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J Mech Behav Mater 2017; 26(1–2): 43–54

Konstantinos A. Lazopoulos* and Anastasios K. Lazopoulos Fractional vector and mechanics https://doi.org/10.1515/jmbm-2017-0012 order of the strain. Many problems have been solved employing those theories concerning size effects, Abstract: Basic equations are studied and lifting of various singularities, porous materials, Aifantis revised under the prism of fractional continuum mechan- [13, 15, 16], Askes and Aifantis [17], Lazopoulos [18, 19]. ics (FCM), a very promising research field that satisfies both Another non-local approach was introduced by Kunin experimental and theoretical demands. The geometry of [20, 21]. the fractional differential has been clarified corrected and There are many studies considering fractional elas- the geometry of the fractional tangent spaces of a manifold ticity theory, introducing fractional strain, Drapaca and has been studied in Lazopoulos and Lazopoulos (Lazopou- Sivaloganathan [22], Carpinteri et al. [23, 24], Di Paola los KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, et al. [25], Atanackovic and Stankovic [26], Agrawal [27], 85–104), providing the bases of the missing fractional dif- Sumelka [28]. Baleanu and his co-workers [29–31] has ferential geometry. Therefore, a lot can be contributed to presented a long list of publications concerning various fractional hydrodynamics: the basic fractional fluid equa- applications of fractional calculus in physics, in control tions (Navier Stokes, Euler and Bernoulli) are derived and theory in solving differential equations and numerical fractional Darcy’s flow in porous media is studied. solutions. In addition Tarasov [32, 33] has presented a Keywords: fractional ; fractional fractional vector fields theory with applications. Darcy flow; fractional ; fractional fluid mechan- Lazopoulos [34] introduced fractional derivatives ics; fractional geometry; fractional Navier Stokes equa- of the strain in the strain energy density function in an tions; fractional Newtonian . attempt to introduce non-locality in the elastic response of materials. Fractional calculus was used by many research- ers, not only in the field of mechanics but mainly in physics 1 Introduction and especially in , to develop the idea of introducing non-locality through. In fact, the history Fractional derivatives and integrals [1–5] have been of fractional calculus is dated since 17th century. Particle applied in many fields, since they are considered as more physics, electromagnetics, mechanics of materials, hydro- advanced mathematical tools for formulating more real- , fluid flow, , , , elec- istic responses in various scientific problems in physics trochemistry and corrosion, are some and engineering [6–8]. Especially in mechanics, research- fields where fractional calculus has been introduced. ers working in disordered (non-homogeneous) materials Nevertheless, the formulation of the various physi- with microstructure, Vardoulakis et al. [9], Wyss et al. cal problems, into the context of fractional mathematical [10], have used fractional analysis for better description analysis, follows a procedure that might be questionable. of mechanics of porous materials, colloidal aggregates, In fact, although the various laws in physics have been ceramics etc., since major factors in determining materials derived through differentials, this is not the case for the are microcracks, voids, and material phases. well known fractional derivatives which are not related to Further viscoelasticity problems have been recently for- differentials. Simple substitution of the conventional dif- mulated applying fractional analysis. ferentials to fractional ones is not able to express realistic Generally, those problems demand non-local theo- behavior of the various physical problems. ries. Just to satisfy that requirement, strain Lazopoulos [35] introduced Leibnitz (L-fractional) theories appeared, Toupin [11], Mindlin [12], Aifantis [13], derivative corresponding to fractional differential. Fur- Eringen [14]. In these theories, the authors introduced thermore, the L-fractional derivative was improved in Laz- intrinsic material lengths that accompany the higher opoulos and Lazopoulos [36], applying it to continuum mechanics. Lazopoulos and Lazopoulos [36, 37] have pre- sented the application of the L-fractional derivative into *Corresponding author: Konstantinos A. Lazopoulos, National the fractional differential geometry, fractional Technical University of Athens (NTUA), 14 Theatrou st., Rafina 19009, Greece,e-mail: [email protected] theory and fractional continuum mechanics. Recently, Anastasios K. Lazopoulos: Mathematical Sciences Department, Lazopoulos et al. [38] have presented application of the Evelpidon Hellenic Army Academy, Vari 16673, Greece L-fractional derivative into the viscoelastic problems. 44 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics

b The present work deals with the study of fractional a 1(fs) If()xd= s (5) fluid flows, introducing the L-fractional derivative into the xb Γα()∫ ()sx− 1−a x various flows concerning the fractional Navier-Stokes equa- tions and the fractional Euler and Bernoulli ones. Further In Eqs. (4, 5) we assume that α is the order of frac- the fractional flows are introduced to Darcy’s flows, just to tional integrals with 0 < α ≤ 1, considering Γ(x) = (x − 1)! study the flows through porous media. There exists a long with Γ(α) Euler’s Gamma function. list of works concerning fractional [39–41]. Thus the left and right Riemann-Liouville (R-L) deriva- However, the use of fractional derivatives, into conventional tives are defined by: fluid flow laws, exhibit the handicap of having no-differen- aad 1− tials. After a brief introduction of the L-fractional derivative aaDfxx()xI= ((fx)) (6) dx and fractional vector field theory, the fractional fluid flows are studied using the L-fractional derivative, trying to intro- and duce non-local fluid mechanics both in time and space con- sidering in homogeneities, porous, cracks, etc. aad 1− xbDf()xI=− ((bx fx)) (7) dx

Pointing out that the R-L derivatives of a constant c 2 Basic properties of fractional are non zero, Caputo’s derivative has been introduced, calculus yielding zero for any constant. Thus, it is considered as more suitable in the description of physical systems. Fractional calculus has recently become a branch of pure In fact Caputo’s derivative is defined by: mathematics, with many applications in physics and 1(x fs′ ) engineering, Tarasov [32, 33]. Many definitions of frac- caDf()xd= s (8) a x Γα(1− ) ∫ ()xs− a tional derivatives exist. In fact, fractional calculus origi- a nated by Leibniz, looking for the possibility of defining n dg 1 and the ­derivative n when n = . The various types of the dx 2 1(b fs′ ) fractional derivatives exhibit some advantages over the caDf()xd=− s (9) xb Γα(1− ) ∫ ()sx− a others. Nevertheless they are almost all non local, con- x trary to the conventional ones. The detailed properties of fractional derivatives may Evaluating Caputo’s derivatives for functions of the be found in Kilbas et al. [42], Podlubny [43] and Samko type: et al. [2]. Starting from Cauchy formula for the n-fold inte- f(x) = (x − a)n or f(x) = (b − x)n we get: gral of a primitive function f(x) ννΓν(1+ ) −α caDx(a−=)(x − a) , (10) x xxnn−1 x2 a x x Γα(1−+ν + ) I nnfx()==fs()()ds dx dx dx ….. fx()dx (1) ∫∫a nn∫∫−−12n ∫ 11 aa aa and for the corresponding right Caputo’s derivative: expressed by a ννΓν(1+ ) −α xbDb()−=xb()− x x Γα(1−+ν + ) 1 − Ifnn()xx=−()sf1 ()sdsx, >∈0, nN (2) a x (1n − )!∫ a Likewise, Caputo’s derivatives are zero for constant functions: and

b f ()xc= . (11) 1 − Ifnn()xs=−()xf1 ()sdsx, >∈0, n N (3) xb (1n − )!∫ Finally, Jumarie’s derivatives are defined by, x 1(dfx sf)(− a) the left and right fractional integral of f are defined as: JaDf()xd= s a x ∫ a Γα(1− ) dx a ()xs− 1(x fs) Ifa ()xd= s (4) a x Γα()∫ ()xs− 1−a a and K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics 45

b acaa Ja 1(dfsf)(− b) dx= Dx()dx (15) Df()xd=− s a x xb Γα(1− ) dx ∫ ()sx− a x hence

ca aaDf()x Although those derivatives are accompanied by some df()xd= a x x. (16) caDx derivation rules that are not valid, the derivatives them- a x selves are valid and according to our opinion are better than Caputo’s, since they accept functions less smooth It is evident that daf(x) is a non-linear function of dx, than the ones for Caputo. Also Jumarie’s derivative is although it is a linear function of dax. That fact suggests zero for constant functions, basic property, advantage, of the consideration of the fractional tangent space that we Caputo derivative. Nevertheless, Caputo’s derivative will propose. Now the definition of fractional differential, Eq. be employed in the present work, having in mind that (16), is imposed either for positive or negative variable dif- Jumarie’s derivative may serve better our purpose. ferentials dαx. In addition the proposed L-fractional (in La honour of Leibniz) derivative a Dfx ()x is defined by,

aLaa (17) df()xD= a x fx()dx 3 The geometry of fractional with the Leibniz L-fractional derivative,

differential ca La Df()x Df()x = a x . (18) a x caDx It is reminded, the n-fold integral of the primitive function a x f(x), Eq. (1), is

x Hence only Leibniz’s derivative has any geometrical of I nnfx()= fs()()ds (12) ∫a physical meaning.

In addition, Eq. 3, is deceiving and the correct form of which is real for any positive or negative increment ds. Eq. (3) should be substituted by,

Passing to the fractional integral La L α f ()xf−=(a)(aaIDxxfx()) x αα x 1−α I ((fx)) = fs()()ds (13) 1(s − a) L α ∫a = Df()sds (19) Γα()Γ(2 − a) ∫ ()xs− 1−a a x a the integer n is simply substituted by the fractional number α. Nevertheless, that substitution is not at all It should be pointed out that the correct forms are straightforward. The major difference between passing defined for the fractional differential by Eq. (17), the from Eq. (11) to Eq. (12) is that although (ds)n is real Leibniz derivative, Eq. (18) and the fractional integral by for negative values of ds, (ds)α is complex. Therefore, Eq. (19). All the other forms are misleading. the fractional integral, Eq. (13), is not compact for any Configurating the fractional differential, along with increment ds. Hence the integral of Eq. (13) is mislead- the first fractional differential space (fractional tangent ing. In other words, the differential, necessary for the space), the function y = f(x) has been drawn in Figure 1, existence of the fractional integral, Eq. (13), is wrong. Hence, a new fractional differential, real and valid for positive and negative values of the increment ds, should f(x) be established. f(x) It is reminded that the a-fractional differential of a function f(x) is defined by Adda [44]:

acaa ∆f(x0) df()xD= a x fx()()dx (14) α d f(x0)

It is evident that the fractional differential, defined by Eq. dx (14), is valid for positive incremental dx, whereas for nega- tive ones, that differential might be complex. Hence con- x sidering for the moment that the increment dx is positive, x x + dx ca 0 0 and recalling that a Dxx ≠ 1, the α-fractional differential of the variable x is: Figure 1: The non-linear differential of f(x). 46 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics with the corresponding first differential space at a point x, 4 Fractional vector calculus according to Adda’s definition, Eq. (14). The tangent space, according to [44] definition, Eq. For Cartesian coordinates, fractional generalizations (14), is configurated by the nonlinear curve dαf(x) versus of the gradient, divergence and rotation operators are dx. Nevertheless, there are some questions concerning defined by (see Lazopoulos and Lazopoulos [36]): the correct picture of the configuration, Figure 1, concern- (A) Grad(α) F(x) ing the fractional differential presented by Adda [44]. caDf()x Indeed, ∇=()ααfg()xxradf()a ()=∇()fx()ee= a i iicaDx i a. The tangent space should be linear. There is not con- a ii = LaDf()xe ceivable reason for the nonlinear tangent spaces. a ii (20) b. The differential should be configured for positive and ca negative increments dx. However, the tangent spaces, where a Di are Caputo’s fractional derivatives of order in the present case, do not exist for negative incre- α and the sub line meaning no contraction. ments dx. (B) Div(α) F(x) α c. The axis d f(x), in Figure 1, presents the fractional dif- ca DF()x ∇⋅()aa==() a kk = La ferential of the function f(x), however dx denotes the FF()xdiv ()xDca a kkF ()x (21) Dx conventional differential of the variable x. It is evident a kk that both axes along x and f(x) should correspond to where the sub-line denotes no contraction. differentials of the same order. (C) Curl(α) F(x)

Therefore, the tangent space (first differential space), caDF curl()aLFe==εεa mn e DFa (22) should be configured in the with axes llmn ca llmn a mn a Dxmm (dαx, dαf(x)). Hence, the fractional differential, defined by Moreover the and the fractional Gauss’ Eq. (17), is configured in the plane (dαx, dαf(x)) by a line, as formula for a field are given by: it is shown in Figure 2. (D) Circulation It is evident that the differential space is not tangent For a vector field (in the conventional sense) to the function at x0, but inter- sects the figure y = f(x) at least at one point x . This space, Fe(,xx , xF)(=+xx, , xF)(e xx, , x ) 0 123111 23 22 123 we introduce, is the tangent space. Likewise, the normal is +e33Fx(,12 xx, 3 ) (23) perpendicular to the line of the fractional tangent. Hence where F (x , x , x ) are absolutely integrable, the circu- we are able to establish fractional differential geometry of i 1 2 3 lation is defined by: curves and surfaces with the fractional field theory. Con- sequently when α = 1, the tangent spaces, we propose, CI()αα()==(,()aa )(dL, )(= IF() dx) LLFFaa∫ F L 11 coincide with the conventional tangent spaces. As a last L ++IF()aa()dxααIF()()dx comment concerning the proposed L-fractional derivative, a2LL2a 33 (24) the physical dimensions of the various quantities remain (E) Fractional Gauss’ formula unaltered from the conventional to any order fractional For the conventional fields theory, let calculus. F = e1F1 + e2F2 + e3F3 be a continuously differentiable real-valued function in a domain W with boundary ∂W. Then the conventional divergence Gauss’ theorem

∆f(x0) is expressed by,

∫∫ Fd⋅=SF∫∫∫ divdV. (25) ∂WW n Since d αf(x) ()α αα αα αα dS=+eee12dxdx32dx31dx+ 31dxdx2 (26) d αx where dax i = 1, 2, 3 is expressed by: x0 i

()αα= αα Figure 2: The virtual tangent space of the f(x) at the point x = x0. dV dx12dxdx3 (27) aL()αα τxx =∇λµV + 2 a Dux aL()αα K.A. Lazopoulos and A.K. Lazopoulos: Fractionalτyy =∇ λµvectorV calculus+ 2 a Dv andy fluid mechanics 47 aL()αα τzz =∇λµV + 2 a Dwz aa LLαα Furthermore τxy ==τµyx []aaDvxy+ Du ca aa==LLαα+ DF()x τxz τµzx []aaDuzxDw divx()aLFx()==a kk δ DFa () ca km a kk (28) aa LLαα a Dxkm τ ==τµ[]Dw+ Dv (33) yz zy aayz

The fractional Gauss divergence theorem becomes: where λ, μ the Lamè constants and V the array

()αα() (ui + vj + wk). FS⋅=dd()ααiv()FdV()α (29) ∫∫∂Ww∫∫∫ a a Remember that the differential dαS = nαdαS, where nα is the 6 The balance principles unit normal of the fractional tangent space as it has been defined in Section 3. Almost all balance principles are based upon Reynold’s transport theorem. Hence the modification of that theorem, just to conform to fractional analysis is presented. The con- 5 Fractional stresses ventional Reynold’s transport theorem is expressed by: ddA AdVd=+VAv dS. (34) In a fractional tangent space the fractional normal vector dt ∫∫dt ∫ n WW ∂W nα does not coincide with the conventional normal vector n (Lazopoulos and Lazopoulos [36]). Hence we should For a vector field A applied upon region W with boundary expect the stresses and consequently the to ∂W and vn is the normal velocity of the boundary ∂W. differ from the conventional ones not only in the values a. The balance of mass but in the directions as well. The conventional balance of mass, expressing the If dαP is a contact force acting on the deformed area mass preservation is expressed by: dαa = nαdαa, lying on the fractional tangent plane where nα d is the unit outer normal to the element of area dαa, then ρdV = 0 (35) dt ∫ the α-fractional stress vector is defined by: W

α α d P In the fractional form it is given by: t = lim α (30) dαa->0 d a ()α d α ρdV= 0 (36) dt ∫ However, the α-fractional stress vector does not have ω W any connection with the conventional one: d where is the total time derivative. dP dt t = lim (31) da->0 da Recalling the fractional Reynold’s transport theorem, we get: since the conventional tangent plane has different orien- ()αα() d ααα tation from the α-fractional tangent plane and the corre- ρρdV=∂((xt, )[+ divdvρ]) V (37) dt ∫∫t sponding normal vectors too. aaVV Following similar procedures as the conventional ones we may establish Cauchy’s fundamental theorem, see [45]. Since Eq. (37) is valid for any volume V, the continuity If tα(nα) is a continuous function of the transplace- equation is: ment vector y, there is an α-fractional Cauchy stress ∂+ρρdiva[]v = 0. (38) tensor field. t where, divα has already been defined by Eq. (28). That T aa=[]τ (32) ij is the expressed in fractional form. For a Newtonian fluid the following relations apply: b. Balance of linear principle It is reminded that the conventional balance of linear τaL=∇λµ()αα+ 2 Du xx V a x momentum is expressed in continuum mechanics by: aL()αα τyy =∇λµV + 2 a Dvy d ()n αα ρδvtVd=+Sdρb V (39) τaL=∇λµ()V + 2 Dw dt ∫∫∫ zz a z ΩΩ∂ Ω aa LLαα τxy ==τµyx []aaDvxy+ Du aa LLαα τxz ==τµzx []aaDuzx+ Dw aa LLαα τyz ==τµzy []aaDwyz+ Dv 48 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics

where v is the velocity, t(n) is the traction on the bound- In Eq. (45) we can write (related to the right side): ary and b is the body force per unit mass. Likewise υ LLDDββσυ=−()σσLLDDββυυ= ()σ that principle in fractional form is expressed by: ixaaiiij xiji ji aaxijixiji L β ()αα() −+()DVσυ=−DD()σσ (47) ij ij ji a xii ji ij ji d ααα ρρvbdV() =+[(divda T )] ()V (40) dt ∫∫ aaΩΩ In Eq. (47) we have:

L β a DDxiυ =+ij Vij Hence the equation of linear motion, expressing the i balance of linear momentum is defined by, where:

aa dv ()aa() ()aa() div []Tb+−ρρ= 0 (41) 11 (48) dt D =∇()vvxx+∇ Vv=∇()xx−∇ v 22

In Eq. (40) we must introduce in the term We consider that: Vijσij = 0, since it is a product of a sym- Tα. This is done by adding the term –pI. Therefore Eq. metric times an antisymmetric tensor. Therefore Eq. (45) (40) becomes: with the help of Eq. (46) and Eq. (47) becomes:

2 aa dv d ρυ divp[]TI−+ρρb −=0 ()aa() La ()a (42) dV+=Ddij συji VDa xi()σ ji d V dt ∫∫VV∫ j dt 2 V + ρυbd()a V ∫ ii (49) It should be pointed out that divα has already been defined by Eq. (28) and is different from the conven- The term LaDd()υσ ()a V in Eq. (49), due to the ∫ a xij ji tional definition of the divergence. V Gauss theorem, takes the form:

Following similar steps as in the conventional case, the La ()aa() a Ddxi()υσji Vn= υσiijjdS balance of rotational momentum yields the symmetry of ∫∫j (50) VS Cauchy stress tensor.

With the help of Eq. (50), Eq. (49) becomes:

d ρυ2 dV()aa+=Ddσυ()VnσρdS()aa+ bdυ ()V 7 First law of dt ∫∫VV2 ij ji ∫∫iijj ii SV (51) This law occurs from Eq. (42). Firstly we replace Ta − pI with Σα. d ρυ2 In Eq. (51) dV()a is the rate of change of dt ∫V 2 α a (43) d Σ =−TIp the kinetic energy of the system KD, σ dV is dt ∫V ij ji α Σ =[]σ (44) d ij the rate of change of the internal energy U , and dt We then derive the equation of υσ nd()aaSb+ ρυd()V from Eq. (42). This is accomplished by multiplying the ∫∫iijj ii is the rate of work of the exter- SV equation Eq. (42) with the velocity υ and the result is inte- i d grated over the volume V: nal forces in the system W . dt

dυ ()aLβ ()aa() For a thermomechanical continuum it is customary to ρυ i dV=+υσDdVbρυ d V (45) ∫VVii∫∫a xij jiV i dt express the time rate of change of internal energy by the integral expression: But the left side of Eq. 45 becomes: dd ()aad () 2 Ue==ρρdV ed V (52) dυ d ρυ dt dt ∫∫dt ρυ i dV()aa= dV() (46) VV ∫VVi ∫ dt dt 2 where e is called the specific internal energy. Moreover, which represents the time rate of change of the kinetic if the vector ci is defined as the heat flux per unit area per energy K in the continuum. unit time by conduction, and z is taken as the radiant heat aL()αα τxx =∇λµV + 2 a Dux K.A. Lazopoulos and A.K. Lazopoulos: FractionalaL vector()αα calculus and fluid mechanics τyy =∇λµV + 2 a Dvy 49 aL()αα τzz =∇λµV + 2 a Dwz aa LLαα constant per unit mass per unit time, the rate of increase τxy ==τµyx []aaDvxy+ Du of total heat into the continuum is given by: aa LLαα τxz ==τµzx []aaDuzx+ Dw aa LLαα d ()aa() τ ==τµ[]Dw+ Dv (60) Qc=− nd Sz+ ρ d V (53) yz zy aayz dt ∫∫ii V S e) The kinetic : Therefore the energy principle for a thermomechani- cal continuum is given by: pp= (,ρ T) (61) dddd Κ +=UW+ Q (54) f) The Fourier law of heat conduction: dt dt dt dt c =−kT∇()α (62) or, in terms of the energy integrals as: g) The caloric equation of state: ddρυ2 dV()aa+=ed()Vnυσ dS()aa+ ρυbd()V ee= (,ρ T) (63) ∫∫VV∫∫iijj ii dt 2 dt SV +−ρzd()aaVcnd()S The system of these 16 equations contains 16 unknowns ∫∫V ii (55) S therefore it is determinate. Usually the studies concerning the Newtonian fluid, are restricted to the equations of con- Converting the surface integrals in Eq. (55) to volume servation of mass (57), linear momentum (58) and consti- integrals by the divergence theorem of Gauss, and again tutive equations (60). the fact that V is arbitrary, leads to the local form of the energy equation:

2 d υ 11LLββ +=eDaaxi()συji+−bDiiυ xiczi + (56) dt 2 ρρj 9 The Euler and Bernoulli equations

The Euler equations occur from Eq. (27) for non-viscous fluid, for which Tα = Ο. This holds because the Lamé con- 8 Navier-Stokes equations stants λ, μ are considered zero in this case. Therefore, Eq. (49) takes the form: The Navier-Stokes equations are consisted by the follow- d a ing equations: ρ vb=−ρ divp[]I (64) dt a) Balance of mass

d a Lets assume a barotropic condition ρ = ρ(p). A pres- ρρ+=div []v 0 (57) dt sure function may be defined: p dpa b) Balance of linear momentum P()ρ = ∫ (65) p ρ d 0 divpaa[]TI−+ρρbv−=0 (58) dt Moreover, the body force may be described by a poten- tial function: c) First law of thermodynamics =−∇()a Ω (66) 2 b d υ 11LLββ +=eDaaxi()συji+−bDiiυ xiczi + (59) dt 2 ρρj With these two conditions Eq. (42) becomes:

L βα() atDPv =−∇+()Ω (67) d) Constitutive equations If Eq. (67) is integrated along a streamline, the result τaL=∇λµ()αα+ xx V 2 a Dux is the Bernoulli equation in the fractional form: aL()αα τyy =∇λµV + 2 a Dvy υ2 d τaL=∇λµ()ααV + 2 Dw ΩΡ++ +=vda xC()t (68) zz a z 2 ∫ dt ii aa LLαα τxy ==τµyx []aaDvxy+ Du aa LLαα τxz ==τµzx []aaDuzx+ Dw aa LLαα τyz ==τµzy []aaDwyz+ Dv 50 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics

∂p γγiiγ −=ρ LLDD((ut)) + cDL ut() (77) 10 Darcy’s law ∂ ft00tt0 x Consider a viscous fluid, flowing in a straight pipe of a constant circular cross section of inner radius R and cross Considering: 2 section of A = πR with perimeter Γ = 2πR. If L is the length ii J()tD=+ρ LLγγ((Dutc)) L Duγ ()t (78) of the considered segment of the pipe defined by positions ft00tt0 x = α and x = b, then L = b – α. The governing Eq. (77) becomes: Denoting Q(x, t) the discharge of the fluid, continu- ∂p ity of flow demands that Q(x, t) be the same at any cross −=Jt() (79) ∂x section of the pipe segment, depending only upon time

Qx(, t)(= Qt) (69) For the case that J(t) = J (constant) between the seg- ments x = α and x = b, the fluid velocity is defined by: The cross sectional velocity of the flow may be com- LLγγ L γ puted by: ρft00DD((ttut)) +=cD0 ut() J = constant (80)

Q L γ VD==0 t ut() (70) Solution of fractional D.E: A In case Δp is the pressure drop from the one end to the other of the pipe segment with: Due to the fluid’s internal friction, shear stresses τ are developed, between the fluid and the inner pipe-wall. ∆αpp=−()bp()< 0 (81a) Similarly to fractional viscoelasticity, the interface friction Then the total discharge is defined by: in circular pipes is equal to: Ap∆ 44µµ QA==V − (81b) τ ==ffVDL γut() (71) cL 0 t RR Eq. (81b) is the famous Darcy’s law. where μf is fluid . The friction stresses may be substituted by a fictitious body force per unit length of the pipe,

L γ (72) 11 Fractional flow in porous media fbf=⋅τπVD= 8(µ 0 t ut) The conservation of linear momentum equation Considering the difference Δp to the hydrostatic we get: yields: ∆ρ=− ∆ pgf h (82) ∂p γγi −−Adxfdx ==ρρAV ADLL((Dut)) (73) ∂x bf ft00t Then we may assume that for the wetted area Av (The area of the voids), Darcy’s law may be applied with: or  ∂p γγi Av ∆p +=fA− ρ LLDD((ut)) (74) Q =− (83) ∂ ft00t cL x Assuming that the surface porosity Φ is equal to Since, according to Poiseille’s law, f is proportional to Α volume porosity Φ we get: the mean flow-velocity

i AV L γ v v f ==cV cDut() (75) ΦΦA =≈= (84) 0 t AV

Furthermore the coefficient c of viscous friction is pro- portional to the fluid viscosity and inversely proportional And thus: to the square of the pipe radius: (85) AAv ≈⋅Φ µf c =⋅π 2 (76) For the specific fluid discharge: R Q q = (86) Thus we obtain the : A K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics 51 we get: Recalling Darcy’s law with:

QQAq Ap∂ µ Vq=≈ =→= ΦV (87) Qc=− , = π2 f (95) AAvvA Φ Cx∂ A

Hence Darcy’s law becomes: or

Qp∆ Ap2 ∂ qK==− (88) Q =− (96) AL 2 ∂ πµf x

Φ with K = c Eliminating the discharge Q, we end up to: Finally, Darcy’s law may be generalized by: ∂p ∂A2 µ ∂A 1 ∂ 1 ∂p f = x (97) qf=− , where = (89) 2 fx∂ K ∂∂txπµ f with p: The pore fluid pressure. Since: q: The specific fluid discharge. ∂∂AA∂∂ppA K: The permeability of the porous medium. ==0 (98) ∂∂tp∂∂txκ μf: The viscosity of the fluid. Consequently, Eq. (97) yields:

2 2 ∂∂pp1  ∂ p 12 Fractional flow in elastic tubes =⋅2 kA⋅+ ⋅⋅κ (99) ∂∂txπµ22 ∂x f Consider a thin elastic tube, at its (unstressed) reference placement, with its inner radius R and its thickness δ  R. However, for small changes in pressure, the linear The elastic tube is considered into the context of linear problem is considered, resulting in: with modulus E. Loading the tube with pressure ∂∂pp2 p, its radius increases by ΔR. Hence its cross-sectional = c (100) ∂t p ∂x2 area becomes:

AR=⋅π∆()+ R 2 (90) where:

3 and its current radius r = R + ΔR is given by: A0 ⋅ κ 2⋅R cp ==2 (101) 2 πµ πµ⋅⋅E ⋅δ R f f rR=+∆RR=+ P (91) ⋅δ E Solution of parabolic equation. Restricting to the where the changes Following just the same procedure, but for the frac- of the radius are infinitesimal, we get: tional time derivative fields, we get the fractional para- bolic equation,  Rp0 ⋅ AA=+00∆AA=+12 (92) 2 E ⋅δ L α ∂ p Dp= c (102) 0 tp∂ 2 x Therefore the relation between the variable cross- section area and the fluid pressure and cross-section area: Furthermore, the initial condition expresses the con- stant pressure value, pR2⋅ AA=⋅1,+= κ (93) 0 κδE ⋅ tp==0: px, ∀ ε[0, L]. (103) 0

Since mass balance is expressed by: Further, at the time =0+, the pressure at the entry point of the tube is increased by Δp, keeping constant the pres- ∂∂AQ +=0 (94) sure value at the exit point of the tube. Consequently the ∂∂tx boundary conditions for t > 0 become, 52 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics

xp==0, pp=+∆p (104) ∆p 10 xp==0, p ±+1,λλ = (115) 1 p 0 xL==, pp0 (105)

=== Nondimensionaling the variables we get: xp1, p2 1

2 ∗∗px∗∗tL px==, 0()≤≤xt1, and ==where tc . (106) Considering first the steady solution: pLoctcp 2 L α ∂ p Dp==00pp==pp+−()px (116) 01t ∂x2 12 Omitting the upper-stars the non dimensional governing equation of the fractional flow become, And introducing a renormalized pressure: 2 α ∂ p L Dp= (107) pp− 0 t ∂x2 pˆ = (117) pp− 12 with the initial condition, From the above formulas occurs:

xp==0; 1, ∀x ε[0, 1] (108) 2 ∂∂ppˆˆ =≤01x ≤ (118) ∂t ∂ 2 and the boundary conditions, x ∆p xp==0, p =+1,λλ = (109) The initial condition is: 1 p 0 phˆ ==()xtx −=1, 0, 0 ≤ (119)

xp==1, p2 = 1. And the boundary conditions are: pˆ ==hx() xt−=1, 0, 01≤≤x (120) ∂∂pp2 = c ∂t p ∂x2 And the boundary conditions: pˆ =>0, tx0, ==0 and x 1 (121) The initial condition for the pressure Applying the separation of variables technique for the tp==0: px, ∀ ε[0, L] (110) 0 fractional diffusion equation we get:

At t = 0+, the pressure at the entry point is increased pˆ(,xt ) = Xx()Tt() (122) by Δp with constant pressure at the exit point. Hence the Following the conventional procedure for the diffusion boundary conditions for t > 0 become: equation we get:

xp==0, pp±+∆p (111) L α 10 DT()t Xx′′() 0 t ==−=λ2 constant (123) Tt() Xx() xL==, pp0

Non-dimensional variables become: Due to appendix, the solution to the equation (101) is

2 given by: ∗∗px∗∗tL px==, 0()≤≤xt1, and == where tc . (112) ∞ pL tc pxˆ(,xt )s=−bnin()πΦα((nπ),2 t). 0 cp ∑n=1 n (124) where: The non-dimensional equation becomes: k 2 Γκ+−α ……… ∂∂ ακ22∞ ( 1 ) t pp (113) Φπ((−=nt), )(−()nπ ) = ∑ k=1 κ−1 ∂t ∂ 2 Γκ(1+…).ΓΓ(3)(2)− α x (125) With the initial conditions. For h = x − 1, (see Appendix) tp==0: 1, ∀xLε[0, ] (114) 21∞ α pˆ ≈− sin(nxπΦ)(()ntπ 2 , ). (126) π ∑n=1 And boundary conditions: n K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics 53

Laϕλ+=2 ϕ 13 Conclusion 0 Dtt () ()t 0 (A.8) Looking for the solution of ϕ(t) into the power series of Since the preliminary elements have been defined expansions of t we get: (Leibniz fractional derivative, fractional gradient, frac- After substituting into the Eq. (A.8) the relation: tional rotation, fractional divergence, fractional circula- tion and fractional Gauss’ theorem), the basic equations ∞∞Γκ(1+−)(Γα2)k+12 κ yt+=Ayt 0 ∑∑n==10κκκ (A.9) of fluid mechanics are reinstated and analyzed. Further, Γκ(1+−α) Fractional Darcy’s flow was studied as an application of Equating the coefficients of the same power of t to zero, the fractional flows into porous media. The main issue we have: in our case is experimental validation of the occurring equations. Γκ(1+−α) y =−Ay2 κκΓκ(1+−)(Γα2)−1

µ+ Γκ(1+−αΓ).…−(1κµ+−α) y =−()Ay21 Appendix κκΓκ(1+…).Γκ(1−+µΓ)(2)− α µ+1 −−µ 1

21µ+ Γκ(1+−α).… yκ =−()Ayκ−1 0 (A.10) The fractional diffusion equation Γκ(1+…).ΓΓ(3)(2)− α Therefore, it occurs: The conventional diffusion equation is: 2 2 =− ∂∂ppˆˆ y10Ay = 2 (A.1) ∂t ∂x 2 − a y =−()Ay22 202 with the initial conditions (3−−aa)(2) y =−()Ay23 pxˆ ==hx() −=1, tx0, 0 ≤≤1 (A.2) 3012

24(4−−α)(3)aa(2 − ) And the boundary conditions: y =−()Ay (A.11) 40288 ∆p xp==0, p +=λλ, 1 (A.3) and p0 ∞ κ Γκ(1+−α).… y =−y ()λ2 === λλ0, ∑ κ=1 κ−1 (A.12) xp1, p2 1 Γκ(1+…).Γα(2 − )

Application of variables technique gives: and

∞ px(, tf)(= xg)(t) (A.4) pbˆ(,xt )s= in()nxπ ∑ n=1 n

∞ κκΓκ(1+−α).… Therefore, the diffusion equation yields: ((− ntπ))2 (A.13) ∑ κ=1 κ−1 2 Γκ(1+…).Γα(2)− ff′′ +=η 0 LaDg()tg+=η2 0 (A.5) 0 t If we define:

ακ∞ Γκ(1+−αΓ).…−(3 α) κ Taking under consideration the boundary conditions the Φπ()()nt22, =−((ntπ)) ∑ κ=1 κ−1 solution is expressed by: Γκ(1+…).ΓΓ(3)(2)− α

∞ (A.14) ptˆ(,xt )s=−bnin()ππxnexp( 222 ) ∑n=1 n (A.6) Then we have:

The fractional diffusion equation is: ∞ α pbˆ = sin(nnπΦxt)(()π 2 , ) (A.15) 2 ∑n=1 n α ∂ px(, t) L = 0 Dpt 2 (A.7) ∂x ∞ For hx()=≤bnsin( πxx), 01≤ ∑ n=1 n (A.16) With its initial and boundary conditions the same as 1 the conventional ones stated above. bh= 2(xn)sin()πxdx (A.17) n ∫0 The solution: pˆ(,xt ) = Xx()Tt() is defined by the solutions the system: With h = x − 1 we get: 54 K.A. Lazopoulos and A.K. Lazopoulos: Fractional vector calculus and fluid mechanics

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