SIMPLE FLOWS OF PSEUDOPLASTIC FLUIDS BASED ON DEHAVEN MODEL

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1 It. J. of Applied Mechacs ad Egieerig, 17, vol., No.4, pp DOI: /ijame SIMPLE FLOWS OF PSEUDOPLASTIC FLUIDS BASED ON DEHAVEN MODEL A. WALICKA Uversity of Zieloa Góra, Faculty of Mechacal Egieerig ul. Szafraa 4, Zieloa Góra, POLAND I this paper three simple flows of visco-plastic fluids of DeHave type or fluids similar to them are cosidered. These flows are: Poiseuille flow i a plae chael, Poiseuille flow through a circular pipe ad rotatig Couette flow betwee two coaxial cyliders. After presetatio DeHave model it was preseted some models of fluids similar to this model. Next it was give the solutios of equatios of motio for three flows metioed above. Key words: DeHave fluids, similar fluids, simple flows. 1. Itroductio I recet years, rheologists have doe a great deal of work o pseudo-plastic fluid flows; the viscosity of these kids of fluids displays a o-liear relatioship betwee the shear stress ad the shear strai rate. To be more precise: i costitutive equatios of these fluids the shear strai rate is a o-liear fuctio of the shear stress. There are may kow formulae to model this relatioship. Oe of the first was a model preseted by Miss S.B. Ellis i 197 [1]. The ext was power-series developmet ad i cosequece polyomials were suggested. The polyomial give by Kraemer ad Williamso [] was later idepedetly proposed by Weisseberg s studet, Rabiowitsch [3]. I the ed of the fifties of the past cetury DeHave [4, 5] proposed his ow model very similar to the model of Miss Ellis (probably he did ot kow her model; ote that the same model as that proposed by Miss Ellis was formulated forty years later by three other researchers, amely by Mr Ellis et al. [6]). A bit later, at the begig of the sixties of the past cetury, Rotem ad Shiar [7, 8] retured to the polyomial represetatio proposig their ow model i 1 ki. (1.1) i1 Similar relatios were proposed by Whorlow [9] or i i1 i k (1.) i1 ki (1.3) i kow as power-series models.

2 136 A.Walicka Each of these models, by suitable choice of material coefficiets reduces to the DeHave model or to the Rabiowitsch model, respectively [1]. Most popular models of fluids which are similar to the DeHave fluid model are preseted i Tab.1. Table 1. Models of fluids similar to the DeHave fluid model [11]. Author(s) Origial model DeHave 1 Model take ito accout i Commets 1 power models Meter Ellis Rotem-Shiar Ree-Eyrig Rabiowitsch Reier power model i i i Philippoff sih 1, Peek-McLea 1 1 Seely e 1 Oe-dimesioal form of the DeHave model may be writte as i i i 6 Cubic models The model has a practical meag for i 1 Quadratic models 1k (1.4) whereas its three-dimesioal form is as follows

3 Simple flows of pseudoplastic fluids based o DeHave model i i A 1k ; (1.5) this form will be used i the ext Sectio to illustrate some flows. The equatios of motio are as follows div v, (1.6) dv divt, (1.7) dt T p1 ; (1.8) it is ecessary to iclude the costitutive Eq.(1.5) i these equatios. Note that the body forces are eglected i Eq.(1.7). The flows of pseudoplastic fluids of DeHave type were studied first by Matsuhisa ad Bird [1], Wadhwa [13] ad thereafter by may rheologists; lately, these flows have bee studied by Keyfets ad Kieweg [14] ad Walicka et al. [15, 16]. I that follows we will cosider three simple flows of the DeHave type which may be frequetly used i practical applicatios.. Poiseuille flow i a plae chael Let cosider the steady lamiar oe-dimesioal flow of a DeHave fluid, due to a pressure gradiet, i a plae chael show i Fig.1. The flow field is give by the followig relatios Fig.1. Chael betwee two parallel plates.,, y, p p z. (.1) x y z z The equatios of motio give by Eqs (1.6) (1.8) reduce to dp dz dyz (.) dy but the costitutive Eq.(1.5) takes the form

4 138 A.Walicka dz i yz 1ki yz. (.3) dy The boudary coditios are stated as follows z z h, y y. (.4) A sigle itegratio of Eq.(.) gives dp yz C1 y. (.5) dz Upo puttig this result ito Eq.(.3), oe obtais the followig expressio d i z dp dp C1 y 1 ki C1 y. (.6) dy dz dz Itroducig here the secod boudary coditio (.5) oe obtais the followig equatio i 1 i 1 C k C, whose uque real root is C1, the Eq.(.5) takes the form dz dp dp y ki y dy dz dz Itegratio of this equatio yields 1. (.7) i 1 y dp ki y dp z C dz dz Upo determiatio of the costat of itegratio from the first boudary coditio (.4), we obtai fially. i 1 h y dp ki h y dp z dz dz The flow rate Q per ut width of the chael is defied as h Q z dy Upo usig relatio (.8), we will obtai. (.8)

5 Simple flows of pseudoplastic fluids based o DeHave model h dp 3kih dp Q 1 3 dz 3 dz i i Note that for the Newtoa flow we have ki ad ; (.9) 3 h dp QQNewt 3 dz. (.1) 3. Poiseuille flow through a circular pipe Let us cosider the steady lamiar flow of a DeHave fluid i a circular pipe of radius R (Fig.). Fig.. Geometry of a circular pipe. The flow field has the form of,, r, p p z. (3.1) r z z The equatios of motio give by Eqs (1.6) (1.8) reduce to dp 1 d rrz (3.) dz r dr whereas the costitutive Eq.(1.5) takes the form dz i rz 1ki rz. (3.3) dr The boudary coditios are stated as follows z z, for r R ad for r. (3.4) r A sigle itegratio of Eq.(3.) gives C1 rdp rz r dz

6 14 A.Walicka ad after itroducig this result ito Eq.(3.3), we have d i z C 1 rdp C 1 rdp 1 k i dr r dz r dz. (3.5) Takig ito accout the secod boudary coditio (3.4), oe obtais that C1 ad Eq.(3.5) takes the form whose itegral is equal to 1 1 dz r dp ki r dp dr dz dz ki z C 1 i r dp r dp 4 dz dz, (3.6) Upo determiatio of the costat of itegratio from the first boudary coditio (3.4), we will obtai fially The flow rate Q is defied us ki z 1 R r dp R r dp 4 dz dz R Q z rdr. Upo usig relatio (3.7), we will fid 4 R i dp 8kiR dp Q dz 4 dz Note that for the Newtoa flow, we have ki ad (3.7) ; (3.8) 4 R dp QQNewt 8 dz. (3.9)

7 Simple flows of pseudoplastic fluids based o DeHave model Rotatig Couette flow betwee two coaxial cyliders Fig.3. Let us cosider the flow of a DeHave fluid i the clearace betwee two coaxial cyliders show i Fig.3. Geometry of the rotatioal flow betwee cylidrical surfaces. The flow field is give by the relatioships, r,, p p r. (4.1) r The equatios of motio (1.6) (1.8) ow take the form z dp, (4.) r dr d r r dr but the costitutive Eq.(1.5) takes the form (4.3) d dr r i r r 1 ki r The boudary coditios are as follows. (4.4) R, for r R,, for r R. (4.5) Upo itegratio of Eq.(4.3), we will obtai C r 1 r i i o

8 14 A.Walicka ad after itroducig this result ito Eq.(4.4), we will have C k C i 1 d 1 i 1 r dr r r r (4.6) because C1. The ext itegratio gives r C1 ki C1 1 1 r r C r O the basis of the secod boudary coditio (4.5) we will obtai. (4.7) C C1 ki C1 1 Ro Ro (4.8) ad fially C1 ki C (4.9) i Ri R o Ri R o The agular velocity r at ay positio r is expressed as r r C1 ki C1 C 1 r r. (4.1) The ut torque actig o the cylidrical surface of radius r is equal to T r C. (4.11) r 1 Note that T s is also the torque which has to be applied to the ier cylider to maitai its motio. Deotig by T r the ati-torque which must be applied to the outer cylider to maitai its rest we have T T T C r 1 C. s 1, (4.1) Itroducig the otatio Ri, R o r r (4.13) R o oe ca write:

9 Simple flows of pseudoplastic fluids based o DeHave model T 1 1 r R o r 1 ki T 1 1 i R o r (4.14) the formula which ca be used for determig the material costats i the DeHave model or similar models from measuremets of torque ad agular velocity i a coaxial aular viscosimeter [1] if it is equipped with a suitable software [17]. 5. Coclusios The simple flows of pseudoplastic fluids based o DeHave model may fid may applicatios i differet braches of techology ad idustry. It ca cite e.g. the theory of lubricatio. Basig o the method of solutio the flow i a plae chael oe may obtai the solutio of the flow i bearig clearaces; the solutio for the flow i a circular pipe may be used to fid the flow i bearig clearaces with porous walls [11]. Nomeclature A 1 the first Rivli-Erickse kiematic tesor e Naperia logarithm base kk, i pseudo-plasticity coefficiets expoetial rheological parameter p pressure Q flow rate shear stress tesor T torque t time v velocity vector k compoets of velocity vector 1 ut tesor shear strai rate extra stress tesor magtude of extra stress tesor shear viscosity, limitig values of shear viscosity fluid desity shear stress agular velocity Refereces [1] Ellis S.B.: Thesis, Lafayette College, Pa, 197. Citted i: Matsuhisa S., Bird R.B. (1965): Aalytical ad umerical solutios for lamiar flow of the o-newtoa Ellis fluid. AiChE Joural, vol.11, No.4, pp [] Kraemer E.O. ad Williamso R.V. (199): Iteral frictio ad the structure of solvated colloids. J. Rheology, vol.1, No.1, pp.76-9.

10 144 A.Walicka [3] Rabiowitsch B. (199): Über die Viskosität ud Elastizität vo Sole (O the viscosity ad elasticity of sols). Zeit. Phys. Chem., A145, pp.1-6. [4] DeHave E.S. (1959): Cotrol valve desig for viscous pseudoplastic fluids. JEC Equipmet ad Desig, vol.51, No.7, pp.63a-66a. [5] DeHave E.S. (1959): Extruder desig for pseudoplastic fluids. Id. Eg. Chem., vol.51, No.7, pp [6] Ellis S.C., Laham A.F. ad Pakhurst K.G.A. (1967): A rotatioal viscometer for surface films. J. Sci. Istr., vol.3, pp [7] Rotem Z. ad Shiar R. (1961): No-Newtoa flow betwee parallel boudaries i liear movemets. Chem. Eg. Sci., vol.15, pp [8] Rotem Z. ad Shiar R. (196): No-Newtoa flow betwee parallel boudaries i liear movemet. II. No- Isothermal Flow. Chem. Eg. Sci., vol.17, pp [9] Whorlow R.W. (199): Rheological Techques (Sec. Editio). New York: Ellis Horwood. [1] Walicka A., Jurczak P. ad Falicki J. (17): Curviliear squeeze film bearig lubricated with a DeHave fluid or with similar fluids. It. J. of Applied Mechacs ad Egieerig, vol., No.3, pp [11] Walicka A. (17): Rheology of fluids i Mechacal Egieerig. Zieloa Góra: Uversity Press. [1] Matsuhisa S. ad Bird R.B. (1965): Aalytical ad umerical solutios for lamiar flow of the o-newtoa Ellis fluid. A.I.Ch. Joural, vol.11, No.4, pp [13] Wadhwa Y.D. (1996): Geeralized Couette flow of a Ellis fluid. AIChE Joural, vol.1, No.5, pp [14] Kheyfets V.O. ad Kieweg S.L. (13): Gravity-drive thi film flow of a Ellis fluid. J. No-Newtoa Fluid Mech., vol., No.1, pp [15] Walicka A., Walicki E., Jurczak P. ad Falicki J. (14): Thrust bearig with rough surfaces lubricated by a Ellis fluid. It. J. Appl. Mech. Eg., vol.19, No.4, pp [16] Walicka A., Walicki E., Jurczak P. ad Falicki J. (16): Curviliear squeeze film bearig with rough surfaces lubrcated by a Rabiowitsch-Rotem-Shiar fluid. Appl. Math. Modell., vol.4, pp [17] Schramm G. (1994): A Practical Approach to Rheology ad Rheometry. Karlsruhe: Haake. Received: Jue 3, 17 Revised: July 5, 17