Systems of Navier-Stokes Equations on Cantor Sets

Abstract

We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media.

1. Introduction

The Navier-Stokes equations [1–3] are commonly used in describing motion of fluids in models relevant to weather, ocean currents, water flow in pipes, and so forth.

However, the turbulent flows may be of fractal character, and the smoothness required in the common problem modelled by the Navier-Stokes equations may be violated in fractal flows. Such distortions exist in both small-scale [4] and large-scale [5, 6] turbulent flows. In this context, Benzi et al. [7] investigated the scaling properties of turbulent flows while She and Leveque [8] stressed the attention on universal scaling laws in fully developed turbulence. Shraiman and Siggia also reported scalar turbulence [9], while the self-similarity in turbulent flows was analyzed by [10]. The fractal dimension of turbulent flows was pointed out in several studies [10–12], and some applications involving the Navier-Stokes equations were suggested [13–15] taking into account the multifractal nature of fully developed turbulence and chaotic systems [16]. In this context, the alpha model of turbulence [17, 18] and its applications to Navier Stokes have been discussed [19].

Based on fractional calculus theory [20–23] time-fractional Navier-Stokes equation has been recently proposed by El-Shahed and Salem [24]. Several analytical solutions have been developed by the Adomian decomposition method [25] and the homotopy perturbation method [26].

Recently, the element of fractal arc length squared in fractal time-space was discussed [27]. Based on it, the Schrödinger equation [28], the heat conduction equation [27, 29], the wave equation [27, 30], and the diffusion equation on Cantor time-space [31] were suggested based on local fractional calculus theory [27, 32, 33]. Carpinteri and Sapora had reported diffusion equation on Cantor space [34]. Kolwankar and Gangal had proposed the Fokker-Planck equation on Cantor space [35].

In this paper we employ the local fractional vector calculus [27], which is set up on fractals, to model systems of Navier-Stokes equations on Cantor sets. The paper is organized as follows. In Section 2, a short introduction to local fractional vector calculus theory is given. Fractal kinematics, continuity equations, and constitutive equations for Newtonian fluid on Cantor sets are discussed in Section 3. Cauchy’s equations of motion and mechanical energy equations on Cantor sets are investigated in Section 4. Systems of Navier-Stokes equations on Cantor sets with Cantorian coordinates systems are presented in Section 5. Finally, in Section 6, the conclusions are presented.

2. Mathematical Tool

Local fractional vector form of function gives [27] 𝐫(𝑥)=𝑀(𝑥)𝐞^𝛼₁+𝑁(𝑥)𝐞^𝛼₂ (1) which is expressed by 𝐫(𝑥)=(𝑀(𝑥),𝑁(𝑥)). (2) For a function of three variables, the vector form can be written in the form [27] 𝐫𝑥,𝑦,𝑧=𝐿𝑥,𝑦,𝑧𝐞^𝛼₁+𝑀𝑥,𝑦,𝑧𝐞^𝛼₂+𝑁𝑥,𝑦,𝑧𝐞^𝛼₃, (3) or 𝐫𝑥,𝑦=𝐿𝑥,𝑦,𝑧,𝑀𝑥,𝑦,𝑧,𝑁𝑥,𝑦,𝑧. (4) Let 𝐮(𝑥,𝑦,𝑧)=𝑢(𝑥,𝑦,𝑧)𝐞^𝛼₁+𝑢₂(𝑥,𝑦,𝑧)𝐞^𝛼₂+𝑢₃(𝑥,𝑦,𝑧)𝐞^𝛼₃ a local fractional vector field and 𝜑(𝑥,𝑦,𝑧) be a local fractional scalar field, the computing rules of Hamilton operator are valid as follows [27]. (1)The local fractional gradient and Laplace operator of a local fractional scalar field are, respectively, defined as [27] ∇^𝛼𝜕𝜑=^𝛼𝜑𝜕𝑥^𝛼₁𝐞^𝛼₁+𝜕^𝛼𝜑𝜕𝑥^𝛼₂𝐞^𝛼₂+𝜕^𝛼𝜑𝜕𝑥^𝛼₃𝐞^𝛼₃,∇^2𝛼𝜕𝜑=^2𝛼𝜑𝜕𝑥₁^2𝛼+𝜕^2𝛼𝜑𝜕𝑥₂^2𝛼+𝜕^2𝛼𝜑𝜕𝑥₃^2𝛼, (5) where local fractional partial derivatives of high order are defined as [27, 31–33] 𝜕^𝑘𝛼𝜕𝑥^𝑘𝛼𝑓(𝑥)=^𝑘times𝜕^𝛼𝜕𝑥^𝛼⋯𝜕^𝛼𝜕𝑥^𝛼𝑓(𝑥). (6)(2)The local fractional divergence and curl of a local fractional vector field are written in the form [27, 31] 𝐝𝐢𝐯^𝛼𝐮=∇^𝛼𝜕⋅𝐮=^𝛼𝑢₁𝜕𝑥^𝛼₁+𝜕^𝛼𝑢₂𝜕𝑥^𝛼₂+𝜕^𝛼𝑢₃𝜕𝑥^𝛼₃,𝐜𝐮𝐫𝐥^𝛼𝐮=∇^𝛼𝜕×𝐮=^𝛼𝑢₃𝜕𝑥^𝛼₂−𝜕^𝛼𝑢₂𝜕𝑥^𝛼₃𝐞^𝛼₁+𝜕^𝛼𝑢₁𝜕𝑥^𝛼₃−𝜕^𝛼𝑢₃𝜕𝑥^𝛼₁𝐞^𝛼₂+𝜕^𝛼𝑢₂𝜕𝑥^𝛼₁−𝜕^𝛼𝑢₁𝜕𝑥^𝛼₂𝐞^𝛼₃. (7)(3)The following equations are valid only in Cantor coordinates [27]: ∇^𝛼(𝐮𝐯)=∇^𝛼𝐮𝐯+𝐮∇^𝛼∇𝐯,_2𝛼𝐀=∇^𝛼∇^𝛼⋅𝐀−∇^𝛼×∇^𝛼×𝐀,𝐀⋅∇^𝛼𝐀=∇^𝛼𝐀⋅𝐀2−∇^𝛼×𝐀×𝐀, (8) where ∇^2𝛼=∇^𝛼⋅∇^𝛼 is local fractional Laplace operator [27].

The local fractional line integral of the function 𝐮(𝑥_𝑃,𝑦_𝑃,𝑧_𝑃) in the local fractional vector form 𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃=𝑢₁𝑥_𝑃,𝑦_𝑃,𝑧_𝑃𝐞^𝛼₁+𝑢₂𝑥_𝑃,𝑦_𝑃,𝑧_𝑃𝐞^𝛼₂+𝑢₃𝑥_𝑃,𝑦_𝑃,𝑧_𝑃𝐞^𝛼₃ (9) along a fractal line 𝑙^𝛼 is written as [27] _𝑙(𝛼)𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅𝑑𝐥^𝛼=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅Δ𝐥^𝛼_𝑃, (10) where for 𝑃=1,2,…,𝑁 and 𝑁 elements of line Δ𝐥^𝛼_𝑃, it is required that all |Δ𝑙^𝛼_𝑃|→0 as 𝑁→∞.

The local fractional surface integral of the given function (9) across a surface 𝑆^(𝛽) is defined as [27] _𝑆(𝛽)𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅𝑑𝐒^(𝛽)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅𝐧_𝑃Δ𝑆_𝑃^(𝛽), (11) where for 𝑃=1,2,…,𝑁 and 𝑁 elements of area Δ𝑆_𝑃^(𝛽) with a unit normal local fractional vector 𝑛_𝑃, it is required that all Δ𝑆_𝑃^(𝛽)→0 as 𝑁→∞.

The local fractional volume integral of the given function (9) in a fractal region 𝑉^(𝛾) is given by [27] _𝑉(𝛾)𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃𝑑𝑉^(𝛾)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃Δ𝑉_𝑃^(𝛾), (12) where for 𝑃=1,2,…,𝑁 and 𝑁 elements of volume Δ𝑉_𝑃^(𝛾), it is required that all Δ𝑉_𝑃^(𝛾)→0 as 𝑁→∞.

Let us consider a local fractional vector field 𝐮=𝑢₁𝐞^𝛼₁+𝑢₂𝐞^𝛼₂+𝑢₃𝐞^𝛼₃, the following results hold [27]: (a)Divergence Theorem of local fractional field states that _𝑉(𝛾)∇^𝛼⋅𝐮𝑑𝑉^(𝛾)=_𝑆(𝛽)𝐮⋅𝑑𝐒^(𝛽). (13)(b)Stokes’ theorem of local fractional field states that _𝑙(𝛼)𝐮⋅𝑑𝐥^𝛼=_𝑆(𝛽)∇^𝛼×𝐮⋅𝑑𝐒^(𝛽). (14)(c)Green’s first theorem in fractal domain states that _𝑆(𝛽)𝜙∇^𝛼𝜑⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜙∇^2𝛼𝜑+∇^𝛼𝜙⋅∇^𝛼𝜑𝑑𝑉^(𝛾). (15)(d)Green’s second theorem in fractal domain states that _𝑆(𝛽)𝜙∇^𝛼𝜑−𝜙∇^𝛼𝜑⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜙∇^2𝛼𝜑−𝜑∇^2𝛼𝜙𝑑𝑉^(𝛾), (16) where the relationships of fractal dimensions of line, surface, and volume fields are 𝛽=2𝛼,𝛾=3𝛼. (17)

3. Fractal Kinematics, Continuity Equations, and Constitutive Equations for Newtonian Fluid on Cantor Sets

In this section, we investigate the fractal kinematics, balance of mass on Cantor sets, and constitutive equations for Newtonian fluid on Cantor sets. We first start with fractal kinematics.

3.1. Fractal Kinematics

For any scalar property 𝜓 associated with an abstract Cantor body Β expressed as 𝜓=𝜓(𝑃,𝑡)=𝜓(𝑋,𝑡)=𝜓(𝑥,𝑡), (18) we define the following Lagrangian and Eulerian temporal and spatial derivatives with local fractional differential operator: ̇𝐷𝜓=^𝛼𝜓𝐷𝑡^𝛼,𝜓^𝛼_,𝑡=𝜕^𝛼𝜓𝜕𝑡^𝛼,∇^𝛼_𝑋𝜕𝜓=^𝛼𝜓𝜕𝑋^𝛼,∇^𝛼_𝑥𝜓=∇^𝛼𝜕𝜓=^𝛼𝜓𝜕𝑥^𝛼. (19) The velocity 𝝊 and the acceleration 𝐚 with local fractional operator are defined through 𝐷𝝊=^𝛼𝝌𝐷𝑡^𝛼𝐷,𝐚=^2𝛼𝝌𝐷𝑡^2𝛼. (20) It immediately follows that the velocity gradient Θ and its symmetric part Λ are expressed through Θ=∇^𝛼⋅𝝊,Θ^𝑇=𝝊⋅∇^𝛼1,Λ=2Θ+Θ^𝑇. (21) The deformation gradient 𝚺 is given by 𝚺=∇^𝛼⋅𝝌, (22) which leads to 𝜕Θ=^𝛼𝚺𝜕𝑡^𝛼𝚺⁻¹. (23) The fractal material derivative of the fluid density 𝜌 is defined as [27] 𝐷^𝛼𝜌𝐷𝑡^𝛼=𝜕^𝛼𝜌𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝜌. (24)

3.2. Balance of Mass on Cantor Sets

The mass of fractal fluid in 𝑉^(𝛾) is defined through [27] 𝑀=_𝑉(𝛾)𝜌𝑑𝑉^(𝛾), (25) which yields that the balance of mass on Cantor sets takes the form [27] 𝜕^𝛼𝑀𝜕𝑡^𝛼=−_𝑆(𝛽)𝜌𝝊⋅𝑑𝐒^(𝛽). (26) Using Divergence Theorem of local fractional field, we deduce to _𝑉(𝛾)𝜕^𝛼𝜕𝑡^𝛼𝜌𝑑𝑉^(𝛾)+_𝑆(𝛽)𝜌𝝊⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼⋅𝜌𝝊𝑑𝑉^(𝛾)=0 (27) which implies that 𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼⋅𝜌𝝊=0. (28) This is called continuity equation on Cantor sets.

It becomes 𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼⋅=𝜕𝜌𝝊^𝛼𝜌𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝜌+𝜌∇^𝛼=𝐷⋅𝝊_𝛼𝜌𝐷𝑡^𝛼+𝜌∇^𝛼⋅𝝊=0, (29) where the fractal material derivative of the fluid density 𝜌 is noted by 𝐷^𝛼𝜌𝐷𝑡^𝛼=𝜕^𝛼𝜌𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝜌. (30) If the fractal fluid is incompressible, we deduce that 𝜕^𝛼𝜌𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝜌𝐷=0⟺^𝛼𝜌𝐷𝑡^𝛼=0or∇^𝛼⋅𝝊=0. (31) It follows that Reynolds transport theorem on Cantor sets states that 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝐹(𝑥,𝑡)𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜕^𝛼𝜕𝑡^𝛼𝐹(𝑥,𝑡)𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐹(𝑥,𝑡)𝝊⋅𝑑𝐒^(𝛽), (32) where 𝝊 is the fractal fluid velocity, 𝑉^(𝛾) is the fractal material volume, and 𝐒^(𝛽) is the fractal surfaces moving with the fractal fluid.

3.3. Constitutive Equations for Newtonian Fluid on Cantor Sets

We will assume a linear relation of the type of fractal Cauchy stress 𝐉=−𝑝𝐈+𝐊∶Λ, (33) where 𝑝 is the thermodynamic pressure, 𝐊 is a fourth-order fractal tensor, Λ is the fractal strain rate tensor, and 𝐈 is unit vector in local fractional field.

The fractal velocity gradient tensor can be decomposed into symmetric and antisymmetric parts [27] ∇^𝛼1⋅𝝊=2Θ+Θ^𝑇+12Θ−Θ^𝑇1=Λ+2Θ−Θ^𝑇. (34) We can write 𝐊∶Λ=2𝜇Λ+𝜆∇^𝛼⋅𝝊𝐈 (35) which leads to 𝐉=−𝑝𝐈+2𝜇Λ+𝜆∇^𝛼⋅𝝊𝐈, (36) where (∇^𝛼⋅𝝊) is the fractal volumetric strain rate and Λ=(1/2)(∇^𝛼⋅𝝊+𝝊⋅∇^𝛼) [27], 𝜆 and 𝜇 are the bulk and shear moduli of viscosity.

The constitutive equation of homogeneous compressible Euler fluid on Cantor sets reads 𝐉=−𝑝𝜌𝐈. (37) The constitutive equation of homogeneous incompressible Euler fluid on Cantor sets is 𝐉=−𝑝𝐈,∇^𝛼⋅𝝊=0. (38) The constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets is written in the form 𝐉=−𝑝𝜌𝐈+2𝜇𝜌Λ+𝜆𝜌∇^𝛼⋅𝝊𝐼. (39) The constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets is expressed as 𝐉=−𝑝𝐈+2𝜇Λ,∇^𝛼⋅𝝊=0. (40) By using Stokes’ assumption 𝜆(𝜌)=−2𝜇(𝜌)/3, the constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets becomes 𝐉=−𝑝𝜌𝐈+2𝜇𝜌∇^𝛼⋅𝝊+𝝊⋅∇^𝛼−23𝜇𝜌∇^𝛼⋅𝝊, (41) while the constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets can obtain that 𝐉=−𝑝𝐈+𝜇∇^𝛼⋅𝝊+𝝊⋅∇^𝛼,∇^𝛼⋅𝝊=0. (42)

4. Cauchy’s Equations of Motion and Mechanical Energy Equations on Cantor Sets

In this section, we consider balance of linear and angular momentums and energy on Cantor sets.

4.1. Balances of Linear and Angular Momentums on Cantor Sets

If the second law of Newton in fractal mechanics is valid, the balance of linear momentum for flows on Cantor sets takes the form [27] 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝝊𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜌𝐛𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐉⋅𝑑𝐒^(𝛽), (43) where 𝐉 denotes the fractal Cauchy stress tensor [27] and 𝐛 denotes the specific fractal body force [27].

By using (32), (43) becomes 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝝊𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜕^𝛼𝜕𝑡^𝛼𝜌𝝊𝑑𝑉^(𝛾)+_𝑆(𝛽)𝜌𝝊𝝊⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜌𝐛𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐉⋅𝑑𝐒^(𝛽) (44) which is rewritten as 𝜕^𝛼𝜕𝑡^𝛼𝜌𝝊+∇^𝛼⋅𝜌𝝊𝝊=𝜌𝐛+∇^𝛼⋅𝐉. (45) In view of (7), (45) yields 𝜕^𝛼𝜌𝜕𝑡^𝛼𝜕𝝊+𝜌^𝛼𝝊𝜕𝑡^𝛼+𝝊𝝊⋅∇^𝛼𝜌+𝜌𝝊∇^𝛼⋅𝝊+𝜌𝝊⋅∇^𝛼𝝊=𝜌𝐛+∇^𝛼⋅𝐉 (46) which implies that 𝝊𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼𝜕𝝊⋅𝜌+𝜌^𝛼𝝊𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝝊=𝜌𝐛+∇^𝛼⋅𝐉 (47) or 𝝊𝜕^𝛼𝜌𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝜌+𝜌∇^𝛼𝜕⋅𝝊+𝜌^𝛼𝝊𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝝊=𝜌𝐛+∇^𝛼⋅𝐉. (48) Therefore, (48) is reexpressed by 𝝊𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼⋅𝜕𝜌𝝊+𝜌^𝛼𝝊𝜕𝑡^𝛼+𝝊⋅∇^𝛼𝝊=𝜌𝐛+∇^𝛼⋅𝐉. (49) Taking continuity equation on Cantor sets and fractal material derivative yields 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=𝜌𝐛+∇^𝛼⋅𝐉. (50) Hence, balance of linear momentum in its local Cantorian form (the Newton’s law in its local Cantorian form) is rewritten as 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=∇^𝛼⋅𝐉+𝜌𝐛 (51) or 𝜌𝜕^𝛼𝝊𝜕𝑡^𝛼=∇^𝛼⋅𝐉+𝜌𝐛−𝜌𝝊∇^𝛼⋅𝝊. (52) It is called as the Cauchy’s equation of motion of flows on Cantor sets.

In the absence of internal couples, the balance of angular momentum on Cantor sets implies that the fractal Cauchy stress is symmetric [27]; that is, 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=∇^𝛼⋅𝐉^𝑇+𝜌𝐛or𝐉=𝐉^𝑇. (53) Taking fractal material derivative, (45) is expressed by 𝜌𝜕^𝛼𝝊𝜕𝑡^𝛼+𝜌𝝊∇^𝛼⋅𝝊=∇^𝛼⋅𝐉^𝑇+𝜌𝐛 (54) or 𝐉=𝐉^𝑇. (55) which yields the Newton’s law in local fractional integration form given by _𝑉(𝛾)∇^𝛼⋅𝜌𝝊⋅𝝊−𝐉^𝑇𝑑𝑉^(𝛾)=_𝑆(𝛽)𝜌𝝊⋅𝝊−𝐉^𝑇𝑑𝐒^(𝛽) (56) if the compressible flow on Cantor sets is steady and body forces are absent. The results are different from [36–38] because of the fractional operators.

In local Eulerian form, Cauchy’s equation of motion of flows on Cantor sets is given by using (37) 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝝊𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜌𝐛𝑑𝑉^(𝛾)−_𝑆(𝛽)𝑝𝐈⋅𝑑𝐒^(𝛽) (57) which implies that 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝𝜌+𝜌𝐛. (58) For compressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (39) 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝝊𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜌𝐛𝑑𝑉^(𝛾)−_𝑆(𝛽)𝑝𝜌𝜌1𝐈+𝜇3∇^𝛼⋅𝝊+𝝊⋅∇^𝛼⋅𝑑𝐒^(𝛽) (59) which is 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝𝜌+13∇^𝛼⋅𝜇𝜌∇^𝛼⋅𝝊+∇^𝛼⋅𝜇𝜌𝝊⋅∇^𝛼+𝜌𝐛. (60) For incompressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (40) 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝝊𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜌𝐛𝑑𝑉^(𝛾)+_𝑆(𝛽)−𝑝𝐈+𝜇∇^𝛼⋅𝝊+𝝊⋅∇^𝛼⋅𝑑𝐒^(𝛽) (61) which leads to 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝+𝜇∇^𝛼𝝊⋅∇^𝛼+𝜌𝐛 (62) or 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝+𝜇∇^𝛼𝝊⋅∇^𝛼. (63)

4.2. Balance of Energy on Cantor Sets

The mechanical energy equation with viscous dissipation on Cantor sets is repeated by 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝜑𝑑𝑉^(𝛾)+_𝑆(𝛽)𝜌𝜑𝝊⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜌𝐛⋅𝝊𝑑𝑉^(𝛾)+_𝑆(𝛽)𝝊⋅𝐉𝑑𝐒^(𝛽)−_𝑉(𝛾)∇^𝛼⋅𝑝𝝊𝑑𝑉^(𝛾)−_𝑉(𝛾)𝜙𝑑𝑉^(𝛾), (64) where the first term is rate of change of kinetic energy in fractal domain, the second term is rate of outflow across fractal boundary, the third term is rate of work by fractal body force, the fourth term is rate of work by surface force, the fifth term is rate of work by volume expansion, and the last term is rate of viscous dissipation.

Local fractional differential form of the mechanical energy equation with viscous dissipation on Cantor sets gives 𝐷^𝛼𝐷𝑡^𝛼𝜌𝜑+∇^𝛼⋅𝜌𝜑𝝊=𝜌𝐛⋅𝝊+𝝊⋅∇^𝛼⋅𝐉+∇^𝛼⋅𝑝𝝊−𝜙, (65) which implies for fractal material derivative of 𝜌𝜑𝜕^𝛼𝜕𝑡^𝛼𝜌𝜑+𝝊⋅∇^𝛼𝜌𝜑+∇^𝛼⋅∇𝜌𝜑𝝊=𝜌𝐛⋅𝝊+𝝊⋅^𝛼⋅𝐉−∇^𝛼⋅𝑝𝝊−𝜙, (66) where the rate of viscous dissipation of kinetic energy per unit volume is 𝜙.

The integration form of the balance of energy on Cantor sets is expressed by 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌𝜃+𝜑𝑑𝑉^(𝛾)=_𝑆(𝛽)𝝊⋅𝐉⋅𝑑𝐒^(𝛽)−_𝑆(𝛽)𝜌𝜃+𝜑𝝊⋅𝑑𝐒^(𝛽)+_𝑉(𝛾)𝜌𝐛⋅𝝊𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐾^2𝛼𝐪⋅𝑑𝐒^(𝛽)−_𝑉(𝛾)𝜙𝑑𝑉^(𝛾) (67) which leads to 𝐷^𝛼𝐷𝑡^𝛼𝜌𝜃+𝜑=𝝊⋅∇^𝛼⋅𝐉+𝜌𝐛⋅𝝊−∇^𝛼⋅𝑝𝝊−∇^𝛼⋅𝜌𝜃+𝜑𝝊−𝐾^2𝛼⋅𝐪−𝜙, (68) where 𝐾^2𝛼 is denoted as fractal heat conduction coefficient and 𝐪 is fractal temperature flied.

The term _𝑆(𝛽)𝜌𝜃+𝜑𝝊⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)∇^𝛼⋅𝜌𝜃+𝜑𝝊𝑑𝑉^(𝛾) (69) vanishes if the term 𝜌𝜃𝝊 is zero at the boundaries and _𝑉(𝛾)𝑝∇^𝛼⋅𝝊𝑑𝑉^(𝛾)=_𝑆(𝛽)𝑝𝝊𝑑𝐒^(𝛽)−_𝑉(𝛾)𝝊⋅∇^𝛼𝑝𝑑𝑉^(𝛾). (70)

The mechanical energy equation with no viscous dissipation on Cantor sets is written in the local Cantorian form as 𝜌𝐷^𝛼𝐷𝑡^𝛼𝜃+𝜑=−∇^𝛼𝑝⋅𝝊+𝝊⋅∇^𝛼⋅𝐉+𝜌𝐛⋅𝝊+𝐾^2𝛼∇^𝛼⋅𝐪, (71) where 𝜑 is the kinetic energy per unit of mass, 𝐛 is the external force per unit of mass, 𝐉 is the fractal Cauchy stress tensor, and 𝜃 is the internal energy per unit of mass.

5. Systems of Navier-Stokes Equation on Cantor Sets

In this section, we investigate the Navier-Stokes equation on Cantor sets and derive systems of Navier-Stokes equation on Cantor sets.

5.1. Navier-Stokes Equation on Cantor Sets

Substituting the constitutive equation into Cauchy’s equation (41) yields 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼⋅𝑝𝐈+∇^𝛼2𝜇∇^𝛼⋅𝝊+𝝊⋅∇^𝛼−23𝜇∇^𝛼⋅𝝊𝐈+𝜌𝐛, (72) where viscosity 𝜇 in this equation can be a function of the thermodynamic state.

This is a general form of the Navier-Stokes equation on Cantor sets, which is the equation of motion for a Newtonian fluid on Cantor sets.

For incompressible fluids ∇^𝛼⋅𝝊=0, we deduce to 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝+𝜇∇^2𝛼𝝊+𝜌𝐛. (73) Applying fractal material derivative of the fluid velocity, 𝐷^𝛼𝝊𝐷𝑡^𝛼=𝜕^𝛼𝝊𝜕𝑡^𝛼+𝝊∇^𝛼⋅𝝊, (74) in compressible fluids, a general form of the Navier-Stokes equation on Cantor sets is stated as 𝜌𝜕^𝛼𝝊𝜕𝑡^𝛼=−∇^𝛼𝑝+𝜇∇^2𝛼𝝊+𝜌𝐛−𝜌𝝊∇^𝛼⋅𝝊. (75) For compressible fluid, the Navier-Stokes equation on Cantor sets reads 𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝+∇^𝛼𝜇∇^𝛼⋅𝝊+𝝊⋅∇^𝛼+∇^𝛼𝜆∇^𝛼⋅𝝊+𝜌𝐛 (76) which becomes for 𝜆=−(2/3)𝜇𝜌𝐷^𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼1𝑝+3𝜇∇^𝛼∇^𝛼⋅𝝊+𝜇∇^2𝛼𝝊+𝜌𝐛 (77) which yields by using (55) 𝜌𝜕^𝛼𝝊𝜕𝑡^𝛼=−∇^𝛼1𝑝+3𝜇∇^𝛼∇^𝛼⋅𝝊+𝜇∇^2𝛼𝝊+𝜌𝐛−𝜌𝝊∇^𝛼⋅𝝊. (78)

5.2. Systems of Navier-Stokes Equation on Cantor Sets

In this section, we consider systems of Navier-Stokes fluid on Cantor sets, which states that the systems consists of the continuity equation, the motion equation, and the energy balance equation on Cantor sets.

By using (29), (71), and (77), systems of compressible Navier-Stokes equations on Cantor sets become as follows: 𝜕^𝛼𝜌𝜕𝑡^𝛼+∇^𝛼⋅𝜌𝐷𝜌𝝊=0,_𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼1𝑝+3𝜇∇^𝛼∇^𝛼⋅𝝊+𝜇∇^2𝛼𝜌𝐷𝝊+𝜌𝐛,_𝛼𝐷𝑡^𝛼𝜃+𝜑=−∇^𝛼⋅𝑝𝝊+𝝊⋅∇^𝛼⋅𝐉+𝜌𝐛⋅𝝊+𝐾^2𝛼∇^𝛼⋅𝐪. (79) By applying (31), (71), and (73), systems of incompressible Navier-Stokes equations on Cantor sets are stated as ∇^𝛼𝜌𝐷⋅𝝊=0,_𝛼𝝊𝐷𝑡^𝛼=−∇^𝛼𝑝+𝜇∇^2𝛼𝜌𝐷𝝊+𝜌𝐛,_𝛼𝐷𝑡^𝛼𝜃+𝜑=−∇^𝛼⋅𝑝𝝊+𝝊⋅∇^𝛼⋅𝐉+𝜌𝐛⋅𝝊+𝐾^2𝛼∇^𝛼⋅𝐪, (80) or in the 3D Cantorian coordinates, systems of Navier-Stokes equations on Cantor sets can be written for each component, 𝑥, 𝑦, and 𝑧 as 𝜕^𝛼𝑣_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑣_𝑦𝜕𝑦^𝛼+𝜕^𝛼𝑣_𝑧𝜕𝑧^𝛼𝜌𝐷=0,_𝛼𝑣_𝑥𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑥^𝛼𝜕+𝜇^2𝛼𝑣_𝑥𝜕𝑥^2𝛼+𝜕^2𝛼𝑣_𝑥𝜕𝑦^2𝛼+𝜕^2𝛼𝑣_𝑥𝜕𝑧^2𝛼+𝜌𝑏_𝑥,𝜌𝐷^𝛼𝑣_𝑦𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑦^𝛼𝜕+𝜇^2𝛼𝑣_𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑣_𝑦𝜕𝑦^2𝛼+𝜕^2𝛼𝑣_𝑦𝜕𝑧^2𝛼+𝜌𝑏_𝑦,𝜌𝐷^𝛼𝑣_𝑧𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑧^𝛼𝜕+𝜇^2𝛼𝑣_𝑧𝜕𝑥^2𝛼+𝜕^2𝛼𝑣_𝑧𝜕𝑦^2𝛼+𝜕^2𝛼𝑣_𝑧𝜕𝑧^2𝛼+𝜌𝑏_𝑧,𝜌𝐷^𝛼𝐷𝑡^𝛼𝜕𝜃+𝜑=−^𝛼𝑝𝑣_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑝𝑣_𝑦𝜕𝑦^𝛼+𝜕^𝛼𝑝𝑣_𝑧𝜕𝑧^𝛼+𝑣_𝑥𝜕^𝛼𝐽_𝑥𝑥𝜕𝑥^𝛼+𝜕^𝛼𝐽_𝑥𝑦𝜕𝑦^𝛼+𝜕^𝛼𝐽_𝑥𝑧𝜕𝑧^𝛼+𝑣_𝑦𝜕^𝛼𝐽_𝑦𝑥𝜕𝑥^𝛼+𝜕^𝛼𝐽_𝑦𝑦𝜕𝑦^𝛼+𝜕^𝛼𝐽_𝑦𝑧𝜕𝑧^𝛼+𝑣_𝑧𝜕^𝛼𝐽_𝑧𝑥𝜕𝑥^𝛼+𝜕^𝛼𝐽_𝑧𝑦𝜕𝑦^𝛼+𝜕^𝛼𝐽_𝑧𝑧𝜕𝑧^𝛼+𝜌𝑏_𝑥𝑣_𝑥+𝜌𝑏_𝑦𝑣_𝑦+𝜌𝑏_𝑧𝑣_𝑧+𝐾^2𝛼𝜕^𝛼𝑞_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑞_𝑦𝜕𝑦^𝛼+𝜕^𝛼𝑞_𝑧𝜕𝑧^𝛼. (81) In the 2D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are rewritten in the form 𝜕^𝛼𝑣_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑣_𝑦𝜕𝑦^𝛼𝜌𝐷=0,_𝛼𝑣_𝑥𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑥^𝛼𝜕+𝜇^2𝛼𝑣_𝑥𝜕𝑥^2𝛼+𝜕^2𝛼𝑣_𝑥𝜕𝑦^2𝛼+𝜌𝑏_𝑥,𝜌𝐷^𝛼𝑣_𝑦𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑦^𝛼𝜕+𝜇^2𝛼𝑣_𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑣_𝑦𝜕𝑦^2𝛼+𝜌𝑏_𝑦,𝜌𝐷^𝛼𝐷𝑡^𝛼𝜕𝜃+𝜑=−^𝛼𝑝𝑣_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑝𝑣_𝑦𝜕𝑦^𝛼+𝑣_𝑥𝜕^𝛼𝐽_𝑥𝑥𝜕𝑥^𝛼+𝜕^𝛼𝐽_𝑥𝑦𝜕𝑦^𝛼+𝑣_𝑦𝜕^𝛼𝐽_𝑦𝑥𝜕𝑥^𝛼+𝜕^𝛼𝐽_𝑦𝑦𝜕𝑦^𝛼+𝜌𝑏_𝑥𝑣_𝑥+𝜌𝑏_𝑦𝑣_𝑦+𝐾^2𝛼𝜕^𝛼𝑞_𝑥𝜕𝑥^𝛼+𝜕^𝛼𝑞_𝑦𝜕𝑦^𝛼. (82) In the 1D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are repeated by the expression 𝜕^𝛼𝑣𝜕𝑥^𝛼𝜌𝐷=0,_𝛼𝑣𝐷𝑡^𝛼𝜕=−^𝛼𝑝𝜕𝑥^𝛼𝜕+𝜇^2𝛼𝑣𝜕𝑥^2𝛼𝜌𝐷+𝜌𝑏,_𝛼𝐷𝑡^𝛼𝜕𝜃+𝜑=−^𝛼𝑝𝑣𝜕𝑥^𝛼𝜕+𝑣^𝛼𝐽𝜕𝑥^𝛼+𝜌𝑏𝑣+𝐾^2𝛼𝜕^𝛼𝑞𝜕𝑥^𝛼. (83)

6. Conclusions

In the present work, we propose the systems of Navier-Stokes equations derived from local fractional vector calculus. These obtained Navier-Stokes equations in one-, two-, and three-dimension Cantorian coordinates are shown to describe the materials as being local fractional continuous and nondifferential functions, which are applied to describe fluid flow in fractal media. Comparing between the fractional result in Navier-Stokes equation in fractal media [39] and local fractional one, the former via fractional calculus is continuous and differential quantities as classical result, however, the latter is local fractional continuous and nondifferential quantities. The classical result is obtained in case of fractal space-time dimension, which is equal to 1.