Fractional constitutive equation (FACE) for non-Newtonian fluid flow: Theoretical description
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec Fractional constitutive equation (FACE) fornon-Newtonian fluid flow: Theoretical description
HongGuang Sun
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Collegeof Mechanics and Materials, Hohai University, Nanjing 210098, China
Yong Zhang
1. College of Mechanics and Materials, Hohai University, Nanjing 210098, China2. Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487,United StatesCorresponding author: [email protected]
Song Wei
College of Mechanics and Materials, Hohai University, Nanjing 210098, China
Jianting Zhu
Department of Civil and Architectural Engineering, University of Wyoming, 1000 E.University Ave., Laramie, WY 82071, United States
Abstract
Non-Newtonian fluid flow might be driven by spatially nonlocal velocity,the dynamics of which can be described by promising fractional derivativemodels. This short communication proposes a space FrActional-order Con-stitutive Equation (FACE) that links viscous shear stress with the velocitygradient, and then interprets physical properties of non-Newtonian fluids forsteady pipe flow. Results show that the generalized FACE model containsprevious non-Newtonian fluid flow models as end-members by simply adjust-ing the order of the fractional index, and a preliminary test shows that theFACE model conveniently captures the observed growth of shear stress forvarious velocity gradients. Further analysis of the velocity profile, frictionalhead loss, and Reynolds number using the FACE model also leads to ana-lytical tools and criterion that can significantly extend standard models inquantifying the complex dynamics of non-Newtonian fluid flow with a wide
Preprint submitted to arxiv.org December 13, 2016 ange of spatially nonlocal velocities.
Keywords:
Fractional constitutive equation, Non-Newtonian fluid flow,Fractional velocity gradient, Velocity profile, Fractional Reynolds number
1. Introduction
Non-Newtonian fluids have many real-world applications [1, 2]. Theoret-ical investigation provides useful information for the analysis and simulationof mass or energy transfer in non-Newtonian fluids [3, 4], although theiranomalous flow behaviors are usually more complex than Newtonian fluids.Extensive experiments and measurements have revealed a nonlinear relation-ship between viscous shear stress and velocity gradient for non-Newtonianfluids such as muddy clay, oils, blood, paints, and polymeric solutions [5, 6].Several empirical or semi-empirical formulas, such as the well-known power-law model, the Bingham model, and the Casson model, have been proposedto successfully quantify non-Newtonian viscosity behaviors observed in mul-tiple disciplines [7]. The present models however suffer from several majordrawbacks. For example, they lack a unified constitutive description for mostnon-Newtonian fluids. In addition, model parameters obtained from one flowsystem usually cannot be extended to another system for the same fluid [8].This communication addresses the first challenge, with the expectation thata generalized constitutive equation may lead to transformative parameters.Complex non-Newtonian fluids may be related to behaviors of their com-ponents, whose dynamics may exhibit complex memory impact in time orspace. Observations and measurements (for systems at either micro- ormacro-scale) show that these fluids are often mixtures of materials with dif-ferent sizes, such as water, solid particles, polymer, oil, and other long chainmolecules [9]. The kinematic and dynamic behavior of these mixed materialscan frequently exhibit long-time memory and (spatially) non-local properties[10, 11]. The Newtonian constitutive equation only involves local influenceof the system, which is captured by an integer-order velocity gradient in themodel. A generalization of the standard, integer-order constitutive equa-tion might be necessary to incorporate memory and non-local properties fornon-Newtonian fluids. Existing models including the power-law model andthe shear rate dependent dynamic viscosity coefficient models can only par-tially describe the time history effect and non-local properties [1, 12, 13, 14].Moreover, shear rate dependent dynamic viscosity coefficient models usually2enerate nonlinear equations which are not easy to get the analytical ex-pressions for real-world applications. Besides, shear rate dependent viscositymodels have various expression forms for different kinds of non-Newtonianfluids, which have caused confusion in applications.Fractional-derivative models were recently suggested by several studiesas a promising tool to characterize non-Newtonian fluids, because fractionalcalculus can well characterize a physical system with long-term memoryand spatial non-locality [15, 17, 18, 19]. For example, the time fractional-derivative based constitutive equation, which can well describe the historydependency in fluid dynamics, was proposed to characterize different types oftime-dependent non-Newtonian fluids [17]. This equation was further appliedto analyze the various dynamic processes of muddy clay [17], seepage flow indual-porosity media [18], blood viscosity [19], and the behavior of Sesbaniagel and xanthan gum [20]. Previous studies of fractional-derivative modelsfor non-Newtonian fluids, however, are mainly from the rheology viewpoint.Most importantly, the possible spatial non-local dependency of flow velocity,which can significantly affect non-Newtonian fluid dynamics, has not beensufficiently addressed, although a few researchers have proposed that spacefractional-derivative models may have the ability to capture non-Newtonianfluids. For example, Ochoa-Tapia et al. [15] provided a brief mathematicalderivation of fractional Newton’s law for viscosity based on Taylor series, toobtain a fractional-order Darcy’s law for describing shear stress phenomenain non-homogeneous porous media. Chen et al. [21] investigated the effectsof model parameters to simulate the boundary layer flow of Maxwell fluids onan unsteady stretching surface, using the time-space fractional Navier-Stokesequation built upon a fractional-derivative based constitutive equation. How-ever, the physical analysis and description of non-Newtonian fluid flow basedon the fractional-derivative constitutive equation, which addresses the non-locality of the velocity field (represented by a strong correlation betweenvelocities), has not been investigated, hindering practical application of thespace-fractional models.This paper proposes and investigates a fractional-derivative constitutiveequation (FACE) by employing a fractional-derivative velocity gradient, inwhich the space fractional-derivative is designed to address the non-local ef-fect of velocity (likely due to complex interactions between non-uniform com-ponents and/or the impact of complex systems) on non-Newtonian fluid flow.Application of the space fractional-derivative is motivated by the studies re-viewed above. We then use the FACE model to reclassify non-Newtonian3uids, and test its applicability using literature data. Related physical quan-tities including velocity profile, frictional head loss, and fractional Reynoldsnumber, are also analyzed for details of non-Newtonian fluid in steady pipeflow, to provide a theoretical description of non-Newtonian fluids necessaryfor real-world applications [1, 22].
2. Fractional-derivative constitutive equation for non-Newtonianfluid
The classical, empirical Newton’s constitutive relationship for shear stressin terms of the velocity gradient can be expressed as τ = µ dudy , (1)in which τ is the viscous shear stress, µ is the dynamic viscosity, u is the flowvelocity, du/dy is the velocity gradient, and y is the direction perpendicularto the fluid flow. Many kinds of fluids (such as slurry, blood, and rubber)however have been found to be non-Newtonian fluids which do not follow theclassical Newton’s law of viscosity (1).Previous investigations have offered several general extensions of Eq. (1)for various non-Newtonian fluids. One of the most commonly used formulasis the following power-law model (which is also called the Ostwaald-de Waelemodel) τ = τ + µ (cid:18) dudy (cid:19) n , (2)in which τ represents the yield stress, n is a parameter which correspondsto different kinds of non-Newtonian fluids. Although this model has provenuseful in describing various kinds of non-Newtonian fluids, its rationality andphysical origin remain obscure. Extensive studies have shown that particlemotion within non-Newtonian fluid has memory, and that the memory rateis related to the physical property of the target non-Newtonian fluid [16, 17].To accurately describe time-dependent flow or rheology behavior of non-Newtonian fluids (or viscoelastic material) such as creep, the following time-fractional constitutive equation has been proposed [17] τ ( t ) = τ + θ λ β d β − ˙ εdt β − , ≤ β ≤ , (3)4here d β − /dt β − is the time fractional integral used to describe the timedependency of non-Newtonian fluids, θ and λ are material constants, and ˙ ε denotes strain rate. This equation has been widely used to construct com-ponent models for Maxwell viscoelastic materials [23], Oldroyd-B fluid [24],and some unsteady flows [25].To address the non-locality of non-Newtonian flow and the potential cor-relation of particles or components with different sizes, here we consider ageneral constitutive relationship by employing the fractional velocity gradi-ent: τ = τ + µ α d α udy α , < α < , (4) τ = µ α d α udy α , < α < , (5)where α is the order of the space fractional derivative, and d α u/dy α repre-sents the fractional velocity gradient. Here the physical origin of non-localitycharacterized by the fractional velocity gradient is likely a result of the mix-ing of non-uniform particles, close interaction of molecules, the existence ofa continuous network of interactions between the elements [26], or the ef-fect of biological and chemical properties on non-Newtonian fluids [13]. Thedefinition of the fractional derivative used in this study is expressed by d α u ( y ) dy α = 1Γ( n − α ) Z y u ( n ) ( τ )( y − τ ) α − n +1 dτ, n − < α ≤ n, (6)where Γ( · ) denotes the Gamma function, and n is the smallest integer greaterthan the order α . In this study, we consider a fractional derivative modelfor non-Newtonian fluid with 0 < α <
2. In order to facilitate analysis, wefurther investigate the relationship between the velocity gradient and viscousshear stress. Hence we rewrite Eq.(5) as τ = τ + µI − α ( dudy ) , < α < ,τ + µ dudy , α = 1 ,τ + µI − α [ ddy ( dudy )] , < α < , (7)5able 1: A reclassification of non-Newtonian fluids based on the FACE model(5).yield stress ( τ ) Fractional derivative order ( α ) Name of fluid τ = 0 α = 0 Elastomer τ = 0 0 < α < τ = 0 α = 1 Newtonian fluid τ = 0 2 > α > τ = C ( C > α = 1 Bingham Fluid (I) τ = C ( C >
0) 0 < α < I − α and I − α represent the fractional integral I γ f ( y ) = 1 Γ ( γ ) Z y ( y − τ ) γ − f ( τ )d τ, γ > . (8)A reclassification based on the FACE (5) for non-Newtonian fluids is givenin Fig. 1 and Table 1. Fig. 1 clearly shows that the distinction of differenttypes of fluids can be well characterized using the order of the fractionalderivative. It is also noteworthy that the limiting case, α = 0 representselastomer and idea fluid when µ = 0.For illustration purpose, we analyze a group of experimental data ofwormlike micelles which have been found to be non-Newtonian fluid [27],and further explore the relationship between shear stress and velocity gradi-ent (or shear rate) using the FACE model (5). Fig. 2 compares fitting resultsusing the FACE model (5), the power-law model, and the Bingham model.It is clear that the power-law and Bingham models give good agreement at ahigh velocity gradient regime, while the FACE model (5) provides accuratedescription for the entire region.
3. Theoretical analysis of steady pipe flow
For application purposes, we conduct theoretical derivations for majorphysical quantities related to non-Newtonian fluid in steady pipe flow.
Here we assume that 1) the non-Newtonian fluid is incompressible, and 2)the pipe flow is laminar. Based on the mechanical analysis of a fluid system6shown by the cylinder with blue color in Fig. 3) in steady flow, the followingforce balance equation can be established in the flow direction p A ′ − p A ′ + ρgA ′ lsinθ − τ · πrl = 0 , (9)where A ′ denotes the cross section area, r is the radius of the cross sectionof the considered cylinder system, ρ is the density of non-Newtonian fluid,and R is the pipe radius. Using the relationship of sinθ = ( z − z ) /l , wecan re-arrange (9) to get( z + p /ρg ) − ( z + p /ρg ) l = 2 τρgr . (10)Finally the viscous shear stress can be expressed as τ = ρgrJ/ , (11)where J (= h f /l ) is the hydraulic gradient, and h f is the frictional head lossexpressed by h f = ( z + p /ρg ) − ( z + p /ρg ).It is clear that the viscous shear stress calculated by the fractional New-tonian constitutive equation Eq.(5) should be equal to Eq. (11) obtained bymechanical analysis. For illustration purposes, firstly we consider the non-Newtonian fluid without the yield stress τ (i.e., τ = 0), which leads to thefollowing result µ d α udr α = − ρgrJ/ . (12)The negative sign on the right-hand side of (12) comes from the oppositedirections of two stress expressions. By employing the property of the Caputofractional derivative [28], we get u ( r ) − u ( r = 0) = − ρgJ r α µ Γ( α + 2) . (13)Since flow velocity at the pipe wall is zero, we have the boundary condition u ( r = R ) = 0, and then we get the following result u ( r = 0) = ρgJ R α µ Γ( α + 2) . (14)7inally, the velocity profile is written as u ( r ) = ρgJ µ Γ( α + 2) ( R α − r α ) . (15)We emphasize that the above result is correct for both 0 < α ≤ < α <
2, since u max = u ( r = 0) and u ′ ( r = 0) = 0. It is also clear that Eq.(15) reduces to the velocity profile of Newtonian fluid when α = 1.Moreover, the maximum velocity and mean velocity can be derived fromEq. (15) u max = ρgJ R α µ Γ( α + 2) , ¯ u = ρgJ µ Γ( α + 2) (1 −
23 + α ) R α , (16)and ¯ uu max = 1 −
23 + α , (17)in which u max and ¯ u represent the maximum and mean velocities in the crosssection, respectively. It is obvious that ¯ u/u max < / < α < u/u max > / < α <
2, and this ratio becomes 1 / α = 1 (i.e.,Newtonian fluid). It implies that the velocity distribution of non-Newtonianfluid may be less or more uniform than Newtonian fluid, which agrees wellwith experimental data [29]. A comparison of the velocity distribution of non-Newtonian fluids in steady pipe flow, as characterized by different fractional-derivative orders, is shown in Fig. 4. Large differences in velocity distribu-tions can be found when different fractional derivative orders are used. InFig. 4, we use the same physical parameters for different non-Newtonianfluids (for direct comparison). It is noteworthy that those parameters onlyaffect the velocity values, while the velocity distribution is determined mainlyby the fractional order α . Further investigations of specific non-Newtonianfluids are needed in a subsequent study. Furthermore, we can get the expression of the frictional head loss ( h f )from the mean velocity in Eq. (16): 8 f = 2 ν Γ( α + 2) g α α l ¯ uR α , (18)in which ν = µ/ρ is the kinematic viscosity. Eq. (18) can be rewrite as h f = 64 − α (3+ α )Γ(1+ α ) ¯ uD α ν lD ¯ u g , (19)where D = 2 R is the diameter of the pipe. Eq. (19) reduces to the expression of the frictional head loss for Newto-nian fluid when α = 1: h f = 64 Re lD ¯ u g , (20)in which Re = ¯ uD/ν is the Reynolds number of Newtonian fluid. Combing(19) and (20), we hereby define the following Reynolds number for non-Newtonian fluid described by the FACE (5): Re α = 2 − α (3 + α )Γ(1 + α ) ¯ uD α ν . (21)This expression indicates that different types of non-Newtonian fluids owntheir critical Reynolds number, which is characterized by the fractional index α corresponding to specific non-Newtonian fluid. For example, the criticalReynolds number of expansion fluid (0 < α <
1) is larger than the pseu-doplastic fluid (1 < α < D = 1 . D plays a more impor-tant role in determining the fractional Reynolds number, in comparison withNewtonian fluid. Extensive experiments are required in a future study todetermine the feasibility of the generalized, fractional Reynolds number (21)in distinguishing flow patters of various non-Newtonian fluids.9 .4. Yield stress From a more general point of view, we further consider non-Newtonianfluid with yield stress τ (i.e., τ = 0). The velocity profile of pipe flowexpressed by Eq. (15) now should be changed to u ( r ) = R α µ Γ(1 + α ) (cid:20) ρgJ R α ) − τ (cid:21) − r α µ Γ(1 + α ) (cid:20) ρgJ r α ) − τ (cid:21) . (22)This means that non-Newtonian fluid flow with nonzero-value yield stressoccurs when the shear stress exceeds a critical value. We also emphasizehere that this expression (22) is only valid for fully developed flow. Variousstructural flows including plug flow may exist for variable viscous shear stressvalues, which can dramatically complicate the above analysis [26, 30]. Thisremains an open research question.
4. Conclusion
This study proposes, explores, and tests the fractional constitutive equa-tion (FACE) for describing non-Newtonian fluids. Results show that theFACE model captures the most common behaviors of non-Newtonian fluidflow, and is sufficiently simple to allow for analytical expressions of the flowfield in steady pipe flow. Although lacking extensive experimental validation,the concept presented in this paper provides the first and fundamental stepin an unpaved path to future development of reliable non-Newtonian fluiddynamic models using the promising fractional derivative.
Acknowledgment
The work was supported by the National Natural Science Foundation ofChina (Grant Nos. 11572112, 41628202, and 11528205). This paper does notnecessarily reflect the view of the funding agency.
References
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0, and µ = 0 .
10 20 30 401020304050607080 Velocity gradient (s −1 ) S hea r s t r e ss ( P a ) Experimental dataFractional derivative modelBingham modelPower−law model