θ^{'} (0) {a n d ϕ}^{'} (0)

obtained in our current analysis with those reported by Muhammad Awais et al. [17]. Importantly, our findings demonstrate a reasonable level of agreement with the results presented by Muhammad Awais, as summarized in Table 1.

3.2. Numerical Example

By introducing two source terms, f₁ and f₂, we propose one numerical example to verify the validity of the numerical schemes. The pertinent parameters are used: Pr = 5, Le = 1, Nb = 1, Nt = 2 and β = 0.8. We consider the equations of the temperature and concentration as follows

f θ^{'} + \frac{1}{P r} θ^{″} + \frac{N t}{P r} {θ^{'}}^{2} + \frac{N b}{P r} θ^{'} \frac{d^{β} ϕ}{d y^{β}} - f_{1} = 0,

(32)

f φ^{'} + \frac{1}{P r L e} \frac{N t}{N b} θ^{″} + \frac{1}{P r L e} \frac{d^{β + 1} ϕ}{d y^{β + 1}} {- f}_{2} = 0,

(33)

with the new conditions:

θ (0) = 0; θ \to 0 a s y \to 1 .

(34)

ϕ (0) = 0; ϕ \to 0 a s y \to 1 .

(35)

Here, f₁ and f₂ are given by

f_{1} = f (4 y^{3} - 6 y^{2} + 2 y) + \frac{1}{P r} (12 y^{2} - 12 y + 2) + \frac{N t}{P r} {(4 y^{3} - 6 y^{2} + 2 y)}^{2} + \frac{N b}{P r} (4 y^{3} - 6 y^{2} + 2 y) O_{1},

f_{2} = f (4 y^{3} - 6 y^{2} + 2 y) + \frac{1}{P r L e} \frac{N t}{N b} (12 y^{2} - 12 y + 2) + \frac{1}{P r L e} O_{2},

in which

O_{1} = \frac{24 y^{4 - β}}{Γ (5 - β)} - \frac{12 y^{3 - β}}{Γ (4 - β)} + \frac{2 y^{2 - β}}{Γ (3 - β)},

O_{2} = \frac{24 y^{3 - β}}{Γ (4 - β)} - \frac{12 y^{2 - β}}{Γ (3 - β)} + \frac{2 y^{1 - β}}{Γ (2 - β)} .

The exact solution is constructed as

θ (y) = y^{2} {(1 - y)}^{2}, ϕ (y) = y^{2} {(1 - y)}^{2} .

Figure 3 shows the temperature and concentration distributions along the y direction obtained by the numerical solution and exact solution. It can be found that the numerical solution agrees well with the exact solution. Table 2 presents the

L_{\infty}

error at various step sizes. The derived convergence order is estimated to be around 1.03, which aligns with our expected convergence rate, confirming the accuracy and reliability of the numerical method.

4. Results and Discussion

In the model under our consideration, fractional Brownian motion exclusively influences temperature and concentration. Consequently, our analysis is centered on probing the influence of crucial physical parameters, including the Prandtl number Pr, the Brownian motion parameter Nb and the thermal Biot number Bi, on temperature, concentration, as well as the Nusselt and Sherwood numbers in the context of fractional Brownian motion.

4.1. Temperature and Concentration Profiles

Figure 4 illustrates the temperature and concentration distributions for various values of β. When the value of the spatial fractional derivative parameter β is smaller, the temperature decreases, resulting in thinner thermal boundary layers, as depicted in Figure 4a. These results indicate that decreasing β enhances heat transfer efficiency. As shown in Figure 4b, as β increases, the concentration near the surface (y < 1.8) significantly intensifies. When y > 1.8, the influence of β on the concentration diminishes. Furthermore, intersection points appear in the concentration curves when moving away from the wall surface (y > 1.8). Unlike integer-order derivatives, fractional-order derivatives encompass more historical information and non-local effects. Consequently, the emergence of intersection points implies a heightened sensitivity of the system to past history.

Figure 5 illustrates the temperature and concentration distributions under different values of Pr for two scenarios where β is set to 0.8 and 1.0. Observing Figure 5, it is noted that the temperature and concentration decrease with an increase in Pr for both scenarios. Physically, the Prandtl number Pr represents the ratio of momentum diffusivity to thermal diffusivity. With a high Pr, heat diffusion in the fluid may slow down, resulting in a thinner thermal boundary layer and potentially a greater temperature difference. Importantly, compared to the integer-order case, at the same Pr, the temperature and concentration are lower when β is 0.8, indicating smaller temperature and concentration boundary layers and higher heat and mass transfer efficiency.

Figure 6 and Figure 7 describe temperature and concentration distributions for different values of Bi and Nb for two scenarios, with β being 0.8 and 1.0 respectively. Figure 6a and Figure 7a indicate that the temperature increases with the growth of Bi and Nb in both scenarios. For the same Bi and Nb values, the temperature in the integer-order case is higher, resulting in a thicker concentration boundary layer and slower mass transfer efficiency. Moreover, it is further observed that the temperature is more sensitive to changes in Bi than in Nb, which is consistent with physical reality. The impact of these two parameters on concentration is illustrated in Figure 6b and Figure 7b. It is observed that the concentration increases with the growth of Bi in both scenarios, but Nb exhibits an opposite trend in its effect on concentration, which is due to the fact that increasing the Brownian motion parameter promotes molecular collisions, thereby accelerating the process of mass transfer. Additionally, under the same Bi and Nb values, the integer-order case consistently shows higher concentrations, resulting in slower mass transfer.

From Figure 4, Figure 5, Figure 6 and Figure 7, it is evident that fractional Brownian motion plays a crucial role in heat and mass transfer. By enhancing molecular collisions and diffusion effects, it results in a thinner temperature and concentration boundary layer, thereby promoting more pronounced variations, particularly in concentration, within nanofluids. Therefore, the introduction of fractional Brownian motion enhances the efficiency of heat and mass transfer in nanofluids, holding significant importance for understanding and optimizing the heat and mass transfer processes in nanofluids.

4.2. Nusselt Number and Sherwood Number

Local Nusselt (Nu) and Sherwood numbers (Sh) are important parameters for reflecting the efficiency of surface heat and mass transfer. As shown in Figure 8 and Figure 9, reducing Nb and increasing Bi increases the numerical values of local Nusselt numbers, but the numerical values of local Sherwood numbers exhibit the opposite trend. Upon examination of Figure 9, it can be observed that Sherwood numbers exhibit insensitivity to variations in Nb and Bi, thereby further confirming the accuracy of the computational results. The variations in local Nusselt numbers and local Sherwood numbers with respect to key physical parameters, including β, Bi, Pr and Nb, are provided in detail in Table 3.

It is noteworthy that a reduction in β leads to an increase in the numerical values of local Nusselt and Sherwood numbers, as depicted in Figure 8 and Figure 9. This observation provides additional confirmation that fractional-order models offer significant advantages in enhancing heat and mass transfer efficiency when compared to their integer-order counterparts.

5. Conclusions

This study explores the mass and heat transfer for a 2D steady Maxwell viscoelastic nanofluid considering convective boundary conditions. The innovation lies in the introduction of a fractional model for the mass flux caused by Brownian diffusion of Maxwell nanofluids. Nonlinear coupled spatial fractional-order energy and mass equations are established by replacing spatial derivatives with Riemann–Liouville fractional derivatives. Numerical solutions are obtained using the finite difference method incorporating the shifted Grünwald formula, with the shooting method and Newton’s iterative technique. The key findings derived from this study are as follows:

The parameter β exhibits a modest influence on temperature fluctuations. In contrast, it markedly influences concentration distributions, with pronounced effects observed in the vicinity of boundary surface.
Intersection points appearing in the concentration curves for different values of β reflect the system’s sensitivity to historical states. As the value of β decreases, the system’s response to historical influences becomes more sensitive.
At β = 0.8, concentrations and temperatures near the wall are lower compared to the integer-order case against the parameters Pr, Bi and Nb, respectively.
At β = 0.8, the local Nusselt number experiences a modest increase, whereas the local Sherwood number demonstrates a significant enhancement contrast to the performance observed in the integer-order case.
Decreasing the value of β results in higher local Nusselt number and Sherwood number, which highlights the enhancement in the efficiency of thermal and mass transfer processes.

At present, the research has predominantly focused on the impact of fractional Brownian motion. Future studies on heat and mass transfer in nanofluids could delve deeper by considering the combined effects of anomalous heat and mass transfer. It is envisioned that the development of more sophisticated numerical methods to simulate the anomalous heat and mass transfer phenomena in nanofluids will further advance the field of nanofluid science. These endeavors are anticipated to lay theoretical groundwork for achieving more efficient energy conversion and utilization.

Author Contributions

Conceptualization, M.S.; methodology, M.S. and H.Z.; investigation, Y.L.; resources, Q.Y.; writing—original draft preparation, Y.L.; writing—review and editing, M.S. and Q.Y.; supervision, H.C.; funding acquisition, M.S. and H.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Natural Science Foundations of Fujian Province, grant number 2021 J01618 and 2023 J01415.

Data Availability Statement

The available data have been presented within this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Perform finite differencing on Equations (24) and (25) and substitute into (31) to obtain the following discrete form:

f_{j} \frac{θ_{j + 1} - θ_{j}}{h} + \frac{1}{P r} \frac{θ_{j + 1} - {2 θ}_{j} + θ_{j - 1}}{h^{2}} \begin{matrix} + \frac{N_{t}}{P r} {(\frac{θ_{j + 1} - θ_{j}}{h})}^{2} + \frac{N_{b}}{P r} \frac{θ_{j + 1} - θ_{j}}{h^{β + 1}} \sum_{l = 0}^{j + 1} w_{l}^{β} ϕ (y_{j - l + 1}) = 0, \end{matrix}

(A1)

\begin{matrix} f_{j} \frac{φ_{j + 1} - φ_{j}}{h} + \frac{1}{P r L e} \frac{N_{t}}{N_{b}} \frac{θ_{j + 1} - {2 θ}_{j} + θ_{j - 1}}{h^{2}} \\ + \frac{1}{P r L e} \frac{1}{h^{β + 1}} \sum_{l = 0}^{j + 1} w_{l}^{β + 1} ϕ (y_{j - l + 1}) = 0, \end{matrix}

(A2)

where Δy = h, j = 1, 2, …, N − 1.

We are using the Newton–Raphson iteration method to solve Equations (32) and (33). Next, we provide the Jacobian matrix J required during the solution process

J = (\begin{matrix} A & B \\ C & D \end{matrix}),

(A3)

where

A = {(a_{j j})}_{j = 1}^{N - 1}, B = {(b_{j j})}_{j = 1}^{N - 1}, C = {(c_{j j})}_{j = 1}^{N - 1}, D = {(d_{j j})}_{j = 1}^{N - 1} .

a_{j j} = - \frac{f_{j}}{h} - \frac{1}{P r} \frac{2}{h^{2}} + \frac{{2 N}_{t}}{h^{2} P r} (θ_{j} - θ_{j + 1}) - \frac{N_{b}}{h^{β + 1} P r} \sum_{l = 0}^{j + 1} w_{l}^{β} ϕ (y_{j + 1 - l}),

a_{j, j + 1} = \frac{f_{j}}{h} + \frac{1}{P r} \frac{1}{h^{2}} + \frac{{2 N}_{t}}{h^{2} P r} (θ_{j + 1} - θ_{j}) + \frac{N_{b}}{h^{β + 1} P r} \sum_{l = 0}^{j + 1} w_{l}^{β} ϕ (y_{j + 1 - l}),

a_{j, j - 1} = \frac{1}{P r} \frac{1}{h^{2}},

b_{j j} = \frac{N_{b} {(θ}_{j + 1} - θ_{j}) w_{1}^{β}}{h^{β + 1} P r}, b_{j, j + 1} = \frac{N_{b} {(θ}_{j + 1} - θ_{j}) w_{0}^{β}}{h^{β + 1} P r}, b_{j, j - 1} = \frac{N_{b} {(θ}_{j + 1} - θ_{j}) w_{2}^{β}}{h^{β + 1} P r},

b_{j k} = \frac{N_{b} {(θ}_{j + 1} - θ_{j}) w_{j - k + 1}^{β}}{h^{β + 1} P r}, w h e r e j - k \geq 2, k = 1,2, \dots N - 1 .

c_{j j} = - \frac{2 N_{t}}{P r L e N_{b} h^{2}}, c_{j, j + 1} = \frac{N_{t}}{P r L e N_{b} h^{2}}, c_{j, j - 1} = \frac{N_{t}}{P r L e N_{b} h^{2}},

d_{j j} = - \frac{f_{j}}{h} + \frac{w_{1}^{β + 1}}{P r L e h^{β + 1}}, d_{j, j + 1} = \frac{f_{j}}{h} + \frac{w_{0}^{β + 1}}{P r L e h^{β + 1}}, d_{j, j - 1} = \frac{w_{2}^{β + 1}}{P r L e h^{β + 1}},

d_{j k} = \frac{w_{j - k + 1}^{β + 1}}{P r L e h^{β + 1}}, w h e r e j - k \geq 2, k = 1,2, \dots N - 1 .

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$Fractalfract 08 00491 g001$

Figure 1. Schematic diagram of the physical model.

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Figure 2. Grid independence test for varying step sizes.

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Figure 3. Comparisons between the exact solutions and numerical solutions.

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Figure 4. Impact of β on temperature (a) and concentration (b) when Pr = 5, Bi = Le = 1, Nt = 0.5 and Nb = 0.7.

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Figure 5. Impact of Pr on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Bi = Le = 1, Nt = 0.5 and Nb = 0.7.

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Figure 6. Impact of Bi on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Pr = 5, Le = 1, Nt = 0.5 and Nb = 0.7.

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Figure 7. Impact of Nb on temperature (a) and concentration (b) for β = 0.8 and β = 1.0 when Pr = 5, Bi = Le = 1 and Nt = 0.5.

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Figure 8. (a,b) Impact of Nb, Bi and β on NuRe_x^1/2.

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Figure 9. (a,b) Impact of Nb, Bi and β on ShRe_x^1/2.

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Table 1. Comparison of results for

θ^{'} (0) a n d ϕ^{'} (0) w h e n λ = 0, N b = 2.5 = N t, β = 1 = L e, P r = 5.0

Table 1. Comparison of results for

θ^{'} (0) a n d ϕ^{'} (0) w h e n λ = 0, N b = 2.5 = N t, β = 1 = L e, P r = 5.0

$B i$	$θ^{'} (0)$		$ϕ^{'} (0)$
$B i$	Present	Muhammad Awais et al. [17]	Present	Muhammad Awais et al. [17]
1	0.1482	0.1476	1.6841	1.6914
10	0.1557	0.1549	1.7046	1.7122
100	0.1564	0.1557	1.7067	1.7144
$\infty$	0.1565	0.1557	1.7070	1.7146

Table 2. The L_∞ error of the space discretization for the temperature and the concentration.

Δy	$θ$	ϕ
Δy	L_∞ Error	L_∞ Error
0.0100	5.06216 × 10⁻⁴	4.68095 × 10⁻⁴
0.0050	2.62164 × 10⁻⁴	2.34746 × 10⁻⁴
0.0025	1.33385 × 10⁻⁴	1.17656 × 10⁻⁴
0.0010	5.39142 × 10⁻⁵	4.71596 × 10⁻⁵

Table 3. Numerical results of NuRe_x^1/2 and ShRe_x^1/2.

β	Bi	Pr	Nb	NuRe_x^1/2 (Actual)	NuRe_x^1/2 (Fitting)	ShRe_x^1/2 (Actual)	ShRe_x^1/2 (Fitting)
0.80	1	10	0.5	0.65486	0.65679	9.81537	9.84340
0.85	1	10	0.5	0.64924	0.64891	7.87356	7.87671
0.90	1	10	0.5	0.64230	0.64102	5.91359	5.91003
0.95	1	10	0.5	0.63369	0.63314	3.93860	3.94334
1.00	1	10	0.5	0.62298	0.62526	1.95130	1.97666
0.9	0.8	10	0.5	0.55484	0.56000	5.96766	5.95982
0.9	0.9	10	0.5	0.60031	0.60051	5.93946	5.93492
0.9	1.0	10	0.5	0.64230	0.64102	5.91359	5.91003
0.9	1.1	10	0.5	0.68116	0.68154	5.88979	5.88513
0.9	1.2	10	0.5	0.71722	0.72205	5.86784	5.86024
0.9	1	8	0.5	0.61279	0.61431	5.65584	5.66084
0.9	1	8.5	0.5	0.62094	0.62099	5.72303	5.72314
0.9	1	9	0.5	0.62854	0.62767	5.78829	5.78543
0.9	1	9.5	0.5	0.63564	0.63435	5.85177	5.84773
0.9	1	10	0.5	0.64230	0.64102	5.91359	5.91003
0.9	1	10	0.40	0.64996	0.64878	5.79001	5.80835
0.9	1	10	0.45	0.64615	0.64490	5.85863	5.85919
0.9	1	10	0.50	0.64230	0.64102	5.91359	5.91003
0.9	1	10	0.55	0.63840	0.63715	5.95862	5.96087
0.9	1	10	0.60	0.63446	0.63327	5.99621	6.01171

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Shen, M.; Liu, Y.; Yin, Q.; Zhang, H.; Chen, H. Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model. Fractal Fract. 2024, 8, 491. https://doi.org/10.3390/fractalfract8080491

AMA Style

Shen M, Liu Y, Yin Q, Zhang H, Chen H. Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model. Fractal and Fractional. 2024; 8(8):491. https://doi.org/10.3390/fractalfract8080491

Chicago/Turabian Style

Shen, Ming, Yihong Liu, Qingan Yin, Hongmei Zhang, and Hui Chen. 2024. "Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model" Fractal and Fractional 8, no. 8: 491. https://doi.org/10.3390/fractalfract8080491

APA Style

Shen, M., Liu, Y., Yin, Q., Zhang, H., & Chen, H. (2024). Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model. Fractal and Fractional, 8(8), 491. https://doi.org/10.3390/fractalfract8080491

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Enhanced Thermal and Mass Diffusion in Maxwell Nanofluid: A Fractional Brownian Motion Model

Abstract

1. Introduction

2. Mathematical Formulation

2.1. Conservation Equations for Nanofluids with Fractional Calculus

2.2. Problem Statement

3. Numerical Technique

3.1. Solution Procedure

3.2. Numerical Example

4. Results and Discussion

4.1. Temperature and Concentration Profiles

4.2. Nusselt Number and Sherwood Number

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A

References

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