Theory and simulation of micropolar fluid dynamics

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1 31 Theoy and simulation of micopola fluid dynamics J Chen*, C Liang, and J D Lee Depatment of Mechanical and Aeospace Engineeing, The Geoge Washington Univesity, Washington, DC, USA The manuscipt was eceived on 9 Octobe 010 and was accepted afte evision fo publication on 1 Januay 011. DOI: / Abstact: This pape eviews the fundamentals of micopola fluid dynamics (MFD), and poposes a numeical scheme integating Choin s pojection method and time-cented split method (TCSM) fo solving unsteady foms of MFD equations. It has been known that Navie Stokes equations ae incapable of eplaining the phenomena at mico and nano scales. On the contay, MFD can natually pick up the physical phenomena at mico and nano scales owingto its additional degees of feedom fo gyation. In this study, the analytical and eact solutions of Couette and Hagen Poiseuille flow ae povided. Though this study is limited to the steady flow cases, the unsteady tem in the MFD has been taken into account. This pesent wok initiates the development of a geneal-pupose code of computational micopola fluid dynamics (CMFD). The discetization scheme in space is demonstated with nealy second-ode accuacy on multiple meshes. Keywods: micopola fluid dynamics (MFD), micofluidics, computational micopola fluid dynamics (CMFD), finite diffeence method, pojection method, time-cente split method (TCSM) 1 INTRODUCTION Reseach activities aiming to eploe fluid physics at nano and mico scales have been inceasing ove the past 0 yeas. Thee ae eisting liteatues that have analysed fluid mechanics in micochannels and micomachined fluid systems (e.g. pumps and valves) using Navie Stokes equations [1]. Fluid flow moves diffeently in the mico scale than that in the maco scale. Thee ae situations in which the Navie Stokes equations, deived fom classical continuum, become incapable of eplaining the mico scale fluid tanspot phenomena []. The eason is that when the channel size is compaable to the molecula size, the spinning of molecules, which have been obseved in molecula dynamics (MD) simulations [3, 4], affects significantly the flow field. This effect of molecula spin is not taken into account in the Navie Stokes equations. A *Coesponding autho: Depatment of Mechanical and Aeospace Engineeing, The Geoge Washington Univesity, Washington, DC, 005, USA jikembo@gwmail.gwu.edu novel appoach, micocontinuum theoy, consisting of micopola, micostetch, and micomophic (3M) theoies, developed by Eingen [5 8] and Lee et al. [9], offes a mathematical foundation to captue such motions. In 3M theoies, each paticle has a finite size and contains a micostuctue that can otate and defom independently, egadless of the motion of the centoid of the paticle. The fomulation of the micopola theoy has additional degees of feedom gyation to detemine the otation of the micostuctue. Hence, the balance law of angula momentum ae given fo solving gyation. This equation intoduces a mechanism to take into account the effect of molecula spin. The micopola theoy thus epesents a pomising altenative appoach to numeically solving mico scale fluid dynamics that can be much moe computationally efficient than the MD simulations. Papautsky [10] was the fist one to adopt the micopola fluid model to eplain the epeimental obsevation of volume flow ate eduction fo the flow in a ectangula micochannel. In addition, Gad-El-Hak [11] eplicitly states that micoscale flows ae essentially diffeent fom flows in the

2 3 J Chen, C Liang, and J D Lee macoscale. The Navie Stokes desciption is incapable of eplaining the obseved effects. The calculated hydodynamic quantities fo a fluid as a classical continuous medium (fom Navie Stokes equations) diffe significantly fom those obtained epeimentally, and the diffeence inceases with the decease of the channel diamete in the flow though naow channels. Thee ae many ecent developments of micopola theoy that have focused on numeical analysis of Hagen Poiseuille flow and its applications on nano- and micofluidics, including Papautsky [10], Ye [1], and Hansen [13]. Howeve, all of the studies consideed only a steady state solution and did not solve fo pessue. Thei methods ae theefoe unable to solve unsteady flow poblems. In this wok, a numeical scheme fo solving the unsteady fom of micopola fluid dynamics (MFD) is developed. A detailed eplanation fo the physical meaning of all coefficients is povided. Analytical and eact solutions fo flat-plate Hagen Poiseuille flow and flat-plate Couette flow ae discussed against numeical solutions. As a numeical eample, lid-diven cavity flow is simulated by solving the micopola equations. Nomenclatue can be found in the Appendi section. MICROPOLAR FLUID THEORY In micocontinuum field theoies, the mateial points of the fluid ae consideed to be small defomable paticles. The macomotion and micomotion of the mateial paticles ae epessed by [5 9] k 5 k ðx; tþ; k51;;3 (1) j k 5 kk ðx; tþ K ; K 51;;3 () Since the mateial paticles ae consideed to be geometical points with mass and inetia, kk ðx; tþ hee epesents the thee defomable diectos attached to the mateial paticles. Fo a mateial body called a micopola continuum, the micomotion is futhe educed to a otation. In othe wods, its diectos ae othonomal and igid, that is kk lk ¼ d kl ; kk kl ¼ d KL (3) Fo fluid flow, defomation-ate tensos ae cucial to the chaacteization of the viscous esistance. Defomation-ate tensos may be deduced by simply calculating the mateial time-ates of the spatial defomation tensos. Fo micopola fluid, two objective defomation-ate tensos ae [5 8] a kl ¼ v l;k 1e lkm v m ; b kl ¼ v k;l (4) v m is the gyation vecto, which is the additional otating degee of feedom fo a paticle. Because the mean fee path of fluid is lage than solid, each fluid molecule has moe space to move aound. When a goup of fluid molecules o a single fluid molecule spins, the effect of the gyation vecto appeas and cannot be obseved in classical continuum theoy. Theefoe, the gy ation vecto is a good candidate fo detemining the physics at the mico scale while adopting the continuum assumption. The balance laws of the micopola continuum can be epessed as [5, 6]: Consevation of mass _1v l;l ¼ 0 (5) Balance of momentum t kl;k 1ðf l _v l Þ¼0 (6) Balance of angula momentum m kl;k 1e lmn t mn 1ðl l i _v l Þ¼0 (7) Consevation of enegy _e t kl ðv l;k 1e lk v Þ m kl v l;k 1 q k;k h ¼ 0 (8) Clausius Duhem inequality ð _c1h_uþ1t kl a kl 1m bl b lk q k u u ;k 0 (9) The linea constitutive equations fo Cauchy stess, moment stess, and heat flu ae deived to be [5, 6] t kl 5 pd kl 1l tða mn [ pd kl 1 D t kl Þd kl 1ðm1kÞa kl 1ma lk m kl 5 a u e klmu ;m 1a tðb mn Þd kl 1bb kl 1gb lk q k 5 K u u ;k1ae klm v m;l (10) Substitute the constitutive equations into all the balance laws, and the govening field equations of MFD can be ewitten as [5 8]: Consevation of mass _1v l;l ¼ 0 (11)

3 Theoy and simulation of micopola fluid dynamics 33 Balance of momentum p1ðl1mþ v1ðm1kþ v 1k3v1f ¼ _v (1) D Dt ð k;k Þ¼v k;l l;k D Dt ð kk Þ¼v kl lk (16) Balance of angula momentum ða1bþ v1g v1kð3v vþ1l ¼ i _v (13) Consevation of enegy _e D t : a T m : b1q h ¼ 0 (14) The micoinetia is defined as i [ hj k j k i R 0 j 5 k j k dv R 0 0 dv 0 [ l (15) and l epesents a hidden length scale, which can be at the level of molecula scale, Kolmogoov mico scale, o Taylo mico scale. These small-scale activities can possibly be measued epeimentally using Lagagian velocities of tace paticles [14,15]. 3 CONNECTION WITH NAVIER STOKES EQUATIONS Voticity is consideed as the ciculation pe unit aea at a point in a fluid flow field. It is a common pactice in geneal vecto analysis to descibe a vecto function of a position having zeo cul as iotational in view of the connection between 3 v and the local otation of the fluid [16]. It has anothe physical intepetation: voticity measues the solid-body-like otation of a mateial point P adjacent to the pimay mateial point P [17]. In micopola fluid dynamics, gyation has a simila concept. One can intepet the motion in MFD using the eath motion as an eample. In the motion of the eath, it not only evolves aound the sun, which esults in seasons, but also spins on its own ais, which makes days. A micopola continuum is consideed as a continuous collection of finite-size paticles. The tanslation of finite-size fluid paticles can be imagined as the eath evolution with, the gyation being simila to the spin of the eath. The mateial time ates of spatial defomation tensos can be obtained as If the micomotion equals the macomotion, that is, l,k = lk, this leads to v k;l ¼ v kl (17) In micopola theoy, the gyation tenso is antisymmetic, that is v kl ¼ e klm v m (18) This leads to v m ¼ 1 e lkmv k;l (19) The physical pictue of equation (19) is simila to the motion of the moon; it always faces the Eath with the same side while evolving aound the Eath. Substitute equation (19) into equation (1) and one can obtain p1ðl1m Þ v1m v1f ¼ _v (0) whee m ¼ m11=k. It is identical to Navie Stokes equations deived fom Newtonian fluid. At this point, the MFD fomulation has been clealy shown as moe geneal than Navie Stokes equations. 4 NUMERICAL SCHEME The time-cente split method (TCSM) was fist developed by Fu and Hodges in 009 fo unsteady advection poblems [18]. Hee the TCSM is futhe etended fo incompessible MFD. The incompessible fluid implies v ¼ 0 and hence the pessue p becomes the Lagange multiplie. The condition v ¼ 0 must be enfoced and indeed it is used to calculate the Lagange multiplie. The Choin s pojection method is incopoated with TCSM to update the pessue gadient tem fo solving the Poisson equation. Also, it is noted that the effect of themomechanical coupling is not consideed. The pojection method was oiginally intoduced to solve time-dependent incompessible Navie Stokes fluid-flow poblems by Choin [19]. In Choin s oiginal vesion of the pojection method, the intemediate velocity v * is eplicitly computed

4 34 J Chen, C Liang, and J D Lee using the momentum equations, ignoing the pessue gadient tem v v n ðt t n Þ ¼ vn v n 1m v n (1) whee v n is the velocity at the nth time step. In the net step, the velocity is updated with v n11 v ðt n11 t Þ ¼ 1 pn11 () In ode to guaantee that v n11 satisfies the continuity equation, taking divegence on both sides of equation () leads to v n11 v ¼ ðtn11 t Þ p n11 (3) Thus, a Poisson equation fo p n11 is obtained as p n11 ¼ ðt n11 t Þ v (4) A distinguished featue of Choin s pojection method is that the velocity field is foced to satisfy the continuity equation at the end of each time step. In this pape, a new method is poposed. It incopoates TCSM into Choin s pojection method fo MFD equations and enfoces the continuity equation to be satisfied in the middle and at the end of each time step. The pocedues of this method ae listed as follows. 1. Neglect the pessue effect and update the velocity fom t n to t * while dealing with the convective tem as v v = v n v * v v n ðt t n Þ 1vn v. Solve p ¼ ðt t Þ v ¼ ðm1kþ v 1k3v n (5) 3. Mach time fom t * to t ** and update velocity as v ¼ v ðt t Þ p which guaantees velocity divegence fee at t **. 4. Solve gyation v ** at t ** using velocity field v ** v v n i ðt t n Þ 1v v ¼ ða1bþ v 1g v 1kð3v v Þ1l (6) 5. Neglect the pessue effect and update velocity fom t ** to t *** while dealing with the convective tem as v v = v ** v ** v v ðt t Þ 1v v ¼ (7) ðm1kþ v 1k3v 6. Solve p n11 ¼ ðt n11 t Þ v 7. Mach time fom t *** to t n11 and update velocity using v n11 ¼ v tn11 t p n11 :v n11 to satisfy the continuity equation at t n Solve gyation v n11 at t n11 using the velocity field n n11 v n11 v i ðt n11 t Þ 1vn11 v n11 ¼ ða1bþ v n11 1g v n11 1 kð3v n11 v n11 Þ1l (8) The pocedues fom step 1 to step 8 complete a physical step of time maching. The advantage of this algoithm is to avoid the non-linea tems in the equations and to povide a set of linea equations with a second-ode accuacy in time evolving [18]. Fo the viscous tems of velocity and gyation, they ae discetized using the cental diffeence method, fo eample v ¼ v ð11; y; zþ v ð; y; zþ1v ð 1; y; zþ 1 v ð; y11; zþ v ð; y; zþ1v ð; y 1; zþ y 1 v ð; y; z11þ v ð; y; zþ1v ð; y; z 1Þ z (9)

5 Theoy and simulation of micopola fluid dynamics 35 v ¼ v ð11; y; zþ v ð; y; zþ1v ð 1; y; zþ 1 v ð; y11; zþ v ð; y; zþ1v ð; y 1; zþ y 1 v ð; y; z11þ v ð; y; zþ1v ð; y; z 1Þ z (30) Fo the convective tems, v v and v v, an upwind scheme is adopted due to the stability issue. Fo eample, at step 1 in the time maching algoithm, the convective tems in momentum equations can be discetized as 8 v n ð;y;zþv j ð;y;zþ vj ð 1;y;zÞ >< if v n v v j; ) ð;y;zþ. 0 v n ð;y;zþv j ð11;y;zþ vj ð;y;zþ >: if v n ð;y;zþ\0 (31) whee j can be, y, o z. Howeve at step 5, v *** is unknown, so the upwind scheme is chosen based on v ** 8 >< v v j; ) >: v v ð; y; zþ v ð; y; zþ v ð; y; zþ v ð 1; y; zþ if v ð; y; zþ. 0 ð11; y; zþ v ð; y; zþ if v ð; y; zþ\0 (3) whee j can be o y o z. The cental diffeence method is also employed to discetize the cul of velocity and gyation. v y v y 5 v ð;y11;zþ v ð;y 1;zÞ y v yð11;y;zþ v y ð 1;y;zÞ ð Þ v ð;y 1;zÞ y v yð11;y;zþ v y ð 1;y;zÞ (35) v y v y 5 v ;y11;z 5 ANALYTICAL AND EXACT SOLUTIONS OF COUETTE FLOW Conside an incompessible fluid with a top plate in height h moving with a velocity U 0, a bottom plate fied, and the following assumptions: (a) no velocity in the y- and z-diections, (b) no gyation in the - and y-diections, (c) fully developed flow (i.e. both -diection velocity and z-diection gyation ae functions of y only), and (d) no body foce. The steady solution fo micopola fluid is k v ¼ Mðm1kÞ C e My C 3 e My 1ðC_1C_3Þy1C4 v z ¼ C e My 1C 3 e My m1k m1k C 1 whee (36) whee j can be, y, oz. At steps 4 and 8, the convective tems in the angula momentum equations ae discetized as 8 >< v v j; ) >: v ð; y; zþ v j ð; y; zþ v j ð 1; y; zþ if v ð; y; zþ. 0 v ð; y; zþ v j ð11; y; zþ v j ð; y; zþ if v ð; y; zþ\0 (33) M 5 kðm1kþ gm1k ð Þ C 1 5 m1k ð m1k C 1C 3 Þ C 5 U 0 k ð 1 emh Þ Mðm1kÞ 1 e Mh e Mh e Mh 1 1 emh C 3 5 e Mh 1 C k C 4 5 ð Mðm1kÞ C C 3 Þ h (37) 8 >< v v j; ) >: v n11 v n11 ð; y; zþ vn11 j ð; y; zþ v n11 j ð 1; y; zþ if v n11 ð; y; zþ. 0 ð; y; zþ vn11 j ð11; y; zþ v n11 j ð; y; zþ if v n11 ð; y; zþ\0 (34) The steady solution of Newtonian fluid is v ¼ y h U 0 v z ¼ 1 v y ¼ U 0 h (38) Taking into account that g is the viscosity coefficient, which tends to stop the otation of the finite

6 36 J Chen, C Liang, and J D Lee Fig. 1 The compaison of angula velocity in Navie Stokes equations and micogyation in MFD (Couette flow) Fig. 3 The compaison of velocity pofiles in Navie Stokes equations and MFD (Poiseuille flow) Figue shows the time evolution of the velocity pofile in Couette flow 6 ANALYTICAL AND EXACT SOLUTIONS OF POISEUILLE FLOW Conside an incompessible fluid in a channel with a unifom pessue gadient G and half channel height h. The steady state solution fo micopola fluid is Fig. Time evolution of velocity in Couette flow. The aow indicates the tansient pocess as time maches on size paticles, one can also define the intenal chaacteistic length l v 5 G ðm1kþ h y 1 kc e Mh 1e Mh e My 1e My Mðm1kÞ v z 5 Gy ðm1kþ 1C e My e My whee M ¼ kðm1kþ gðm1kþ ; C ¼ Gh ðe Mh e Mh Þðm1kÞ The steady solution of Newtonian fluid is (40) (41) l ¼ g k m1k 1 (39) m1k Note that g = 0 leads to l = 0. Figue 1 shows the gyation plot with the change of l. It can be obseved that as g deceases, the gyation effect intensifies. It should be mentioned that the gyation is nomalized by the angula velocity solved fom Navie Stokes equations. Utilizing the poposed numeical method, the tansient pocess of Couette flow can now be tackled. The fluid is initially at est. The time step is set as and the esult is output evey 100 steps. v ¼ G m ðh y Þ v z ¼ 1 v y ¼ Gy (4) m It can be obseved that k is the connection between velocity and gyation, which indicates the stength of the coupling effect. In Figs 3 and 4, m1k keeps constant while k is changing. It is obvious that when k is dominating in m1k, the coupling effect is so stong that the cente velocity of MFD is quite diffeent fom the velocity obtained fom the Navie Stokes equations. The plot of gyation and cente

7 Theoy and simulation of micopola fluid dynamics 37 compaed. Diffeent numbes of gid point (including 6 3 6, , and 1 3 1) ae tested. Figue 5 plots the L 1 and L eo fo the cente velocity and gyation. The eos all decay as gid points incease. The compaison is also listed in Table 1. The aveage odes of L 1 eo of velocity ae 1.94 and 1.55 fo velocity and gyation, espectively. The aveage ode of L eo of velocity is 1.55, and gyation is The accuacies of both velocity and gyation ae nealy second-ode in space. Fig. 4 Fig. 5 The compaison of angula velocity in Navie Stokes equations and micogyation in MFD (Poiseuille flow) Table 1 Eo analysis of the numeical scheme Eo analysis of Poiseuille flow velocity ae in Figs 3 and 4, espectively, while the cente velocity and the gyation ae nomalized by the velocity and angula velocity, espectively, fom the Navie Stokes equations. 7 NUMERICAL ACCURACY STUDY Velocity L 1 eo Ode L eo Ode Micogyation L 1 eo Ode L eo Ode Unifom mesh is utilized to analyse numeical accuacy, while the numeical and analytical solutions of the flat-plate Hagen Poiseuille flow ae 8 LAMINAR LID-DRIVEN CAVITY FLOW The cavity is a squae bo with side length d = 0.1. The velocity on the top of the bo, u N, is 1. The mateial constants ae set as follows: m ¼ 10 4 ; k ¼ ; g ¼ Consequently, the Reynolds numbe, Re =(u N d/( m 1 k), is 10. The mesh numbe is The time step is set as 0.005, while the Couant condition equies that t t ma = /u N = Note that this poposed numeical scheme is semi-implicit so that the Couant condition does not have a significant influence on the numeical method. Figue 6 plots the cente velocity vecto. A big eciculation egion can be clealy seen. Figue 7 shows the pessue distibution. The bounday condition of pessue is set unde Neumann bounday conditions, eactly on the wall, while the efeence point is set in the cente of the bo. In addition, the nomalized pessue u is 0.1. Figue 8 shows the gyation in this cavity. It is appaently seen that the fluid paticles spin clockwise below the top of the bo and they spin counteclockwise at both sides of the bo. Howeve, the maimum of both spinning diections (clockwise and counteclockwise) does not occu on the sides because gyation is set to zeo unde bounday conditions. Based on the cente velocity, it is staightfowad to calculate the voticity as shown in Fig. 9. It should be emphasized that the voticity is the otation of the fluid molecule elative to its neighbouing fluid molecules but the gyation is the selfspinning of the fluid molecule. The total velocity in the micopola theoy is defined as v k ð; z; tþ ¼v k ð; tþ1e kml v m z l (43) Figue 10 shows the total velocity. It is plotted on a fine mesh and calculated based on equation (43). The maimum kkof j the finite-size paticle is set to be the half diagonal of the element. Theefoe, the size of the element is as lage as the fluid paticle. Hence the size of simulated cavity is as lage as that

8 38 J Chen, C Liang, and J D Lee Fig. 9 Voticity in cavity flow Fig. 6 Cente velocity in cavity flow Fig. 7 Pessue distibution in cavity flow Fig. 10 Total velocity in cavity flow of 0 fluid paticles, 033 makjk, while a fluid paticle efes to eithe a fluid molecule o a goup of fluid molecules. Fo the ovelapping egions, the value of total velocity is aveaged. Using the concept of total velocity, it enables obsevation of the gyation effect. In this eample, the gyation tends to induce the fomation of votices in the bottom cones (see Figue 10). 9 Fig. 8 Gyation in cavity flow CONCLUDING REMARKS Micocontinuum field theoies povide additional degees of feedom to incopoate the micostuctue of the continuous medium. In this pape, the micopola theoy is biefly intoduced. Eta

9 Theoy and simulation of micopola fluid dynamics 39 otating degees of feedom not only widen the physical backgound of micofluidics andthe fluid mechanics at mico- and nanoscales, but also enlage the capacity to addess vaious featues missing fom the Navie Stokes equations. The second-ode accuate TCSM successfully incopoated with Choin s pojection method to solve the MFD. This wok discusses only the steady flow cases. Nevetheless, the unsteady tems in the MFD ae taken into account igoously and completely in the poposed numeical scheme. The developed discetization schemes in space ae demonstated with nealy second-ode accuacy on multiple meshes. This study initiates the development of a geneal pupose numeical solve fo computational MFD. Inteested eades may adopt the numeical methods developed in this pape to eploe the feasibility of micopola fluid dynamics on multiscale fluid mechanics poblems. Ó Authos Gad-El-Hak, M. The fluid mechanics of micodevices. J. Fluid Engng, 1999, 11, Ye, S., Zhu, K., and Wang, W. Lamina flow of micopola fluid in ectangula micochannels. Acta Mechanica Sinica, 006,, Hansen, J. S., Davis, P. J., and Todd, B. D. Molecula spin in nano-confined fluidic flows. Micofluidics and Nanofluidics, 009, 6, Heinloo, J. Fomulation of tubulence mechanics. Phys. Rev. E, 004, 69, Modant, N., Metz, P., Michel, O., and Pinton, J.- F. Measuement of Lagangian velocity in fully developed tubulence. Phys. Rev. Lett., 001, 87, Batchelo, G. K. An intoduction to fluid dynamics, 1967 (Cambidge Univesity Pess, Cambidge) pp Panton, R. L. Incompessible flow, 005 (John Wiley & Sons, New Jesey) pp Fu, S. and Hodges, B. R. Time-cente split method fo implicit discetization of unsteady advection poblems. J. Engng Mech, 009, 135, Choin, A. J. Numeical solution of the Navie Stokes equations. Math. Comput., 1968,, REFERENCES 1 Body, J. P. and Yage, P. Low Reynolds numbe mico-fluidic devices. In the Poceedings of Solid- State Senso and Actuato Wokshop, South Caolina, USA, June 1999, pp Holmes, D. B. and Vemeulen, J. R. Velocity pofiles in ducts with ectangula coss sections. Chem. Engng Sci., 1968, 3, Kucaba-Pietal, A., Walenta, Z., and Peadzynski, Z. Molecula dynamics compute simulation of wate flows in nanochannels. Bulltn Polish Acad. Sci.: Tech. Sci., 009, 57, Delhommelle, J. and Evans, J. D. Poiseuille flow of a micopola fluid. Molec. Phys., 00, 100, Eingen, A. C. Simple micofluids. Int. J. Engng Sci., 1964,, Eingen, A. C. Theoy of micopola fluids. J. Appl. Math. Mech., 1966, 16, Eingen, A. C. Micocontinuum field theoies I: foundations and solid, 1999 (Spinge, New Yok) pp Eingen, A. C. Micocontinuum field theoies II: fluent media, 001 (Spinge, New Yok) pp Lee, J. D., Wang, X., and Chen, J. An oveview of micomophic theoy in multiscaling of synthetic and natual systems with self-adaptive capability, 010, pp (National Taiwan Univesity of Science and Technology Pess). 10 Papautsky, I., Bazzle, J., Ameel, T., and Fazie, A. B. Lamina fluid behavio in micochannel using micopola fluid theoy. Sensos and Actuatos, 1999, 73, APPENDIX Notation e intenal enegy density f k body foce tenso h enegy souce density i micoinetia K/u Fouie heat-conduction coefficient l l body moment density m kl coupled stess tenso p pessue q k heat vecto t kl Cauchy stess tenso defomation gadient k,k a, b, g total velocity d kl, d KL Konecke delta e klm pemutation symbols h entopy density u absolute tempeatue l, m, k viscosity coefficients fo stess n k cente velocity n k,l velocity gadient v k total velocity mass density kk micomotion c = e hu Helmholtz fee enegy v k gyation vecto gyation tenso v kl