FEEDBACK BOUNDARY OF 4:1 ROUNDED CONTRACTION SLIP FLOW FOR OLDROYD-B FLUID BY FINITE ELEMENT

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1 Joural of Egieerig Sciece ad Techology Vol. 13, No. 3 (2018) School of Egieerig, Taylor s Uiversity FEEDBACK BOUNDARY OF 4:1 ROUNDED CONTRACTION SLIP FLOW FOR OLDROYD-B FLUID BY FINITE ELEMENT NAWALAX THONGJUB Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techology, Thammasat Uiversity, Pathumthai 12120, Thailad t_awalax@hotmail.com Abstract The drive force feedback i viscoelastic flow is a techique to eforce coverged solutio of Oldroyd-B fluid o the basis of semi-implicit Taylor- Galerki fiite elemet method for 4:1 cotractio geometry with rouded corer meshes. Meawhile i the umerical computatio, the Pha-Thie slip rule is applied to complete the slip velocity alog the die wall. After applicatio of the slip coditio, the severe stress ear die exit ad the vortex size aroud cotractio corer are clearly reduced. This simulatio is modeled with the Navier-Stokes ad Oldroyd-B equatios i two-dimesioal plaar isothermal icompressible creepig flow. The o-liear differetial models are discretised to system of liear equatios with semi-implicit Taylor-Galerki fiite elemet method. I additio, the Streamlie-Upwid/Petrov-Galerki ad velocity gradiet recovery are accuracy ad the stability schemes to stabilize a approximate solutio. The optimal slip velocity is modified step by step by critical slip coefficiet to set the slip coditio at the wall ad whe the solutios of slip ad o- slip cases are compared with aalytic solutio, the outcome of slip coditio shows better coformity to real results. Keywords: Feedback, Slip effect, 4:1 cotractio flow, Oldroyd-B fluid, Slip velocity. 1. Itroductio This research is focused o feedback boudary of 4:1 rouded cotractio slip flow for the Oldroyd-B fluid by a semi-implicit Taylor-Galerki pressurecorrectio fiite elemet method (STGFEM) uder two-dimesioal plaar system. The kiematic behaviors of this viscoelastic fluid are strog elogatio 725

2 726 N. Thogjub Nomeclatures (No-dimesioal system) D Deformatio tesor rate II Secod ivariat Critical secod ivariat II crit II w L Secod ivariat at the wall Characteristic legth Time step 1 2 Half time step 1 Full time step P Pressure Time step Re Reyolds umber t Time U Velocity U mea Mea velocity of flowrate for o slip U slip Slip velocity V Characteristic velocity We Weisseberg umber Greek Symbols (No-dimesioal system) Desity Viscosity 0 Zero-shear viscosity p Polymeric viscosity s 1 xy Abbreviatios Solvet viscosity Polymeric compoet of the extra-stress tesor Relaxatio time Shear stress Shear rate Slip coefficiet FEM Fiite Elemet Method J&S Johso, M.W. ad Segalma, D. [13] STGFEM Semi-implicit Taylor-Galerki Pressure-correctio Fiite Elemet Method ad violet shear stress whe the flow cofrots cotractio sectio abruptly the the streamlie path suddely chages trajectory. For this pheomeo, the slip velocity at cotractio wall is calculated i order to reduce shear stress at sharp corer.

3 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B I experimetal observatio for viscoelastic fluids [1], the Rheogoiometer was used to measure the rheological properties i a abrupt 2 to 1 cotractio domai. The sigificat compariso was emphasized the differece betwee the accomplishmet result ad umerical predictio of power-law model. The equipmet [2] for plaar cotractio flows was implemeted for Boger fluid. After acute umerical solutio was revealed, a circular cotractio flow for both Newtoia ad No-Newtoia fluids [3] has bee set up to bechmark with experimetal result. To avoid icoveiet calculatio from aalytical data, the mathematical model of viscoelastic problem is simulated i form of o-liear partial differetial equatios uder coservatio law of mass ad mometum. I this case, a complex problem is elimiated by meas of umerical techiques such as fiite elemet method, fiite volume method ad fiite differece method. Recetly, the semi-lagragia fiite volume method [4] was applied for solvig a 4:1 plaar cotractio of creepig flow via Oldroyd-B model. The iertial flows ad the cosequece procedure [5] of a ew coditio by chagig Cartesia coordiates with axisymmetric cylidrical flow were expaded with fixed grids i Euleria methods. A semi-implicit Taylor-Galerki fiite elemet method [6] has bee brought to solve stick-slip flow of Oldroyd-B problem alog with the modificatio for free surface method at die exit boudary which was determied for Die-swell flow i terms of swellig ratio as a fuctio of relaxatio time. Cosequetly, the Pha- Thie/Taer fluid i complete die of pressure-toolig problem [7] has bee simulated uder the same STGFEM icludig security techique of Streamlie- Upwid Petrov/Galerki for stabilizig covergig solutio. To utilize fiite elemet method ad fiite volume method i the same code, the ew hybrid scheme of fiite volume/elemet method [8, 9] by meas of cell-vertex discretisatio techique has bee proposed i order to have calculated cooperative stress ad flow field for Oldroyd-B ad Pha-Thie/Taer fluids with both rouded ad sharp corer cotractio flows. Collectig experimetal ad umerical data to study fluid flows through solid wall has demostrated that slip velocity appears o solid surface ad hece reduces shear rate. To make solutio look closer to real problems, a umber of literatures have set up various umerical methods to estimate slip velocity [10] at the wall o free surface. For large slip speed o die wall of HDPE ad LLDPE, the deformatio ca be oticed uder surface melt fracture [11]. I additio, the aalysis solutios for capillary tubes [12] have bee calculated after settig the slip velocity. I this study, the slip coditio has bee employed i the problem of 4:1 cotractio for Oldroyd-B model uder the two-dimesioal plaar isothermal icompressible flow. A semi-implicit Taylor-Galerki pressure-correctio fiite elemet method has trasformed Navier-Stokes equatio to a system of simple liear equatios ad all solutios have bee stabilized by meas of the Streamlie-Upwid Petrov/Galerki ad velocity gradiet recovery techiques. The solutios of slip coditio ad o slip situatio have bee compared after optimizatio of the slip coefficiet with rouded corer geometries.

4 728 N. Thogjub 2. Goverig Equatios The goverig equatios are the Navier-Stokes equatios i dimesioless form of the coservatio of mass ad mometum for viscoelastic fluid based o ogravity ad steady icompressible flow. Cosiderig the stadard values of the fluid properties, the o-dimesioal system of the cotiuity equatio (1) ad Navier-Stokes equatio (2) are represeted as U 0 (1) U Re T ReU U P (2) t Re VL (3) 0 The o-dimesioal parameters are Re 0, p 8 0 9, ad s T 2 s D (4) U + U D 2 t (5) The o-dimesioal costitutive equatio of Oldroyd-B fluid is represeted as t We t We 2 p U U U D (6) 1 V We L The aalytical values (J&S) of the shear stress [13] of Oldroyd-B fluid is demostrated as a fuctio of viscosity ad shear rate where a is a scalar parameter o iterval (0, 2) as below (7) p xy s 2 a(2 a)( p) 1 (8) 3. Computer Programme A fiite elemet method is employed to compute the o-liear differetial equatios as the Navier-Stokes equatio ad the costitutive equatio of Oldroyd-B model. The goverig equatios are discretised to three time stages by a semiimplicit Taylor-Galerki pressure-correctio scheme. Ad the the derivative equatios are coverted to liear equatios ad solved by the iterative method ad Cholesky decompositio scheme. The Feedback STGFEM is employed to approach the coverged solutio ad be stabilized by the streamlie-upwid Petrov/Galerki ad velocity gradiet recovery techiques. For studyig slip effect, Pha-Thie slip

5 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B rule is applied to compute the appropriate slip velocity at wall to reduce the severe stress at cotractio geometries as detailed i previous paper [14] Semi-implicit Taylor-Galerki pressure-correctio fiite elemet method Semi-implicit Taylor-Galerki pressure-correctio fiite elemet method is the itegratio of time-splittig step scheme ad fiite elemet method. This method is based o a semi-implicit fractioal step method to discretise o-dimesioal Navier- Stokes equatio (2) ad costitutive equatio (6) to three stages per time step. The fiite differece discretisatio is used to trasform the iitial ad boudary value problem accomplished with the partial differetial equatios (2) ad (6). The time derivative is replaced by Taylor series ad the space (spatial) term is expaded by the weight residual of Galerki fiite elemet method. After that the three stages per time step are coverted to the system of liear equatios, the approximate solutio of steps 1a, 1b ad 3 are computed with Jacobi iterative method whereas step 2 is calculated by Cholesky decompositio algorithm. Step 1a: 2Re ( 12 ) t U U 12 2 Re P ( sd U U s D D ) (9) 2 We ( 12 ) t 2 We ( ( ) t p D U U U ) (10) Step 1b: Re ( * ) t U U 1 2 * 2 P Re ( s s ) D U U D D (11) We ( 1 ) t 2 We ( ( ) t ) 12 p D U U U (12) Step 2: 2 1 2Re ( P P ) U * (13) t Step 3: 2Re ( 1 * ) ( 1 P P ) t U U (14) where * is the itermediate time step.

730 N. Thogjub 3.2. Pha-Thie slip rule The slip effect is the pheomeo that the slip velocity occurs at wall, observed i idustrial process ad experimet i sciece.

6 730 N. Thogjub 3.2. Pha-Thie slip rule The slip effect is the pheomeo that the slip velocity occurs at wall, observed i idustrial process ad experimet i sciece. Pha-Thie slip rule [15] is effective to fit slip velocity at wall because the outcome is close to experimetal result [16]. The slip velocity is expressed as a fuctio of wall shear stress ad the slip velocity will be calculated whe a critical shear stress is less tha the wall shear stress. The behavior of secod ivariat is similar to the shear rate [14], so the secod ivariat value is used to calculate the slip velocity as below. II II U U U e w crit slip mea mea (15) 3.3. The drive force feedback To eforce the umerical solutio to coverged solutio, feedback of pressuredrive velocity flow [17] is used to set the boudary coditio at ilet each step util the ilet flow approaches steadiess. For viscoelastic fluid, this techique is effective by settig the itermediate boudary to cotrol the approximate solutio. After that the umerical result will adjust smoothly ad elimiate the case of jumpig to the diverged solutio. This scheme uses the differetial time step at 0.1 ad the differetial time step will be reduced step by 0.1 whe the umerical error reaches to termiate this the differetial time. After that we started to set the iitial colum (C1) from cosiderig the stress, pressure ad velocity at odes alog C2, C3 ad C4 as see i Fig 1. Cosiderig the values of C4, it is foud that the maximum error of C4 is less tha the other colums, the values odes betwee 22 ad 28 will be set up to istate the values of odes 1-7 i C1. This cocept is still operated to fid the boudary coditio at ilet util the umerical solutio is coverged. The effective use uder the drive force feedback flow for velocity ad pressure is developed i order to reduce the time step from the repetitio of the computatio ad approach steadily aalytical solutio, as outlied i Appedix A. Fig. 1. Example of odes ad colums i the drive force feedback flow for the velocity ad pressure. 4. Results ad Discussio The effect of 4:1 cotractio problem such as shark ski pheomeo ad severe stress whe especially viscoelastic fluids pass the sharp corer happe clearly i egieerig ad idustrial process. I this research, the acute geometry is curved to rouded corer i order to reduce large stress values ad lip vortex by

Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B.... 731 Aboubacar et al. [8, 9].

I Fig 2(a), the symmetric geometry of plaar 4:1 cotractio is located at the ceter lie.

medium, ad fie meshes are mesh1, mesh2 ad mesh3 as show i Fig. 2(b), 2(c) ad 2(d), respectively.

7 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B Aboubacar et al. [8, 9]. I additio, the slip velocity is applied o the chael wall by meas of Pha-Thie slip rule [14, 17]. I Fig 2(a), the symmetric geometry of plaar 4:1 cotractio is located at the ceter lie. The half chael legth of 4L ad L chael widths are log eough to develop Poiseuille flow at ilet ad exit sectios which are 27. 5L ad 49L, respectively. 3 2 U ( y) (16 y ), V 0 x 128 y (16) 2 U 2We x U xx p, yy 0, ad x y xy p y The three differet patters of coarse, medium, ad fie meshes are mesh1, mesh2 ad mesh3 as show i Fig. 2(b), 2(c) ad 2(d), respectively. The subdomai of every mesh is cut at positio x = 22 to x = 30 from the full domai. The smallest elemet size of ubias mesh structure is geerated ear rouded corer ad the mesh specificatios are defied i Table 1. (17) (a) the domai of mesh2 (b) mesh1 (c) mesh2 (d) mesh3 Fig. 2. Mesh patters.

8 732 N. Thogjub Table 1. Mesh specificatio of rouded corer. mesh1 mesh2 mesh3 Elemets Nodes Degree of Freedom The Miimum of Mesh Size Effect of Slip Coditio Problem The results for viscoelastic flows of rouded corer meshes uder the coditio of o slip ad slip effect are cosidered ad the best mesh is selected to display fial solutio that duplicatio would reduce as well as savig computer times. After the best mesh had bee selected, it was applied to operate o slip case ad slip coditio for further details as above. A stick or o slip problem has bee studied by collectig stresses ad shear rate so the highest values of ormal stress, shear stress ad shear rate o bottom dowstream wall are compared i order to choose a optimal mesh. As such, mesh2 was select as a model of fial solutio. For Oldroy-B fluids, the peak values of i Table 2 are cosidered with oslip o wall for various We. The peak values o bottom dowstream wall of for mesh1 vary as a fuctio of We but i cotrast to mesh2 ad mesh3. As such, mesh1 is termiated by icosistet value whe compared i group. Maxima of all stresses i Table 3 look alike i mesh refiemet. For this problem, the rouded corer mesh2 i a similar fashio to mesh3 is picked up for further rus because it is simple to access ad the results are i close agreemet with the outcome of mesh3 but the time step of mesh3 is more tha the time step of mesh2 for all We values. We have selected mesh2 to simulate the slip case for reducig the calculated time. Table 2. The peak values of o the bottom dow-stream wall with o-slip. We mesh1 mesh2 mesh Table 3. The peak values of stresses o the bottom dow-stream with o-slip at We 1. mesh1 mesh2 mesh3 xx xy yy I the cotext of viscoelastic fluid, the peak values of ormal stress o bottom dowstream wall with o-slip for differet values of We ca be observed i Fig. 3. Sice all oscillatio graphs look similar i patter but higher values of We climb up fast ad drop quickly, this ca be assiged to the fact that higher We values have loger relaxatio time. This implies that the tred keeps higher

9 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B memory profile of its path ad the its memory of origial positio leads to its drop i close viciity of the begiig locatio. Fig. 3. o the bottom dowstream wall without slip of Oldroyd-B fluid for mesh2. To fid the appropriate slip velocity, we have foud the optimum slip coefficiet 0.1 for selectig the critical II for all We. Whe we have set 0.1, the highest values of shear rate alog the wall are reduced. II=3.3, II = 2.3, II = 2.3 ad II = 2.4 are the critical II for We = 0.25, 0.5, 0.75 ad 1, respectively as illustrated i Fig 4. I additio, the shear rate lie of the best of slip velocity clearly becomes lower tha o-slip coditio ad it is give the smoothest lie whe it is compared with other slip velocity values. I a similar maer to sharp corer mesh, the previous algorithm is set up to imply the least shear rates of all We values ad fially the critical secod ivariats are adjusted to the values i Table 4. This table reflects that whe the critical II is modified properly, the appropriate is 0.1 for ay values of We. The streamlie cotour for We 1 of mesh2 with slip at 0.1 ad II 2.4 is depicted i Fig 5. The vortex size of slip problem is smaller tha o-slip as well as We = 0.25, 0.5 ad There is slight alteratio to a rouded corer because the stability of a rouded corer is more tha a sharp corer whilst the stress value of the sharp corer is higher tha that of a rouded corer at cotractio. I additio, the lie plot betwee the shear stress versus the shear rate of slip, o slip ad aalytical result are compared i Fig 6 ad the aalytical lie is deoted by J&S. The solutio of the slip effect that is ru by mesh2, is slightly differet from J&S with the percetage error % for We 0.25 ad % for We 1 ad it is closer to J&S more tha o-slip coditio with the error % for We 0.25 ad % for We 1, as depicted i Fig 6. The slip coditio has less approximate error tha its o- slip couterpart. Table 4. The least shear rate for proper ad suitable II of Oldroyd-B fluid with mesh2. We II crit

10 734 N. Thogjub (a) 0.1 ad We 0.25 (b) 0.1 ad We 0.5 (c) 0.1 ad We 0.75 (d) 0.1 ad We 1 Fig. 4. The peak of versus II for mesh2 alog bottom dowstream wall.

(a) o slip (b) slip at 0.1 ad II 2.4 Fig. 5. Streamlie cotour for mesh2 alog bottom dowstream wall (a) We 0.25 (b) We 1 Fig. 6.

11 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B Comparig the appropriate slip values at the wall ad without slip, we foud that the slip ifluece ca slightly reduce the peak xx ad the high shear rate as displayed i Table 5. (a) o slip (b) slip at 0.1 ad II 2.4 Fig. 5. Streamlie cotour for mesh2 alog bottom dowstream wall (a) We 0.25 (b) We 1 Fig. 6. The compariso of xy versus with J&S for mesh2 o bottom dowstream wall of Oldroyd-B fluid. Table 5. The peak value of ad xx o the bottom dowstream wall for mesh2. We xx o slip slip o slip slip For the o slip coditio, the fluid velocity is zero at wall boudary whilst the real pheomeo makes us kow that the velocity would ot be zero so the slip coditio is set to predict optimal velocity from the critical secod ivariat. Uder slip situatio, the stresses are reduced ad jump to the aalytical value (J&S) more tha the o slip case, thus this effect leads the way to dimiish vortex size. The solutios of slip coditio at We 1 are depicted i Fig 7. The highest values of stress are located alog rouded corer geometry same as sharp corer cotractio. The maximum velocity i x-directio appears alog symmetry lie while the maximum velocity of y-directio is placed ear rouded corer.

12 736 N. Thogjub Maximum value of pressure started at ilet ad the gradually decreases util it is vaished at outlet. (a) xx (b) xy (c) yy (d) U x (e) V y (f) P Fig. 7. Lie cotour with slip alog chael wall at =1, = 2.4, ad We =1. 5. Coclusios For steady-state viscoelastic flow i 4:1 cotractio rouded geometry through plaar isothermal Oldroyd-B model, the semi-implicit Taylor-Galerki pressure-correctio fiite elemet scheme ad the drive force feedback flow are employed to solve the oliear partial differetial equatio for stick before Pha-Thie slip rule is added to calculate velocity at chael wall. The streamlie patter, stress visualizatio ad velocity field cotours were depicted to compare rouded corer meshes with sharp corer meshes. All figures gave the same tred whe varied Weisseberg umbers but the sigificat differeces of vortex sizes were oticed coveietly. After appropriate critical II was adjusted, the slip coefficiet of Viscoelastic fluids with rouded corer meshes is ivestigated for the optimum value of 0.1. The appropriate values of the slip coefficiet ad the secod ivariat cause the peak of shear rate lower tha o-slip case ad vortex size of sharp corer domai was decreased more tha rouded corer shape. I case of the right selectio for secod ivariat, this proper slip coefficiet well reduces the peak of shear rate ad vortex size sice the velocity at wall forces fluid flow with smooth path ad

13 Feedback Boudary of 4:1 Rouded Cotractio Slip Flow for Oldroyd-B stable outcome. I additio, the higher ad II preseted more oscillatios that brought to the pheomeo of shark ski. Refereces 1. Boger, D.V.; ad Ramamurthy, A.V. (1972). Flow of viscoelastic fluids though a abrupt cotractio. Rheologica Acta, 11, Walters, K.; ad Rawliso, D.M. (1982). O some cotractio flows for boger fluid. Rheologica Acta, 21, Boger, D.V. (1987). Viscoelastic flows through cotractios. Aual Review of Fluid Mechaics, 19, Phillips, T.N.; ad Williams, A.J. (1999). Viscoelastic flow through a plaar cotractio usig a semi-lagragia fiite volume method. Joural of No- Newtoia Fluid Mechaics, 87, Phillips, T.N.; ad Williams, A.J. (2002). Compariso of creepig ad iertial flow of a oldroyd b fluid through plaar ad axisymmetric cotractios. Joural of No-Newtoia Fluid Mechaics, 108(1-3), Ngamaramvaraggul, V.; ad Webster, M.F. (2001). Viscoelastic simulatio of stick-slip ad die-swell flows. Iteratioal Joural for Numerical Methods i Fluids, 36, Ngamaramvaraggul, V.; ad Webster, M.F. (2002). Simulatio of pressuretoolig wire-coatig flow with Pha-Thie/Taer models. Iteratioal Joural for Numerical Methods i Fluids, 38, Aboubacar, M.; ad Webster, M.F. (2001). A cell-vertex fiite volume/elemet method o triagles for abrupt cotractio viscoelastic flows. Joural of No-Newtoia Fluid Mechaics, 98, Aboubacar, M.; Matallah, H. ad Webster, M.F. (2002). Highly elastic solutios for Oldroyd-b ad Pha-Thie/Taer fluids with a fiite volume/elemet method: Plaar cotractio flows. Joural of No- Newtoia Fluid Mechaics, 103, Sillima, W.J.; ad Scrive, L.E. (1980). Separatig flow ear a static cotact lie: slip at a wall ad shape of a free surface. Joural of Computatioal Physics, 34, Ramamurthy, A.V. (1986). Wall slip i viscous fluids ad ifluece of materials of costructio. Joural of Rheology, 30, Jiag, T.Q.; ad Youg, A.C. (1986). The rheological characterizatio of HPG gels: measuremet of slip velocities i capillary tubes. Rheologica Acta, 25(4), Johso, M.W.; ad Segalma, D. (1977). A model for viscoelastic fluid behavior which allows o-affie deformatio. Joural of No-Newtoia Fluid Mechaics, 2, Thogjub, N.; Puagkird, B. ad Ngamaramvaraggul, V. (2013). Simulatio of slip effect with 4:1 cotractio flow for Oldroyd-B fluid. Iteratioal Joural of Applied Sciece ad Techology, 6(3), Pha-Thie, N. (1988). Ifluece of wall slip o extrudate swell: A boudary elemet ivestigatio. Joural of No-Newtoia Fluid Mechaics, 26,

14 738 N. Thogjub 16. Ngamaramvaraggul, V.; ad Webster, M.F. (2000). Simulatio of coatig flows with slip effects. Iteratioal Joural for Numerical Methods i Fluids, 33, Ngamaramvaraggul, V.; ad Thogjub, N. (2014). Newtoia fluid through the abrupt 4:1 cotractio flow of rouded corer geometry with feedback pressure-drive velocity flow. Iteratioal Joural of Iformatio Techology ad Computer Sciece, 15(2), Appedix A Flowchart of the Drive Force Feedback Flow The below flowchart is set up to ru the optimum solutio (pressure, velocity, stress) at ilet boudary uder the drive force feedback techique. The approximate value will be estimated ad retur the values back to the ext vertical odes as below demostratio workflow. START Read Iput Data Set E, Δt = 0.01 Calculate solutio with FEM Retur the optimum values of pressure, velocity, stress to ilet boudary Calculate solutio with FEM Compute Error E =10-1 E Δt =10-1 Δt t Error < 10-5 NO Prit out Result Error < E END Fig. A. Flow chart of feedback of force-drive velocity flow.

THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT

Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN 56-586 THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE Atif Nazir, Tahir Mahmood ad