˘ ¸ II ANALELE UNIVERSITAT BUCURES ¸ TI ˘ ANO L, 2001 MATEMATICA, Annual National Conference ”Caius Iacob” of Fluid Mechanics and Technical Applications Dumitru-Ion ARSENIE and Ichinur OMER, Determination of the pressure in a pipeline with uniformly distribution of the discharge ˘ ˘ Florin BALT ARET ¸ U, Numerical prediction of air flow pattern in a ventilated room 11 ˘ ˘ ˘ A, ˘ Numerical simulation of a horizontal Florin BALT ARET ¸ U, Cornel MIHAIL buoyant jet deflected by the Coand˘ a effect 17 Galina CAMENSCHI, The effects of the temperature - dependent viscousity on flow in cooled channel 23 Claude CARASSO and Ruxandra STAVRE, Numerical simulation of a jet of ink 31 Adrian CARABINEANU, Numerical and qualitative study of the problem of incompressible jets with curvilinear walls 37 Mircea Dimitrie CAZACU, On partial differential equations of the viscous liquid relative flow through the turbomachine blade channel 45 Mircea Dimitrie CAZACU and Loredana NISTOR , Numerical solving of the bidimensional unsteady flow of a viscous liquid, generated by displacement of a flat plate 53 − ˘ Eduard - Marius CRACIUN, Behaviour of the 2m piezoelectric crystal containing a crack in antiplane state 61 Liviu Florin DINU , An example of interaction between two gasdynamic objects: a piecewise constant solution and a model of turbulence 67 Alexandru DUMITRACHE , An Analytical Model of the Glass Flow during the Pressing Process 79 C FALUP-Precurariu, D MINEA, Oana FALUP-PRECURARIU, Laura DRACEA, The dynamic of mechanical forces on lung properties 87 Constantin FETECAU , Nonsteady shearing flow of a fluid of Maxwellian type 93 Sumar Sommaire Contents ˘ and Lucian IORGA, An adaptive method for structured Florin FRUNZULICA meshes in CFD problems 99 Stelian ION, A Numerical Method for Richards’ Equation 109 Mirela KOHR, Boundary Integral Method for an Oscillatory Stokes Flow Problem 115 Mircea LUPU, Adrian POSTELNICU, Ernest SCHEIBER, Analytical Method for maximal drag airfoils optimization in cavity flows 123 Dorin MARINESCU, On a Boltzmann Model of Fermions 141 Anca Marina MARINOV, A Two Dimensional Mathematical Model for Simulating Water and Chemical Transport in an Unsaturated Soil 149 Alexandru M MOREGA, A Numerical Analisys of Laminar Transport Processes in Ducts with Cross-Sectional Periodicity 159 Sebastian MUNTEAN, Romeo F SUSAN-RESIGA, Ioan ANTON and Victor ANCUS ¸ A, Domain Decomposition Approach for 3D Flow Computation in Hydraulic Francis Turbine 169 Elena PELICAN, Constantin POPA, Approximate orthogonalization of linearly independent functions with applications to Galerkin-like discretization techniques 179 Dumitru POPESCU, Stelian ION and Maria Luiza FLONTA, Appearance of pores through black lipid membranes due to collective thermic movement of lipid molecules 185 Mihai POPESCU, Optimality and Non-Optimality Criteria for Singular Control193 Lucica ROSU, Liliana SERBAN, Dan PASCALE, Cornel CIUREA and Carmen MAFTEI, The study of the water stability in canals with rectangular cross section by means of frequency analysis 199 Valeriu Al SAVA, Translation flows of non-local memory-dependent micropolar fluids 205 Romeo F SUSAN-RESIGA and Hafiz M ATASSI, Nonreflecting Far-Field Conditions for Unsteady Aerodynamics and Aeroacoustics 211 Victor TIGOIU, On the Uniqueness of the Solution of the Initial and Boundary Value Problem for Third Grade Fluids 223 Annual National Conference ”Caius Iacob” of Fluid Mechanics and Technical Applications anniversary session Institute of Applied Mathematics ”Caius Iacob” –ten years In the Romanian scientific life it is acknowledged as a tradition the fact that the researchers in the Fluid Mechanics field participate at a scientific meeting every year in October This scientific event, also known as National Colloquium on Fluid Mechanics and its Technical Applications for a long time, initiated since 1959 by the Society of Mathematical Science has become, thanks to all fluid mechanics researchers’ effort, an annual meeting It take place in various economic or universitary centers all over the country In addition to the usual significance of this event, the meeting that will ghater us this year has a special meaning: it celebrate ten years since the Institute of Applied Mathematics ”Caius Iacob” of Romanian Academy was founded The process of setting up such an institute has begun after the breaking up of the scientific institutes of Romanian Academy and has been carried on by the Colloquies As for this year’s Conference, we consider it necessary that this tradition should be preserved and that an old goal should be achieved: ”that a large amount of productive units, as well as the youth in Universities should be involved in such scientific events: [ ] that the Fluid Mechanics course should be introduced in every University” (Caius Iacob and C Ciobanu, preface of the Proceedings of National Colloquium on Fluid Mechanics, Iasi, 1978) Analele Universitˇa¸tii Bucure¸sti, Matematicˇ a Anul L(2001), pp 5–10 Determination of the pressure in a pipeline with uniformly distribution of the discharge Dumitru Ion ARSENIE and Ichinur OMER November 2, 2001 Abstract - In the literature, the most studied is the samples pipelines in the steady and unsteady conditions We propose to study the waterhammer phenomenon in the pipeline with uniformly distribution of the discharge By using the impulse variation theorem, we obtain the variation of the medium velocity, the variation of the velocity of elastic wave propagation and the variation of the pressure along the pipeline We have determined these when in the final section occurs a quick totally close manoevre of the valve We applie the results into a numerical example, using an original program Key words and phrases : pipeline with uniformly distribution of the discharge, pressure, discharge, velocity of the elastic wave propagation, waterhammer Mathematics Subject Classification (2000) : 76A05 Introduction In the literature the most studied pipeline is the simple pipeline, in the steady and unsteady flow, less treated being specials pipelines In this paper we propose to study the waterhammer in the pipeline with uniformly distribution of the discharge in the case of the quick totally close manoeuvre of the valve which is in the final section of the pipeline This pipe represent one of the simplified models of water supply pipes, irrigations etc We shall obtain the variation of the medium velocity, the variation of the velocity of elastic wave propagation and the variation of the pressure along the pipeline In the end of the paper we present a numerical example Useful notations : p - the pressure, β - ununiform coefficient of the rate (Boussinesq), ρ - water density, v - flow velocity, A - area of the transversal section, Q - discharge, t - the time, V - the volume, ∆p-the medium pressure variation on the segment, vx,t - the velocity in the x section and at t time, ∆x cj = - the medium velocity of elastic wave propagation on the segment ∆t D.-I Arsenie , I Omer j, j = 1, n − 1,the segment j being delimited by the sections i and i+1, vi+1,i q0 ∆x q0 B = , νi+1 = vx,0 = v1,0 + (L − x) - the velocity in the x 2A B A section at initial time before the stop manoeuvre of the valve For the section x of the segment j and after the moment when the waterhammer wave has arrived in this section, the velocity is: vx,t = vi,t + q0 A 1+ ∆pj [L − (i − 1) ∆x − x] p0 In the numerical example, the pipeline with the uniformly distribution of the discharge was divided into equals lengths segments ∆x In the x - coordinate sections which delimit these segments, we applied the precedents formulas, considering: x = i∆x; i = 1, n The first section is the final section (going out of the pipeline with the uniformly distribution of the discharge) and the last section is the initial section (entrance of the water in the pipeline with the uniformly distribution of the discharge), the sections order being in the same direction like the waterhammer propagation direction One of the used hypothesis refers to the way of the discharge variation which leaves the pipe and which has an uniformly distribution in the initial steady flow We considered that the specifically discharge q0 , corresponding to the pressure p0 , is changed when the steady condition, respective when the pressure in the pipeline become p0 + ∆p, by the orifice-nozzle type relation: q0 = √ k p0 , qx = k =⇒ qx = q0 1+ p0 + ∆px ∆px p0 (1) Mathematical model To determine the proposed relation, we applie the variation of impulse theorem on the control volume which delimit the liquid comprised into the sections i and i+1 (the segment j) In the projection on the water flow direction (the horizontally axis), the scalar relation is −∆pi A = βρvi Qi − βρvi+1 Qi+1 + ρ V ∂vx dV ∂t (2) Determination of the preassure Figure 1: The calculus scheme − → Because the pipeline’s axis is horizontal, the weight of the liquid ( G ) − → and the reaction of the pipeline’s wall ( R ) don’t appear in this equation Replacing the expression of the velocity’s derivative by a finit difference: ∂vx ∼ vx,t − vx,0 = ∂t ∆t (3) and solving the integral, the relation (2)becomes: ∆pi+1 = ρcj v1,0 + −ρcj q0∆ x 2A q0 A 1+ Making the follow notation yi+1 = thus: i− ∆x − vi,i−1 (4) ∆pj p0 ci+1 , the relation (4) can be write cj D.-I Arsenie , I Omer 1+ ∆pj v1,0 vi+1,0 yi+1 ∆vi+1 = + 2i − − + p0 B B B (5) Rasing to square the relation (5) and solving the equation in yi+1 , we obtain the calculus relation for this parameter:   ∆p vi+1,0  v1,0 j yi+1 = (6) + 2i − − + 1+ νi+1 B B p0 Because the segments have small length, we admitted that between the ∆p medium value + p0j and the values of the segments extremities is one relation corresponding to the linear variation: 1+ ∆pj = p0 1+ ∆pi + p0 1+ ∆pi+1 p0 (7) In the relation (6) result that have appeared two unknown variables yi+1 and ∆pi+1 Between these two variables exists the Jukovski relation: ∆pi+1 = −ρci+1 ∆vi+1 = −ρcj yi+1 ∆vi+1 (8) For solve this problem we used an iterative procedure, considering the initial approach ∆pj ∼ = ∆pi , which permits to calculate yi+1 , respective ci+1 Then with the relation (8) we determine ∆pi+1 and thus we may calculate again ∆pj in the next approach In the numerical example treated, the iterative process stopped when the relative error of the value ∆pj lowers under 1% The velocity vi+1 is: vi+1,i = B( + +2 + ∆p1 ∆p2 +2 1+ + p0 p0 ∆pi + p0 1+ ∆pi+1 ) p0 (9) and the variation of velocity: ∆vi+1 = vi+1,i − vi,0 For the final segment 1, the previous relations become (10) Determination of the preassure y1 = + ∆v2 = B 1+ 1 + ν2 ν2 ∆p1 + p0 1+ 1+ ∆p1 p0 ∆p2 −2 p0 (11) − v1,0 (12) Figure 2: The distribution of velocity’s variation, pressure variation and velocity of elastic wave propagation along the pipeline Numerical example We consider a pipeline with the uniformly distribution of the discharge for which we know the follows characteristically elements: d = 250 mm, l = 200m, Qi = 80 l/s, q0 = 0,0002m3 /ms, c1 = 1000m/s, p1 =4 bari (400000 N/m2 ).We divided the pipeline into equals lengths segments ∆x = 10m and 10 D.-I Arsenie , I Omer we determined the variation of the medium velocity, the variation of the velocity of elastic wave propagation and the variation of the pressure along the pipeline (for four segments) Conclusions We observe in the Figure (2) that the medium velocity, the velocity of elastic wave propagation and the pressure diminish along the pipeline, strating from the final section of the pipeline where is the valve It is possible that in some situations the pressure’s variations becomes so small, that they may be neglected The quality aspect (the attenuation of the waterhammer along the pipeline) doesn’t meet in the simple pipeline References [1] Trofin, P., Water Supply E.D.P., Bucure¸sti, 1983 [2] Jeager, Ch., Fluid Transients in Hydro-Electric Engineering Practice Blackie, London, 1977 [3] Cioc, D Hydraulic E.D.P., Bucure¸sti, 1975 Dumitru Ion Arsenie Ovidius University Constantza, B-dul Mamaia nr.124, 8700-Constanta, ROMANIA E-mail: icky@univ-ovidius.ro Ichinur Omer Ovidius University Constantza, B-dul Mamaia nr.124, 8700-Constanta, ROMANIA E-mail: icky@univ-ovidius.ro 220 R F Susan-Resiga and H M Atassi method, due to the accurate non-reflecting condition employed, and its careful numerical implementation Conclusions The paper is focused on formulating and implementing exact boundary nonreflecting conditions for exterior aeroacoustic problems Although the boundary value problem is formulated for the acoustic velocity potential, the radiation conditions are imposed on the unsteady pressure On the other hand, the exact nonreflecting condition employed here is written as a nonlocal Dirichlet-to-Neumann map for a circle, thus coupling all the presure values on the outer boundary The main features of our FEM implementation of the nonreflecting conditions can be summarized as follows: • When computing the pressure normal derivative, the second order radial derivative of the velocity potential is replaced by a second order tangential derivative As a result, only first order radial (normal) derivative of the velocity potential need to be computed allowing the use of standard Finite Element techniques • Instead of the radial derivative of the velocity potential we are using the local normal flux, which is the natural quantity computed on the boundary in the FEM formulation The tangential derivatives for both potential and potential flux are easily computed on the circle using second order schemes The scaling factors introduced when numerically computing the unsteady pressure and its normal derivative keep the matrix entries of order one, thus enhancing the accuracy of the solution The above nonreflecting conditions can be applied for any loaded airfoil aeroacoustic problem, provided that the outer boundary is chosen to be a circle after the Prandtl-Glauert coordinate transformation is employed, and the circle radius is large enough to have a practically uniform mean flow outside the computational domain To assess the accuracy of the boundary conditions and their Finite Element implementation we solve the model problem of a thin unloaded airfoil in a transverse gust Our numerical solution is compared with a solution obtained by solving the Possio integral equation The agreement is excellent for both near-field and far-field unsteady pressure Nonreflecting Far-Field Conditions 221 References [1] H M Atassi, Unsteady Aerodynamics of Vortical flows: early and recent developments In Aerodynamics and Aeroacoustics, pages 121–172 ed Fung, K.-Y., World Scientific, 1994 [2] H M Atassi, M Dusey, and C M Davis, Acoustic radiation from a thin airfoil in nonuniform subsonic flow AIAA Journal, 31(1):12–19, 1993 [3] S Balay, W D Gropp, Lois C McInnes, and B F Smith PETSCc users manual Technical Report ANL-95/11 - Revision 2.1.0, Argonne National Laboratory, 2000 [4] O G Ernst, A finite-element capacitance matrix method for exterior helmholtz problems Numerische Mathematik, 75:175–204, 1996 [5] D Givoli, Numerical Methods for Problems in Infinite Domains Springer-Verlag, New York, 1998 [6] M.E Goldstein, Aeroacoustics McGraw-Hill International Book Company, New York, 1976 [7] F Ihlenburg, Finite Element Analysis of Acoustic Scattering SpringerVerlag, New York, 1998 [8] J B Keller and D Givoli, Exact non-reflecting boundary conditions Journal of Computational Physics, 82:172–192, 1989 [9] C Possio, L’azione aerodinamica sul profilo oscillante in un fluido compressible a velocit´a iposonora L’Aerotehnica, 18(4), 1938 [10] J R Scott and H M Atassi, A finite-difference frequency domain numerical scheme for the solution of the gust response problem Journal of Computational Physics, 119:75–93, 1995 [11] R F Susan-Resiga and H M Atassi, Domain-Decomposition Method for Time-Harmonic Aeroacoustic Problems AIAA Journal, 39(5):802– 809, 2001 Romeo F Susan-Resiga Universitatea “Politehnica” Timi¸soara, Bd Mihai Viteazu 1, ˆ 1900-Timi¸soara, ROMANIA E-mail: resiga@acad-tim.utt.ro 222 R F Susan-Resiga and H M Atassi Hafiz M Atassi University of Notre Dame, Notre Dame, IN 46556, U.S.A E-mail: atassi@nd.edu Analele Universitˇa¸tii Bucure¸sti, Matematicˇ a Anul L(2001), pp 223–232 On the Uniqueness of the Solution of the Initial and Boundary Value Problem for Third Grade Fluids Victor TIGOIU November 2, 2001 Abstract - This paper shows that the solution of the initial and boundary value problem for a third grade fluid (different from the model employed by Fosdick and Rajagopal, see Tigoiu [9]) is unique This result completes the oldest one from [9] and produce an a priori estimate which is usefulness in the proof of the asymptotic stability of the rest state also Key words and phrases : third grade fluids, uniqueness, asymptotic stability Mathematics Subject Classification (2000) : 76A05, 76A10 Introduction In the last years some new results concerning with polynomial fluids of second and third grade are made into evidence On a part the existence, uniqueness and dependance on the initial data are discussed in papers like Dunn and Fosdick [2], Cioranescu and Guirault [1], Galdi and Sequeira [4], Passerini and Patria [7], for second grade fluids and Fosdick and Rajagopal [3], Passerini and Patria [7], Tigoiu [9], [10] for third grade fluids On the other part the problem of the asymptotic stability of the rest state has been also discussed (especially in connection with the question of Joseph [5] on the ”nonexistence” of polynomial fluids of grade greater then ) in [3], Patria [8] and Tigoiu [10], [11] It was proved in [10], [11] that there is a strong connection (for instance) between some stability results and some a priori estimate obtained in the proof for uniqueness (for weakly perturbed flows) We shall develop in the first part of our paper this ideas In the second part we shall employ with the proof of uniqueness for the solution of the initial and boundary value problem for the general class of third grade fluids introduced in [9] in the nonlinear case 223 224 V Tigoiu Uniqueness for the solution of the initial and boundary value problem Linear case Let an incompressible third grade fluid given by (see [9]) T = −pI + µA1 + α1 (A2 − A21 ) + β1 A3 + β2 (A1 A2 + A2 A1 )+ (1) +β3 (trA21 )A1 , with β1 < 0, µ ≥ 0, β1 + 2(β2 + β3 ) ≥ In (1) A1 , A2 , A3 are the first three Rivlin - Ericksen tensor fields µ, α1 , β1 , β2 , β3 are constant constitutive modules which obey the above thermodynamic restrictions (see [9]) and p is the hydrostatic pressure field In this section we consider a weakly perturbed (from the rest state) flow Consequently, the response of the fluid (1) will be insensitive to the second order (nonlinear) terms The flow and continuity equations will be given then by ∂v ∂ ∂2 − µdivA1 − α1 divA1 − β1 divA1 + gradp = ρb, ∂t ∂t ∂t (2) divv = The attached initial and boundary value problem is v(0, x) = vo (x), ∂v (0, x) = ao (x), ∂t (3) v(t, x)|∂Ω = where Ω is supposed to be a rigid and fixed domain ( Ωt = Ω for any t ∈ [0, ∞), see also for such problems [3], [5]) Let be (vi , pi ), i = 1, two solutions of the problem (2) - (3) corresponding to the initial data (vio , aoi ), i = 1, 2, respectively Like is habitual in this kind of problems, we denote by v ≡ v1 − v2 ; vo ≡ v1o − v2o ; ao ≡ ao1 − ao2 ; p ≡ p1 − p2 ; b ≡ b1 − b2 (4) Due to the linearity of equations (2), the problem verified by (v, p) is similar but with null initial data We adapt, in a nontrivial manner, for our problem, a known technique (see Lions [6]) We suppose for this that v ∈ (Wo1,2 (Ω))3 ∂v ∂A1 with condition (2)2 and with ∈ (L2 (Ω))3 , ∈ (L2 (Ω))6 , ∂t ∂t Uniqueness 225 ∂ A1 ∈ (L2 (Ω))6 , p ∈ W 1,2 (Ω) and b ∈ (L2 (Ω))3 , a conservative field ∂t2 After some long but straightforward calculations (multiplying (2)1 with v ∂v and with and integrating over Ω) we arrive at ∂t ρd dt β1 d dt v ∂A1 ∂t + µ − β1 + µd dt α1 d dt + ∂A1 ∂t = 0, + A1 A1 A1 + β1 d2 dt2 A1 − (5) α1 + 2ρc2o ∂A1 ∂t ≥ 0, where we have employed Friedrichs and Korn’s inequalities and we denoted co the corresponding domain dependent constant (see also [10]) Relations (5) lead us to the a priori inequality d β1 d dt dt µ + A1 A1 2 + α1 + µ A1 α1 − β1 + 2ρc2o + 2 + ρ v ≥ 0, ∂A1 ∂t + β1 ∂A1 ∂t + (6) for all t ∈ (0, To ) We remark that the quantity in accolades is not, strictly speaking, an energy For this reason we shall call our method ”a quasienergetic method” So, we denote E(t) ≡ + β1 d dt β1 A1 ∂A1 ∂t 2 + α1 + µ ξβ1 − A1 A1 2 + ρ 2 v + (7) , ˙ for all ξ ∈ R and we evaluate the quantity E(t) + ξE(t) ˙ E(t) + ξE(t) ≥ + −ξ β1 + ξ(α1 + µ + 2ρc2o ) − 2µ A1 ξβ1 − 2(α1 − β1 + 2ρc2o ) ∂A1 ∂t + (8) We simply remark now that, if ξ ∈ (−∞, −λ), λ > and −λ ≡ 2(α1 − β1 + 2ρc2o ) α1 + µ + 2ρc2o + , β1 (α1 + µ + 2ρc2o )2 − 8µβ1 , 2β1 226 V Tigoiu ˙ then it follows E(t) + ξE(t) ≥ Employing the constitutive inequality β1 < and Friedrichs’s inequality we obtain d dt A1 α1 + µ + 2ρc2o β1 + A1 ≤ F (0)e−ξt , β1 (9) α1 + µ + 2ρc2o and putting into β1 evidence that ξo < and ξo − ξ > 0, we obtain from (9), for each t ∈ [0, T0 ), the a priori estimate for the L2 norm of the field A1 for all t ∈ [0, To ) Denoting now ξo ≡ A1 (t) − e−(ξo −ξ)t F (0)+ β1 ξo − ξ ≤ A1 (0) e−(ξo −ξ)t e−ξt (10) − e−(ξo −ξ)t α1 + µ − ξβ1 ; P ≡ 1+ ; ξo − ξ β1 M (t) ≡ 2M1 (t); N (t) ≡ P M1 (t) + e−(ξo −ξ)t , and we easily remark that < M1 (t) < 1/(ξo − ξ); P > Consequently we can state the following lemma We denote now M1 (t) ≡ Lemma Each solution of the problem (2) - (3) verifies the a priori estimate A1 (t) 2≤ ∂A1 (t) ∂t N ≤ N A1 (0) A1 (0) +M +M ∂A1 (0) ∂t ∂A1 (0) ∂t e−ξt , (11) e−ξ1 t , for all t ∈ [0, To ) and where N’, M’ depend on N, M, on constitutive modules µ, α1 , β1 and on To also From Lemma we obtain the main result of this part of the paper Theorem (uniqueness) Let it be (v1 , p1 ) and (v2 , p2 ) two solutions of the problem (2) − (3) corresponding to the same initial data Then v1 (t, x) = v2 (t, x) a.e in Ω, for all t ∈ [0, To ) Moreover, independent on the statement of the theorem it results also that ∂v2 ∂v1 (t, x) = (t, x) a.e in Ω and for all t ∈ [0, To ) ∂t ∂t Remark The a priori estimate (11) has been used in [11] in order to prove the asymptotic stability of the rest state Uniqueness 227 Uniqueness for the solution of the initial and boundary value problem Nonlinear case Here we shall employ a known stability criterion (see [9], Chap III.B) and consequently (without a significant lost of generality) we shall restrict to the class of third grade fluids with 3β1 + 2β2 = 0, α1 > The constitutive law is then given by T = −pI + µA1 + α1 (A2 − A21 ) + β1 A3 − β1 (A1 A2 + A2 A1 )+ + β1 (trA21 )A1 , (12) with µ ≥ 0, α1 > 0, β1 < The flow equations are ρv˙ = divT + ρb (13) where T is given in (12) and b is the body forces field (supposed to be conservative) The initial value problem is given in (3)1 and for boundary conditions we have v(x, t) |∂Ω = 0, ∇v(x, t) |∂Ω = L0 (t, x) (14) Like in the previous section we shall consider a pair of solutions (vi , pi ), i = 1,2 and we shall employ the notations (4) We remark that the initial and boundary conditions are given by v(x, t) = 0, ∂v (x, t) = 0, v(x, t) |∂Ω = 0, ∇v(x, t) |∂Ω = ∂t (15) We introduce (12) (written for v1 and v2 ) into the flow equations, ∂v we multiply by v and , respectively, we integrate over Ω and after ∂t subtracting the resulting relations we arrive to ρ ˜ ∂v ˜ ) · A1 (v)dx = 0, (T1 − T · vdx + ρ ∇v2 [v] · vdx + ∂t Ω Ω ρ Ω ∂v ∂v ∂v ∂v · dx + ρ ∇v[v] · dx + ρ ∇v[v2 ] · dx+ ∂t ∂t ∂t ∂t Ω Ω +ρ ∇v2 [v] · Ω Ω ∂v ˜ ˜ ) · A1 ( ∂v )dx = 0, dx + (T1 − T ∂t ∂t Ω (16) 228 V Tigoiu ∂v ∂v ∂A1 ∂v where A1 (v) = ∇v + (∇v)T , and A1 ( ) = ∇ + (∇ )T = ∂t ∂t ∂t ∂t ˜ and T ˜ stand for the values of the effective stress from (12), Here T evaluated for v1 and v2 respectively In order that relations (16) make sense we suppose ∂v ∈ L∞ (0, T ; W01,4 (Ω) ∩ W 2,4 (Ω), ∂t v ∈ L∞ (0, T ; W01,4 (Ω) ∩ W 3,4 (Ω)), ∂2v ∂A1 ∈ L2 (0, T ; W01,2 (Ω)), ∈ L∞ (0, T ; W01,4 (Ω) ∩ W 2,4 (Ω), ∂t ∂t (17) ∂ A1 ∈ L2 (0, T ; W01,2 (Ω)) ∂t2 A very long and carefully made calculus lead us to obtain from (16) − β1 d dt ∇A1 + ρ 2 L2 (Ω) − β1 v d dt L2 (Ω) ≤ − ∇A1 [v2 ] β1 β1 d dt L2 (Ω) ∇A1 L2 (Ω) + 2µ + α1 + B2 (T, v, v2 ) β1 A1 − β1 B3 (T, v, v2 ) ∇A1 − β1 B6 (T, v, v2 ) ∇A1 [v2 ] − L2 (Ω) β1 ∇A1 [v] ∂A1 ∂t L2 (Ω) L2 (Ω) − L2 (Ω) + L2 (Ω) A1 L2 (Ω) + − + (18) ρ + B1 (T, v, v2 ) β1 B5 (T, v, v2 ) L2 (Ω) 2µ + α1 A1 − L2 (Ω) v ∇A1 [v] β1 B4 (T, v, v2 ) ∂A1 ∂t − L2 (Ω) − L2 (Ω) where Bi (T, v, v2 ) are combinations of ci (T, v2 ), cj (T, v)andc20 (Ω) which are given bellow c1 (T, v2 ) = sup ess{max | v2 (t, x) |}, c2 (T, v2 ) = sup ess{max | ∇L(2) |}, t∈(0,T ) x∈Ω c3 (T, v2 ) = sup ess max | t∈(0,T ) x∈Ω t∈(0,T ) x∈Ω ∂L(2) | , c4 (T, v2 ) = sup ess{max | L(2) |}, x∈Ω ∂t t∈(0,T ) Uniqueness c5 (T, v2 ) = sup ess max | x∈Ω t∈(0,T ) ∂v2 ∂ (2) | , c6 (T, v2 ) = sup ess max | ∇A1 | , x∈Ω ∂t ∂t t∈(0,T ) ∂v ∂ | , c8 (T, v) = sup ess max | ∇A1 | x∈Ω ∂t ∂t t∈(0,T ) c7 (T, v) = sup ess max | x∈Ω t∈(0,T ) 229 Here c20 (Ω) is again the constant which results from the application of Friedrichs and Korn inequalities In order to obtain (18) the imbedding theorems of Sobolev and Kondrachew have also been employed If we chose now c (T, v, v2 ) = max{Bi (T, v, v2 ); i = 1, 6} then the inequality (18) is transformed into − β1 d dt β1 d + dt L2 (Ω) ≤ ∇A1 ∂A1 (t) ∂t d f (t) + c (T, v, v2 )f (t)+ dt L2 (Ω) (19) exp(−B4 (T, v, v2 )t) exp(B4 (T, v, v2 )t), where we denoted β1 d dt A1 L2 (Ω) + ρ 2 L2 (Ω) − − β1 f (t) = v ∇A1 [v2 ] β1 + 2µ + α1 ∇A1 L2 (Ω) + β1 A1 L2 (Ω) − ∂A1 ∂t L2 (Ω) β1 + ∇A1 [v] L2 (Ω) − L2 (Ω) We are now ready to give the following lemma Lemma If the estimate (19) is true then, there is T1 > such that d f (t) + c (T, v, v2 )f (t) ≥ dt (20) for all t ∈ (0, T1 ) It is to remark, for the proof, that from (17) the map t −→ ∇A1 (t) 2L2 (Ω) is from C ((0,T)) As ∇A1 (t) 2L2 (Ω) ≥ for all t > it results that there d is T0 > such that ∇A1 2L2 (Ω) ≥ , for all t ∈ (0, T0 ) Similarly, dt d ∂A1 there is T0 > such that (t) 2L2 (Ω) exp(−B4 (T, v, v2 )t) ≥ 0, dt ∂t 230 V Tigoiu for all t ∈ (0, T0 ) We denote now T1 ≡ min{T0 , T0 } and then the estimate (20) is true for all t ∈ (0, T1 ) We remark now that from the initial data (3)1 and from (20) we obtain ≤ + β1 d dt L2 (Ω) A1 + 2µ + α1 A1 L2 (Ω) + (21) β1 − 4c (T, v, v2 ) v 2L2 (Ω) − ρ + 2c (T, v, v2 ) ∇A1 (t) L2 (Ω) Now we can state the following principal lemma Lemma For any c > there is T2 > 0, such that for any t ∈ (0, T2 ) the inequality d dt −c A1 (t) L2 (Ω) ∇A1 (t) L2 (Ω) − (22) M2 t) − exp(− c exp(−M1 t ≤0 holds In (22) we have 2µ + α1 + 2(ρ + 2c (T, v, v2 ))c20 (Ω) > 0, −β1 β1 − 4c (T, v, v2 ) M2 (T, v, v2 ) = > β1 M1 (T, v, v2 ) = (23) For the proof, it results from (21), with Friedrichs and Korns’s inequalities that d dt A1 (t) L2 (Ω) − M1 (T, v, v2 ) − M2 (T, v, v2 ) A1 (t) L2 (Ω) ∇A1 (t) − L2 (Ω) (24) ≤ 0, where we have employed the notations (23) Then for any c > it results that d dt d +c dt A1 (t) L2 (Ω) ∇A1 (t) −c L2 (Ω) ∇A1 (t) L2 (Ω) exp(−M1 t) + M2 exp −(M1 + )t c exp M2 t c (25) ≤ Uniqueness 231 M2 )t c is C ((), T )) Then a similar reasoning to those performed for (20) gives: there is < T2 < T1 such that for any t ∈ (0, T2 ) we shall have d M2 ∇A1 (t) 2L2 (Ω) exp −(M1 + )t ≥ and consequently we obdt c tain (22) Employing now the boundary conditions (16) we arrive to We see now that the map t −→ A1 (t) L2 (Ω) −c ∇A1 (t) ∇A1 (t) L2 (Ω) L2 (Ω) exp −(M1 + − exp(− M2 t) ≤ 0, c (26) for any t ∈ (0, T2 ) and c > On the other hand, for any fixed t ∈ (0, T2 ) the map c −→ A1 (t) L2 (Ω) −c ∇A1 (t) L2 (Ω) − exp(− M2 t) c ≡ F (c) (27) is a continuous function of c ∈ R+ Let us now consider a sequence cn > 0, cn −→ As F (cn ) ≤ for any n, it results that lim F (cn ) ≤ and it n→∞ follows immediately that A1 (t) 2L2 (Ω) ≤ for any t ∈ (0, T2 ) (then by continuity we prolong the result on the whole interval) So we arrived to the following main result Theorem (uniqueness) Let be (vi , pi ), with i = 1,2 two solutions of the problem (12), (13), (3)1,2 and (14) Then, for T > 0, we have v1 (x, t) = v2 (x, t), (28) a.e in Ω × [0, T ) Moreover, the equality is established also between the firsts Rivlin - Ericksen tensors (1) (2) A1 (x, t) = A1 (x, t), (29) a.e in Ω × [0, T ) References [1] Cioranescu D and Girault V., Weak and classical solutions of a family of second grade fluids, Int J Non -Linear Mechanics, 32, 2, (1997), 317-335 232 V Tigoiu [2] Dunn J.E and Fosdick R L., Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade, Arch Rat Mech Anal., 56(1974), 191 - 252 [3] Fosdick R L and Rajagopal K R., Thermodynamics and stability of fluids of third grade, Proc Roy Soc London, A, 339(1980), 351 - 377 [4] Galdi G P and Sequeira A., Further existence results for classical solutions of the equations of second grade fluids, Arch Rat Mech Anal., 128 (1994), 297 [5] Joseph D D., Instability of the rest state of fluids of arbitrary grade greater then one, Arch Rat Mech Anal., 75(1981), 251 - 256 [6] Lions J.L., Quelques M´ethodes de R´esolution des Probl`emes aux Limites non Lin´eaires, Dunod, Gauthier-Villard,Paris, 1969 [7] Passerini A and Patria M C., Existence, uniqueness and stability of steady flows of second and third grade fluids in an unbounded ”pipelike” domain Int J Non-Lin Mech., 35(2000), 1081- 1103 [8] Patria M C., Stability questions for a third grade fluid in exterior domains, Int J Non - Linear Mechanics, 24, 5(1989), 451 -457 [9] Tigoiu V., Wave propagation and thermodynamics for third grade fluids, St Cerc Mat., 39, 4(1987), 279 - 347 [10] Tigoiu, V., Weakly perturbed flows in third grade fluids , ZAMM, 80, 6(2000), 423- 428 [11] Tigoiu V., Does a Polynomial Fluid of Third Grade Exist ? (A Hint on the Validity of Using Coleman and Noll’s Approximation Theorem), Proceedings of the 5-th Int Sem ”Geometry, Continua and Microstructures”, (2001), 207 - 220 Victor Tigoiu Facultatea de Matematic˘a, Universitatea din Bucure¸sti Str Academiei 14, 70109 Bucure¸sti, Romania E-mail: tigoiu@math.math.unibuc.ro NOTES FOR AUTHORS This journal is primarly devoted to papers in pure and applied mathematics It will also publish informations about the scientific life of the Faculty, reviews of books, monographs, proceedings and miscellaneea Contributors shoud submit two copies of their papers to Chief Editor, or to the Secretary of the Editorial Committee of ” Analele Universitˇ a¸tii Bucure¸sti, Seria Matematicˇ a ” Bucharest University, Department of Mathematics, Str Academiei 14, R-70109 Bucure¸sti, Romania Submisson of the paper for this journal will be taken to imply that it contains original work not copyrighted or published, in part or in the whole, and that, if accepted for publication, it will not be published elsewhere in the same form, in any language, without the written consent of the University of Bucharest Each paper requires a short abstract summarizing the significant coverage and findings, the AMS clasifications ( see Math Reviews, Annual Index 1982, 1991, 1995 ) and the key words and phrases References shoud appear within the text between brakets, the full list shoud be collected and typed at the end of the paper in alphabetical order, according to the abreviations of Mathematical Reviews All papers are refered, those not accepted for publication are not returned to the authors Author(s) will receive 10 reprints free of charge The authors’ complete names should be indicated, the family name in capital letters and first names in lowercase letters A disk containing the LATEX source file of the article submitted for publication is requested The source file for the article must be prepared in the simplest version with standard LATEX options for the style of the article to allow the editors of this Journal to make only minor changes in the text in order to put the issue in the final form We appreciate it, if the text does not contain \overfull and/or \underfull boxes, if equations not exceed the indicated text width, if hyphenations have been checked as well as the page breaks Apart from the following basic commands, please don’t change any default settings (e.g no offsets) and don’t use additional fonts and font sizes (\textwidth= 127mm, \textheight=190mm) Tiparul s-a executat de firma EUROGEMA EXIM, Bucure¸sti 2002