# Systems of Navier-Stokes Equations on Cantor Sets

Authors: Xiao-Jun Yang; Dumitru Baleanu; J. A. Tenreiro Machado
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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]
(1)r(x)=M(x)e1α+N(x)e2α
which is expressed by
(2)r(x)=(M(x),N(x)).
For a function of three variables, the vector form can be written in the form [27]
(3)r(x,y,z)=L(x,y,z)e1α+M(x,y,z)e2α+N(x,y,z)e3α,
or
(4)r(x,y)=(L(x,y,z),M(x,y,z),N(x,y,z)).
Let u(x,y,z)=u(x,y,z)e1α+u2(x,y,z)e2α+u3(x,y,z)e3α 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]
(5)∇αφ=∂αφ∂x1αe1α+∂αφ∂x2αe2α+∂αφ∂x3αe3α,∇2αφ=∂2αφ∂x12α+∂2αφ∂x22α+∂2αφ∂x32α,
where local fractional partial derivatives of high order are defined as [27, 31–33]
(6)∂kα∂xkαf(x)=∂α∂xα⋯∂α∂xα︷ktimesf(x).(2)
The local fractional divergence and curl of a local fractional vector field are written in the form [27, 31]
(7)divαu=∇α·u=∂αu1∂x1α+∂αu2∂x2α+∂αu3∂x3α,curlαu=∇α×u=(∂αu3∂x2α-∂αu2∂x3α)e1α+(∂αu1∂x3α-∂αu3∂x1α)e2α+(∂αu2∂x1α-∂αu1∂x2α)e3α.(3)
The following equations are valid only in Cantor coordinates [27]:
(8)∇α(uv)=(∇αu)v+u∇αv,∇2αA=∇α(∇α·A)-∇α×(∇α×A),A·∇αA=∇α(A·A2)-(∇α×A)×A,
where ∇2α=∇α·∇α is local fractional Laplace operator [27].The local fractional line integral of the functionu(xP,yP,zP) in the local fractional vector form
(9)u(xP,yP,zP)=u1(xP,yP,zP)e1α+u2(xP,yP,zP)e2α+u3(xP,yP,zP)e3α
along a fractal line lα is written as [27]
(10)∫l(α)u(xP,yP,zP)·dlα=limN→∞∑P=1Nu(xP,yP,zP)·ΔlPα,
where for P=1,2,…,N and N elements of line ΔlPα, it is required that all |ΔlPα|→0 as N→∞.The local fractional surface integral of the given function (9) across a surface S(β) is defined as [27]
(11)∬S(β)u(xP,yP,zP)·dS(β)=limN→∞∑P=1Nu(xP,yP,zP)·nPΔSP(β),
where for P=1,2,…,N and N elements of area ΔSP(β) with a unit normal local fractional vector nP, it is required that all ΔSP(β)→0 as N→∞.The local fractional volume integral of the given function (9) in a fractal region V(γ) is given by [27]
(12)∭V(γ)u(xP,yP,zP)dV(γ)=limN→∞∑P=1Nu(xP,yP,zP)ΔVP(γ),
where for P=1,2,…,N and N elements of volume ΔVP(γ), it is required that all ΔVP(γ)→0 as N→∞.Let us consider a local fractional vector fieldu=u1e1α+u2e2α+u3e3α, the following results hold [27]: (a)
Divergence Theorem of local fractional field states that(13)∭V(γ)∇α·udV(γ)=∯S(β)u·dS(β).(b)
Stokes’ theorem of local fractional field states that(14)∮l(α)u·dlα=∬S(β)(∇α×u)·dS(β).(c)
Green’s first theorem in fractal domain states that(15)∯S(β)ϕ∇αφ·dS(β)=∭V(γ)(ϕ∇2αφ+(∇αϕ)·(∇αφ))dV(γ).(d)
Green’s second theorem in fractal domain states that(16)∯S(β)(ϕ∇αφ-ϕ∇αφ)·dS(β)=∭V(γ)(ϕ∇2αφ-φ∇2αϕ)dV(γ),
where the relationships of fractal dimensions of line, surface, and volume fields are
(17)β=2α,γ=3α.


## 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 propertyψ associated with an abstract Cantor body Β expressed as
(18)ψ=ψ(P,t)=ψ^(X,t)=ψ~(x,t),
we define the following Lagrangian and Eulerian temporal and spatial derivatives with local fractional differential operator:
(19)ψ˙=Dαψ^Dtα,ψ,tα=∂αψ~∂tα,∇Xαψ=∂αψ^∂Xα,∇xαψ=∇αψ=∂αψ~∂xα.
The velocity υ and the acceleration a with local fractional operator are defined through
(20)υ=DαχDtα,a=D2αχDt2α.
It immediately follows that the velocity gradient Θ and its symmetric part Λ are expressed through
(21)Θ=∇α·υ,ΘT=υ·∇α,Λ=12(Θ+ΘT).
The deformation gradient Σ is given by
(22)Σ=∇α·χ,
which leads to
(23)Θ=∂αΣ∂tαΣ-1.
The fractal material derivative of the fluid density ρ  is defined as [27]
(24)DαρDtα=∂αρ∂tα+υ·∇αρ.


### 3.2. Balance of Mass on Cantor Sets

The mass of fractal fluid inV(γ) is defined through [27]
(25)M=∭V(γ)ρdV(γ),
which yields that the balance of mass on Cantor sets takes the form [27]
(26)∂αM∂tα=-∯S(β)ρυ·dS(β).
Using Divergence Theorem of local fractional field, we deduce to
(27)∭V(γ)∂α∂tαρdV(γ)+∯S(β)ρυ·dS(β)=∭V(γ)[∂αρ∂tα+∇α·(ρυ)]dV(γ)=0
which implies that
(28)∂αρ∂tα+∇α·(ρυ)=0.
This is called continuity equation on Cantor sets.It becomes(29)∂αρ∂tα+∇α·(ρυ)=∂αρ∂tα+υ·(∇αρ)+ρ(∇α·υ)=DαρDtα+ρ(∇α·υ)=0,
where the fractal material derivative of the fluid density ρ  is noted by
(30)DαρDtα=∂αρ∂tα+υ·(∇αρ).
If the fractal fluid is incompressible, we deduce that
(31)∂αρ∂tα+υ·(∇αρ)=0⟺DαρDtα=0or∇α·υ=0.
It follows that Reynolds transport theorem on Cantor sets states that
(32)DαDtα∭V(γ)F(x,t)dV(γ)=∭V(γ)∂α∂tαF(x,t)dV(γ)+∯S(β)F(x,t)υ·dS(β),
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(33)J=-pI+K:Λ,
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]
(34)∇α·υ=12(Θ+ΘT)+12(Θ-ΘT)=Λ+12(Θ-ΘT).
We can write
(35)K:Λ=2μΛ+λ(∇α·υ)I
which leads to
(36)J=-pI+2μΛ+λ(∇α·υ)I,
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(37)J=-p(ρ)I.
The constitutive equation of homogeneous incompressible Euler fluid on Cantor sets is
(38)J=-pI,∇α·υ=0.
The constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets is written in the form
(39)J=-p(ρ)I+2μ(ρ)Λ+λ(ρ)(∇α·υ)I.
The constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets is expressed as
(40)J=-pI+2μΛ,∇α·υ=0.
By using Stokes’ assumption λ(ρ)=-2μ(ρ)/3, the constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets becomes
(41)J=-p(ρ)I+2μ(ρ)(∇α·υ+υ·∇α)-23μ(ρ)(∇α·υ),
while the constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets can obtain that
(42)J=-pI+μ(∇α·υ+υ·∇α),∇α·υ=0.


## 3.1. Fractal Kinematics

For any scalar propertyψ associated with an abstract Cantor body Β expressed as
(18)ψ=ψ(P,t)=ψ^(X,t)=ψ~(x,t),
we define the following Lagrangian and Eulerian temporal and spatial derivatives with local fractional differential operator:
(19)ψ˙=Dαψ^Dtα,ψ,tα=∂αψ~∂tα,∇Xαψ=∂αψ^∂Xα,∇xαψ=∇αψ=∂αψ~∂xα.
The velocity υ and the acceleration a with local fractional operator are defined through
(20)υ=DαχDtα,a=D2αχDt2α.
It immediately follows that the velocity gradient Θ and its symmetric part Λ are expressed through
(21)Θ=∇α·υ,ΘT=υ·∇α,Λ=12(Θ+ΘT).
The deformation gradient Σ is given by
(22)Σ=∇α·χ,
which leads to
(23)Θ=∂αΣ∂tαΣ-1.
The fractal material derivative of the fluid density ρ  is defined as [27]
(24)DαρDtα=∂αρ∂tα+υ·∇αρ.


## 3.2. Balance of Mass on Cantor Sets

The mass of fractal fluid inV(γ) is defined through [27]
(25)M=∭V(γ)ρdV(γ),
which yields that the balance of mass on Cantor sets takes the form [27]
(26)∂αM∂tα=-∯S(β)ρυ·dS(β).
Using Divergence Theorem of local fractional field, we deduce to
(27)∭V(γ)∂α∂tαρdV(γ)+∯S(β)ρυ·dS(β)=∭V(γ)[∂αρ∂tα+∇α·(ρυ)]dV(γ)=0
which implies that
(28)∂αρ∂tα+∇α·(ρυ)=0.
This is called continuity equation on Cantor sets.It becomes(29)∂αρ∂tα+∇α·(ρυ)=∂αρ∂tα+υ·(∇αρ)+ρ(∇α·υ)=DαρDtα+ρ(∇α·υ)=0,
where the fractal material derivative of the fluid density ρ  is noted by
(30)DαρDtα=∂αρ∂tα+υ·(∇αρ).
If the fractal fluid is incompressible, we deduce that
(31)∂αρ∂tα+υ·(∇αρ)=0⟺DαρDtα=0or∇α·υ=0.
It follows that Reynolds transport theorem on Cantor sets states that
(32)DαDtα∭V(γ)F(x,t)dV(γ)=∭V(γ)∂α∂tαF(x,t)dV(γ)+∯S(β)F(x,t)υ·dS(β),
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(33)J=-pI+K:Λ,
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]
(34)∇α·υ=12(Θ+ΘT)+12(Θ-ΘT)=Λ+12(Θ-ΘT).
We can write
(35)K:Λ=2μΛ+λ(∇α·υ)I
which leads to
(36)J=-pI+2μΛ+λ(∇α·υ)I,
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(37)J=-p(ρ)I.
The constitutive equation of homogeneous incompressible Euler fluid on Cantor sets is
(38)J=-pI,∇α·υ=0.
The constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets is written in the form
(39)J=-p(ρ)I+2μ(ρ)Λ+λ(ρ)(∇α·υ)I.
The constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets is expressed as
(40)J=-pI+2μΛ,∇α·υ=0.
By using Stokes’ assumption λ(ρ)=-2μ(ρ)/3, the constitutive equation of homogeneous compressible Navier-Stokes fluid on Cantor sets becomes
(41)J=-p(ρ)I+2μ(ρ)(∇α·υ+υ·∇α)-23μ(ρ)(∇α·υ),
while the constitutive equation of homogeneous incompressible Navier-Stokes fluid on Cantor sets can obtain that
(42)J=-pI+μ(∇α·υ+υ·∇α),∇α·υ=0.


## 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]
(43)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)+∯S(β)J·dS(β),
where J denotes the fractal Cauchy stress tensor [27] and b denotes the specific fractal body force [27].By using (32), (43) becomes
(44)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)∂α∂tα(ρυ)dV(γ)+∯S(β)ρυυ·dS(β)=∭V(γ)ρbdV(γ)+∯S(β)J·dS(β)
which is rewritten as
(45)∂α∂tα(ρυ)+∇α·[ρυυ]=ρb+∇α·J.
In view of (7), (45) yields
(46)∂αρ∂tαυ+ρ∂αυ∂tα+υυ·∇αρ+ρυ∇α·υ+ρυ·∇αυ=ρb+∇α·J
which implies that
(47)υ(∂αρ∂tα+∇α(υ·ρ))+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J
or
(48)υ(∂αρ∂tα+υ·∇αρ+ρ∇α·υ)+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J.
Therefore, (48) is reexpressed by
(49)υ(∂αρ∂tα+∇α·(ρυ))+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J.
Taking continuity equation on Cantor sets and fractal material derivative yields
(50)ρDαυDtα=ρb+∇α·J.
Hence, balance of linear momentum in its local Cantorian form (the Newton’s law in its local Cantorian form) is rewritten as
(51)ρDαυDtα=∇α·J+ρb
or
(52)ρ∂αυ∂tα=∇α·J+ρb-ρυ(∇α·υ).
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,
(53)ρDαυDtα=∇α·JT+ρborJ=JT.
Taking fractal material derivative, (45) is expressed by
(54)ρ∂αυ∂tα+ρυ(∇α·υ)=∇α·JT+ρb
or
(55)J=JT.
which yields the Newton’s law in local fractional integration form given by
(56)∭V(γ)∇α·(ρυ·υ-JT)dV(γ)=∯S(β)[ρυ·υ-JT]dS(β)
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)
(57)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)-∯S(β)pI·dS(β)
which implies that
(58)ρDαυDtα=-∇αp(ρ)+ρb.
For compressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (39)
(59)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)-∯S(β)(p(ρ)I+μ(ρ)(13(∇α·υ)+υ·∇α))·dS(β)
which is
(60)ρDαυDtα=-∇αp(ρ)+13∇α·[μ(ρ)(∇α·υ)]+∇α·(μ(ρ)υ·∇α)+ρb.
For incompressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (40)
(61)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)+∯S(β)(-pI+μ(∇α·υ+υ·∇α))·dS(β)
which leads to
(62)ρDαυDtα=-∇αp+μ∇α(υ·∇α)+ρb
or
(63)ρDαυDtα=-∇αp+μ∇α(υ·∇α).


### 4.2. Balance of Energy on Cantor Sets

The mechanical energy equation with viscous dissipation on Cantor sets is repeated by(64)DαDtα∭V(γ)(ρφ)dV(γ)+∯S(β)(ρφυ)·dS(β)=∭V(γ)(ρb·υ)dV(γ)+∯S(β)υ·JdS(β)-∭V(γ)∇α·(pυ)dV(γ)-∭V(γ)ϕdV(γ),
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(65)DαDtα(ρφ)+∇α·(ρφυ)=ρb·υ+υ·(∇α·J)+∇α·(pυ)-ϕ,
which implies for fractal material derivative of ρφ(66)∂α∂tα(ρφ)+υ·∇α(ρφ)+∇α·(ρφυ)=ρb·υ+υ·(∇α·J)-∇α·(pυ)-ϕ,
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(67)DαDtα∭V(γ)ρ(θ+φ)dV(γ)=∯S(β)υ·J·dS(β)-∯S(β)ρ(θ+φ)υ·dS(β)+∭V(γ)(ρb·υ)dV(γ)+∯S(β)K2αq·dS(β)-∭V(γ)ϕdV(γ)
which leads to
(68)DαDtα[ρ(θ+φ)]=υ·(∇α·J)+ρb·υ-∇α·(pυ)-∇α·[ρ(θ+φ)υ-K2α·q]-ϕ,
where K2α is denoted as fractal heat conduction coefficient and q is fractal temperature flied.The term(69)∯S(β)ρ(θ+φ)υ·dS(β)=∭V(γ)∇α·[ρ(θ+φ)]υdV(γ)
vanishes if the term ρθυ is zero at the boundaries and
(70)∭V(γ)(p(∇α·υ))dV(γ)=∯S(β)(pυ)dS(β)-∭V(γ)[υ·(∇αp)]dV(γ).The mechanical energy equation with no viscous dissipation on Cantor sets is written in the local Cantorian form as(71)ρDαDtα(θ+φ)=-∇αp·υ+υ·(∇α·J)+ρb·υ+K2α∇α·q,
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.


## 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]
(43)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)+∯S(β)J·dS(β),
where J denotes the fractal Cauchy stress tensor [27] and b denotes the specific fractal body force [27].By using (32), (43) becomes
(44)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)∂α∂tα(ρυ)dV(γ)+∯S(β)ρυυ·dS(β)=∭V(γ)ρbdV(γ)+∯S(β)J·dS(β)
which is rewritten as
(45)∂α∂tα(ρυ)+∇α·[ρυυ]=ρb+∇α·J.
In view of (7), (45) yields
(46)∂αρ∂tαυ+ρ∂αυ∂tα+υυ·∇αρ+ρυ∇α·υ+ρυ·∇αυ=ρb+∇α·J
which implies that
(47)υ(∂αρ∂tα+∇α(υ·ρ))+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J
or
(48)υ(∂αρ∂tα+υ·∇αρ+ρ∇α·υ)+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J.
Therefore, (48) is reexpressed by
(49)υ(∂αρ∂tα+∇α·(ρυ))+ρ(∂αυ∂tα+υ·∇αυ)=ρb+∇α·J.
Taking continuity equation on Cantor sets and fractal material derivative yields
(50)ρDαυDtα=ρb+∇α·J.
Hence, balance of linear momentum in its local Cantorian form (the Newton’s law in its local Cantorian form) is rewritten as
(51)ρDαυDtα=∇α·J+ρb
or
(52)ρ∂αυ∂tα=∇α·J+ρb-ρυ(∇α·υ).
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,
(53)ρDαυDtα=∇α·JT+ρborJ=JT.
Taking fractal material derivative, (45) is expressed by
(54)ρ∂αυ∂tα+ρυ(∇α·υ)=∇α·JT+ρb
or
(55)J=JT.
which yields the Newton’s law in local fractional integration form given by
(56)∭V(γ)∇α·(ρυ·υ-JT)dV(γ)=∯S(β)[ρυ·υ-JT]dS(β)
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)
(57)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)-∯S(β)pI·dS(β)
which implies that
(58)ρDαυDtα=-∇αp(ρ)+ρb.
For compressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (39)
(59)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)-∯S(β)(p(ρ)I+μ(ρ)(13(∇α·υ)+υ·∇α))·dS(β)
which is
(60)ρDαυDtα=-∇αp(ρ)+13∇α·[μ(ρ)(∇α·υ)]+∇α·(μ(ρ)υ·∇α)+ρb.
For incompressible fluid, Cauchy’s equation of motion of flows on Cantor sets is obtained by using (40)
(61)DαDtα∭V(γ)ρυdV(γ)=∭V(γ)ρbdV(γ)+∯S(β)(-pI+μ(∇α·υ+υ·∇α))·dS(β)
which leads to
(62)ρDαυDtα=-∇αp+μ∇α(υ·∇α)+ρb
or
(63)ρDαυDtα=-∇αp+μ∇α(υ·∇α).


## 4.2. Balance of Energy on Cantor Sets

The mechanical energy equation with viscous dissipation on Cantor sets is repeated by(64)DαDtα∭V(γ)(ρφ)dV(γ)+∯S(β)(ρφυ)·dS(β)=∭V(γ)(ρb·υ)dV(γ)+∯S(β)υ·JdS(β)-∭V(γ)∇α·(pυ)dV(γ)-∭V(γ)ϕdV(γ),
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(65)DαDtα(ρφ)+∇α·(ρφυ)=ρb·υ+υ·(∇α·J)+∇α·(pυ)-ϕ,
which implies for fractal material derivative of ρφ(66)∂α∂tα(ρφ)+υ·∇α(ρφ)+∇α·(ρφυ)=ρb·υ+υ·(∇α·J)-∇α·(pυ)-ϕ,
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(67)DαDtα∭V(γ)ρ(θ+φ)dV(γ)=∯S(β)υ·J·dS(β)-∯S(β)ρ(θ+φ)υ·dS(β)+∭V(γ)(ρb·υ)dV(γ)+∯S(β)K2αq·dS(β)-∭V(γ)ϕdV(γ)
which leads to
(68)DαDtα[ρ(θ+φ)]=υ·(∇α·J)+ρb·υ-∇α·(pυ)-∇α·[ρ(θ+φ)υ-K2α·q]-ϕ,
where K2α is denoted as fractal heat conduction coefficient and q is fractal temperature flied.The term(69)∯S(β)ρ(θ+φ)υ·dS(β)=∭V(γ)∇α·[ρ(θ+φ)]υdV(γ)
vanishes if the term ρθυ is zero at the boundaries and
(70)∭V(γ)(p(∇α·υ))dV(γ)=∯S(β)(pυ)dS(β)-∭V(γ)[υ·(∇αp)]dV(γ).The mechanical energy equation with no viscous dissipation on Cantor sets is written in the local Cantorian form as(71)ρDαDtα(θ+φ)=-∇αp·υ+υ·(∇α·J)+ρb·υ+K2α∇α·q,
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
(72)ρDαυDtα=-∇α·(pI)+∇α[2μ(∇α·υ+υ·∇α)-23μ(∇α·υ)I]+ρb,
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
(73)ρDαυDtα=-∇αp+μ∇2αυ+ρb.
Applying fractal material derivative of the fluid velocity,
(74)DαυDtα=∂αυ∂tα+υ(∇α·υ),
in compressible fluids, a general form of the Navier-Stokes equation on Cantor sets is stated as
(75)ρ∂αυ∂tα=-∇αp+μ∇2αυ+ρb-ρυ(∇α·υ).
For compressible fluid, the Navier-Stokes equation on Cantor sets reads
(76)ρDαυDtα=-∇αp+∇α[μ(∇α·υ+υ·∇α)]+∇α(λ(∇α·υ))+ρb
which becomes for λ=-(2/3)μ(77)ρDαυDtα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb
which yields by using (55)
(78)ρ∂αυ∂tα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb-ρυ(∇α·υ).


### 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:
(79)∂αρ∂tα+∇α·(ρυ)=0,ρDαυDtα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb,ρDαDtα(θ+φ)=-∇α·(pυ)+υ·(∇α·J)+ρb·υ+K2α∇α·q.
By applying (31), (71), and (73), systems of incompressible Navier-Stokes equations on Cantor sets are stated as
(80)∇α·υ=0,ρDαυDtα=-∇αp+μ∇2αυ+ρb,ρDαDtα(θ+φ)=-∇α·(pυ)+υ·(∇α·J)+ρb·υ+K2α∇α·q,
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
(81)∂αvx∂xα+∂αvy∂yα+∂αvz∂zα=0,ρDαvxDtα=-∂αp∂xα+μ(∂2αvx∂x2α+∂2αvx∂y2α+∂2αvx∂z2α)+ρbx,ρDαvyDtα=-∂αp∂yα+μ(∂2αvy∂x2α+∂2αvy∂y2α+∂2αvy∂z2α)+ρby,ρDαvzDtα=-∂αp∂zα+μ(∂2αvz∂x2α+∂2αvz∂y2α+∂2αvz∂z2α)+ρbz,ρDαDtα(θ+φ)=-(∂α(pvx)∂xα+∂α(pvy)∂yα+∂α(pvz)∂zα)+vx(∂αJxx∂xα+∂αJxy∂yα+∂αJxz∂zα)+vy(∂αJyx∂xα+∂αJyy∂yα+∂αJyz∂zα)+vz(∂αJzx∂xα+∂αJzy∂yα+∂αJzz∂zα)+ρbxvx+ρbyvy+ρbzvz+K2α(∂αqx∂xα+∂αqy∂yα+∂αqz∂zα).
In the 2D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are rewritten in the form
(82)∂αvx∂xα+∂αvy∂yα=0,ρDαvxDtα=-∂αp∂xα+μ(∂2αvx∂x2α+∂2αvx∂y2α)+ρbx,ρDαvyDtα=-∂αp∂yα+μ(∂2αvy∂x2α+∂2αvy∂y2α)+ρby,ρDαDtα(θ+φ)=-(∂αpvx∂xα+∂αpvy∂yα)+vx(∂αJxx∂xα+∂αJxy∂yα)+vy(∂αJyx∂xα+∂αJyy∂yα)+ρbxvx+ρbyvy+K2α(∂αqx∂xα+∂αqy∂yα).
In the 1D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are repeated by the expression
(83)∂αv∂xα=0,ρDαvDtα=-∂αp∂xα+μ∂2αv∂x2α+ρb,ρDαDtα(θ+φ)=-∂α(pv)∂xα+v∂αJ∂xα+ρbv+K2α∂αq∂xα.


## 5.1. Navier-Stokes Equation on Cantor Sets

Substituting the constitutive equation into Cauchy’s equation (41) yields
(72)ρDαυDtα=-∇α·(pI)+∇α[2μ(∇α·υ+υ·∇α)-23μ(∇α·υ)I]+ρb,
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
(73)ρDαυDtα=-∇αp+μ∇2αυ+ρb.
Applying fractal material derivative of the fluid velocity,
(74)DαυDtα=∂αυ∂tα+υ(∇α·υ),
in compressible fluids, a general form of the Navier-Stokes equation on Cantor sets is stated as
(75)ρ∂αυ∂tα=-∇αp+μ∇2αυ+ρb-ρυ(∇α·υ).
For compressible fluid, the Navier-Stokes equation on Cantor sets reads
(76)ρDαυDtα=-∇αp+∇α[μ(∇α·υ+υ·∇α)]+∇α(λ(∇α·υ))+ρb
which becomes for λ=-(2/3)μ(77)ρDαυDtα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb
which yields by using (55)
(78)ρ∂αυ∂tα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb-ρυ(∇α·υ).


## 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:
(79)∂αρ∂tα+∇α·(ρυ)=0,ρDαυDtα=-∇αp+13μ∇α((∇α·υ))+μ∇2αυ+ρb,ρDαDtα(θ+φ)=-∇α·(pυ)+υ·(∇α·J)+ρb·υ+K2α∇α·q.
By applying (31), (71), and (73), systems of incompressible Navier-Stokes equations on Cantor sets are stated as
(80)∇α·υ=0,ρDαυDtα=-∇αp+μ∇2αυ+ρb,ρDαDtα(θ+φ)=-∇α·(pυ)+υ·(∇α·J)+ρb·υ+K2α∇α·q,
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
(81)∂αvx∂xα+∂αvy∂yα+∂αvz∂zα=0,ρDαvxDtα=-∂αp∂xα+μ(∂2αvx∂x2α+∂2αvx∂y2α+∂2αvx∂z2α)+ρbx,ρDαvyDtα=-∂αp∂yα+μ(∂2αvy∂x2α+∂2αvy∂y2α+∂2αvy∂z2α)+ρby,ρDαvzDtα=-∂αp∂zα+μ(∂2αvz∂x2α+∂2αvz∂y2α+∂2αvz∂z2α)+ρbz,ρDαDtα(θ+φ)=-(∂α(pvx)∂xα+∂α(pvy)∂yα+∂α(pvz)∂zα)+vx(∂αJxx∂xα+∂αJxy∂yα+∂αJxz∂zα)+vy(∂αJyx∂xα+∂αJyy∂yα+∂αJyz∂zα)+vz(∂αJzx∂xα+∂αJzy∂yα+∂αJzz∂zα)+ρbxvx+ρbyvy+ρbzvz+K2α(∂αqx∂xα+∂αqy∂yα+∂αqz∂zα).
In the 2D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are rewritten in the form
(82)∂αvx∂xα+∂αvy∂yα=0,ρDαvxDtα=-∂αp∂xα+μ(∂2αvx∂x2α+∂2αvx∂y2α)+ρbx,ρDαvyDtα=-∂αp∂yα+μ(∂2αvy∂x2α+∂2αvy∂y2α)+ρby,ρDαDtα(θ+φ)=-(∂αpvx∂xα+∂αpvy∂yα)+vx(∂αJxx∂xα+∂αJxy∂yα)+vy(∂αJyx∂xα+∂αJyy∂yα)+ρbxvx+ρbyvy+K2α(∂αqx∂xα+∂αqy∂yα).
In the 1D Cantorian coordinates, systems of incompressible Navier-Stokes equations on Cantor sets are repeated by the expression
(83)∂αv∂xα=0,ρDαvDtα=-∂αp∂xα+μ∂2αv∂x2α+ρb,ρDαDtα(θ+φ)=-∂α(pv)∂xα+v∂αJ∂xα+ρbv+K2α∂αq∂xα.


## 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.



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*Source: 769724-2013-06-19.xml*