J Mech Behav Mater 2017; 26(1–2): 43–54 Konstantinos A. Lazopoulos* and Anastasios K. Lazopoulos Fractional vector calculus and fluid mechanics https://doi.org/10.1515/jmbm-2017-0012 order derivatives of the strain. Many problems have been solved employing those theories concerning size effects, Abstract: Basic fluid mechanics 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 continuum mechanics; fractional fractional vector fields theory with applications. Darcy flow; fractional derivative; 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 fluids. 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 quantum mechanics, 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- dynamics, fluid flow, rheology, viscoelasticity, optics, elec- istic responses in various scientific problems in physics trochemistry and corrosion, chemical physics 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 deformation 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, gradient 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 vector field 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 fluid dynamics [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.