Analytical and numerical solutions for torsional flow between coaxial discs with heat transfer

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1 Analytical and numeical solutions fo tosional flow between coaxial discs with heat tansfe David O. Olagunju, Anand B. Vyas, and Shangyou Zhang Novembe, 2007 Abstact We conside non-isothemal tosional flow between two coaxial paallel plates with heat tansfe at the uppe otating plate, constant tempeatue on the lowe stationay plate, and no heat loss at the fluid-ai inteface. Viscous heating is modelled by a Nahme law with exponential dependence on tempeatue. Due to the highly nonlinea natue of the govening equations an exact solution is not feasible. Theefoe we solve the poblem using both numeical and petubations methods. Specifically, analytical solutions ae obtained using asymptotic expansions based on the aspect atio and the Nahme-Giffith numbe, a measue of viscous heating, as petubation paametes. The numeical solutions ae obtained by a finite element method. Good ageement is found between the analytical and numeical solutions in appopiate paamete ange. In viscometic applications the toque exeted by the fluid on the lowe plate is an impotant quantity. Fo isothemal flow the dimensionless toque can be easily calculated. In this pape we obtain an analytical fomula that can be used to calculate non-isothemal coection to the toque. Keywods. Paallel-plate flow, viscous heating, axisymmetic finite elements.inf-sup condition, singula vetex, AMS subject classifications (2000). 76D07, 76M0, 65F0. Intoduction The poblem of viscous heating in viscometic flow of Newtonian and non-newtonian liquids is of both theoetical and pactical inteest. An exact solution was obtained fo flow of a Newtonian fluid between two infinite paallel plates by Nahme in 940 []. The viscosity was modelled by an exponential function of tempeatue. A simila esult was obtained by Keasley [2] fo the pessue gadient flow of a Newtonian fluid in a tube. Bid and Tuian [3] analyzed the viscous heating poblem fo the flow of a Newtonian fluid between a cone and a plate. The equations fo the tempeatue and velocity wee uncoupled by assuming an isothemal velocity pofile. The vaiational fom fo the tempeatue was then solved numeically. Isothemal bounday conditions wee imposed at the physical boundaies and zeo heat loss imposed at the ai-liquid inteface. Thei analysis showed that viscous heating can lead to obsevable eos in the cone-and-plate viscomete. In a subsequent pape, Tuian and Bid [4] extended the theoetical investigation to plane Couette flow of Newtonian fluids with tempeatue dependent viscosity and themal conductivity. The themal conductivity was assumed Depatment of Mathematical Sciences, Univesity of Delawae, DE 976. olagunju@math.udel.edu.

2 to be a linea function of tempeatue while the viscosity was assumed to obey a Nahme law with exponential dependence on tempeatue. A egula petubation solution in powes of the Binkman numbe was obtained fo the velocity and tempeatue. A petubation solution in powes of Binkman numbe wee late obtained fo non-newtonian liquids descibed by the powe-law and Ellis models [5]. The bounday conditions wee the same as those used in [4]. Exact analytical solutions have also been obtained by Matin [6] fo flows between infinite concentic cylindes and infinite paallel plates fo Newtonian and powe law fluids. Two types of bounday conditions wee consideed, one in which both sufaces wee isothemal and the othe in which one was isothemal and the othe adiabatic. Closed fom solutions simila to those of Nahme and Keasley wee obtained fo isothemal and adiabatic bounday conditions by Gavis and Lauence [7]. These ealy analytical studied wee motivated by the need to quantify the deviation fom isothemal flow in viscometes when viscous dissipation is significant and to povide simple fomulas to coect fo such deviations in quantities such as the toque on the stationay plate in paallel-plate flow. In ecent yeas enewed inteest in this matte has been spued by expeiments in elastic instabilities of viscoelastic fluids. It has been obseved that viscous heating could lead to qualitatively and quantitatively significant deviation fom isothemal theoy in the stability popety of a viscoelastic fluid. Expeimental study of the stability of isothemal flow of viscoelastic tosional flow was fist epoted by Lason et al [8] and McKinley et al [9]. Linea stability analysis was fist caied out by Oztekin and Bown [0] fo flow between paallel-plate flow in which the bounday condition at the fluid-ai inteface was neglected. Linea stability esults fo flow in a finite domain incopoating bounday conditions at the fee suface have also been consideed [, 2, 3]. Expeiments on the effect of viscous heating on the stability of tosional flow of viscoelastic fluids was fist epoted by Rothstein and McKinley [4]. The esults wee found to be emakably diffeent fom those of isothemal flows. It was shown that viscous heating tended to stabilize the flow. A linea stability fo the non-isothemal poblem was late analyzed by Olagunju et al [5] which agees qualitatively with thei expeimental esults. In that pape isothemal bounday conditions on the plates wee used while those at the fee suface wee neglected. Howeve, as noted by Aigo [6], it is pactically impossible to contol the tempeatue on uppe otating cone (o plate in case of paallel plate tosional viscomete). He also notes that the uppe otating cone (o plate) is cooled by convection of ambient ai at Celsius. A moe ealistic set of bounday conditions is to teat the bottom plate as isothemal, the fee suface as an insulated bounday and the top plate as a themal mass. The insulation bounday condition suface can be justified on the gounds that fo a small gap thickness, the suface available fo heat tansfe to the ambience though the adial inteface is pactically negligible. It is hoped that this will give esults that ae in quantitative ageement with expeiments. A heat tansfe bounday condition was used fo the viscoelastic Taylo-Couette poblem by Al-Mubaiyedh et al [7]. In thei study, the heat tansfe bounday condition was used to numeically simulate the expeiments of Baumet and Mulle [8, 9]. Anothe assumption that was made in [5] is that the paallel plates ae infinite in extent. In ode to obtain bette ageement between theoy and expeiments we think that it is necessay to elax these assumptions. As a fist step in this diection we study the effect of the finite geomety and the moe ealistic bounday conditions on the base flow. In [20], Olagunju showed that fo tosional flow of a viscoelastic fluid the base flow is not always puely cicumfeential. He showed that viscous heating leads to seconday flows with eciculating oll cells in the base solution. In this pape, we obtain petubation and numeical solutions fo the flow between two 2

3 paallel plates of a Newtonian fluid with tempeatue dependent viscosity. Specifically we will assume an exponential dependence of the Nahme type. As noted above this poblem has been solved fo flow between two infinite paallel plates [, 4]. In this limit the poblem educes to two coupled odinay diffeential equations fo the tempeatue and azimuthal velocity. We popose to solve the poblem in a finite geomety with a fluid-ai inteface. In this case we obtain two coupled patial diffeential equations fo the tempeatue and velocity. Bid and Tuian [5] have also analyzed the poblem in a finite geomety between a cone and a plate. Howeve they assumed that the velocity pofile was isothemal theeby educing the poblem to a single patial diffeential equation fo the tempeatue. In all pevious wok that we know the bounday conditions ae eithe isothemal on both plates o isothemal on one plate and zeo heat tansfe on the othe. We will adopt the moe ealistic bounday conditions descibed above, namely isothemal condition on the stationay plate, heat tansfe on the uppe plate and zeo heat tansfe at fluid-ai inteface. To the best of ou knowledge exact analytical solutions fo this poblem have not been peviously epoted. Having an analytical solution fo this poblem will enable one to estimate eos in the toque calculations due to heat tansfe and edge effects if needed. This is impotant in viscomety. Analytical solutions can also be used to validate numeical calculations. Fo viscoelastic flows in which seconday flow in the base flow is weak o non-existent the solution povided hee povides an accuate appoximation to the base flow needed in any linea stability analysis. 2 Govening equations We conside the flow of a fluid in the egion between two coaxial paallel plates of adius a and sepaation h in which the top plate otates at a constant angula speed ω and the bottom plate is stationay. Following Olagunju [2], the nondimensionalized equations govening the pimay flow fo the azimuthal velocity W and a scaled tempeatue Θ ae given in cylindical coodinates as, 2 W ( 2 W 2 2 ( Θ Θ with bounday conditions, + W W ) 2 = Θ W + 2 Θ ( W W ) [ ( W ) Θ 2 ( W = Na 0 e Θ + 2 W ) ) 2 ] at z = 0, W = 0, Θ = ϑ w (3) Θ at z =, W =, + B Θ = Bϑ a (4) at = 0, W = 0, Θ < (5) W at =, W = 0, Θ = 0 (6) Hee ϑ w and ϑ a ae the scaled tempeatue at the stationay plate and the ambient. The aspect atio = h/a and the modified Biot numbe B is defined in Appendix A.. The Nahme-Giffith numbe Na 0 which is a measue of viscous heating in the fluid is zeo fo isothemal flows. It is defined as, Na 0 (η 0 δa 2 ω 2 )/(kt 0 ). The quantity η 0 is the isothemal viscosity, δ is a themal sensitivity paamete, k the themal conductivity and T 0 a efeence () (2) 3

4 tempeatue [4]. Since the equations ae nonlinea finding an exact analytical solution valid fo all paamete values is impactical. Theefoe we will solve the equations numeically using a finite element method. We will also obtain analytical solutions using petubation expansions in Nahme-Giffith numbe Na and the aspect ation. 3 Analytical solutions 3. An exact solution fo = 0 and B = 0 An exact solution of equations ()-(6) can be found fo = 0, B = 0 and all values of Na 0. This coesponds to plane Couette flow with the uppe plate insulated [5]. This solution does not satisfy the bounday conditions at the fluid-ai inteface = 0. Howeve, we will show that it povides an excellent appoximation to the solution fo small except vey close to =. In addition we will show that the toque exeted by the fluid on the lowe plate is vey well appoximated by this exact solution when the aspect atio is small. An analytical fomula is povided which can be used to calculate non-isothemal coection to the toque, as is often equied in heomety. Note that the exact solution obtained by Nahme [] coesponds to = 0 and B =. Setting and B to zeo, equation () - (6) educe to the following. d 2 W dz 2 = dθ dw dz dz (7) The bounday conditions ae d 2 Θ dz 2 = Na 0e Θ ( dw dz ) 2 (8) W = 0, Θ = ϑ w, fo z = 0 (9) and Θ W =, = 0, fo z =. (0) It is staightfowad to obtain the solution which is given by whee Na = Na 0 e ϑw, and W = 2 tanh[e( 2z)], µna () [ ] Θ = ϑ w + ln ( + 2 Na 8 )sech2 [E( 2z)] (2) µ = E = tanh (µna/2) ] [2Na( Na) 2. The dimensionless toque on the lowe plate is defined ( ) dw T = 2 ddz. 0 0 dz z=0 4

5 Using the solution obtained above, a seies solution fo T valid fo Na < 2 is given by T = n=0 k=0 ( ) k ( Na 2 )n+k (n + k)! n!k! (2n + )(2n + 2k + 4) (3) Fo othe values of Na the integal can easily be computed numeically. These solutions will be compaed to asymptotic and numeical solutions below. 3.2 Asymptotic solution fo Na : ϑ w = ϑ a We seek a egula expansion in Nahme numbe fo W and Θ as follows. W = W 0 + Na W + O(Na 2 ), Θ = Θ 0 + Na Θ + O(Na 2 ) (4) Hee and in what follows Na = Na 0 e ϑw. The govening equations fo W 0 and Θ 0 ae, 2 ( W W W 0 W ) 0 2 = Θ 0 W 0 + Θ ( 2 0 W0 W ) 0 (5) 2 ( Θ Θ ) Θ 0 = 0 (6) with the bounday conditions, at z = 0, W 0 = 0, Θ 0 = ϑ w (7) Θ 0 at z =, W 0 =, + B Θ 0 = B ϑ w (8) at = 0, W 0 = 0, Θ 0 < (9) W 0 at =, W 0 = 0, Θ 0 = 0 (20) The leading ode solution fo Na = 0 gives the isothemal solution W 0 = z, (2) Θ 0 = ϑ w (22) Note that this solution is valid fo all values of The solution at ode Na coesponding the fist non-isothemal coection satisfies the following equations. 2 ( W W 2 + W W ) 2 = Θ (23) 2 ( Θ Θ ) Θ = 2 (24) with bounday conditions, at z = 0, W = 0, Θ = 0 (25) at z =, W = 0, Θ + BΘ = 0 (26) 5

6 at = 0, W = 0, Θ < (27) W at =, W = 0, Θ = 0 (28) Note that equations (23) - (24) ae now uncoupled. The equations can be solved exactly by sepaation of vaiables as follows. and Θ = Γ n λ 2 n= n [ ( 2 3 I λn ) ] 0 ( λ n I λn ) sin(λ n z) (29) W = λ 2 n F m () sin(mπz). (30) m= whee λ n, n =, 2, ae positive solutions of the tanscendental equation tan(λ n ) + λ n B = 0, (3) and Γ n = 2 4( cos(λ n )) 2λ n sin(2λ n ). This equation has infinitely many positive oots. Hee I n is the modified Bessel function of the fist kind. This solution is also valid fo all values of the aspect atio. The expession fo F m () involves complicated integals of Bessel functions (see the Appendix fo details). 3.3 Asymptotic solution fo Na : ϑ w ϑ a The govening equations fo W 0 and Θ 0, W and Θ ae the same as in the pevious section. At zeoth ode in Na we have the isothemal solution The equations at ode Na ae Θ 0 = χz + ϑ w whee, χ B(ϑ a ϑ w ) + B (32) ( e χz ) W 0 = e χ (33) 2 ( W W 2 + W W ) 2 = χ eχz e χ Θ + χ W (34) 2 Θ 2 with bounday conditions, + 2 ( 2 Θ 2 + Θ ) = χ2 2 e χz (e χ ) 2 (35) on z = 0, W = 0, Θ = 0 (36) Θ on z =, W = 0, + BΘ = 0 (37) on = 0, W = 0, Θ < (38) on =, W W = 0, Θ = 0 (39) 6

7 Although these equations can also be solved exactly the solutions ae athe too complicated. Theefoe, we will obtain the solution as an expansion in. This limit has applications in heometic devices whee is typically less than 0.. Fo the case = O() the solution will be computed numeically. Because the limit 0 is singula we use the method of matched asymptotic expansion. Thus we seek an oute solution, an inne solution and then obtain a composite expansion fo the solution by matching. Oute Solution Fo the oute solution we expand as follows. W = W o 0 + W o + O( 2 ), Θ = Θ o 0 + Θ o + O( 2 ) (40) whee the supescipt (o) efes to the oute solution. The govening equations at zeoth ode in ae, 2 W o 0 2 = χ eχz e χ Θ o 0 o + χ W 0 (4) 2 Θ o 0 2 with the coesponding bounday conditions, = χ2 2 e χz (e χ ) 2 (42) at z = 0, W o 0 = 0, Θ o 0 = 0 (43) at z =, W0 o Θ o 0 = 0, + BΘ 0 = 0 (44) The solution satisfying the above govening equations and the bounday conditions ae, Θ o 2 [ 0 = (e χ ) 2 e χz + z (χ + ] B)eχ B (45) + B W o 0 = χ 3 ] [ e2χz (e χ ) 3 2χ + ((χ + B)eχ B) eχz (χz ) ( + B)χ 2 χeχz 3 [ ] e 2χ (e χ ) 4 + ((χ + B)e χ + (χ )eχ B) 2χ ( + B)χ 2 3 [ e 2χ e χ (e χ ) 4 eχ ((χ + B)e χ ] B) 2 + B (46) (47) Futhe, it is also detemined that, W o = 0 Θ o = 0. (48) Inne solution Fo the inne expansion we intoduce the stetched vaiable, ξ, and seek an expansion of the fom W i = W0 i + W i + O( 2 ), Θ i = Θ i 0 + Θ i + O( 2 ). (49) 7

8 The govening equations and the bounday conditions at zeoth ode in ae, 2 W0 i W0 i ξ 2 χ W 0 i = χeχz (e χ ) ( ) Θ i 0 (50) 2 Θ i Θ i 0 ξ 2 = χ2 e χz (e χ ) 2 (5) at z = 0, W i 0 = 0, Θ i 0 = 0 (52) at z =, W i 0 = 0, Θ i 0 + BΘi 0 = 0 (53) at ξ = 0, W i 0 ξ = 0, It is staightfowad to obtain the following expessions, Θ i 0 = (e χ ) 2 Θ i 0 ξ [ e χz + z (χ + B)eχ B + B = 0 (54) ] (55) W i 0 = ] χ [ e2χz (e χ ) 3 2χ + ((χ + B)eχ B) eχz (χz ) ( + B)χ 2 [ ] χeχz e 2χ (e χ ) 4 + ((χ + B)e χ + (χ )eχ B) 2χ ( + B)χ 2 [ e 2χ e χ (e χ ) 4 eχ ((χ + B)e χ ] B) 2 + B (56) (57) The ode equations ae 2 Θ i Θ i ξ 2 2 W i W i ξ 2 χ W i = W i 0 = 2ξχ2 e χz (e χ ) 2 (58) ( ) Θ i 0 + W i ( ) 00 Θ i (59) The bounday conditions ae the same as above. The solution of the equations is Θ i = 2ξΘ i 0 2 n= Γ n e λn( ) λ 3 n sin(λ n z), (60) W i = 3ξW i m= n= m= 2Ãme Λmξ e χz 2 Λ 2 m Γ n Bmn λ 2 n(λ 2 m λ 2 n) sin(mπz) (e λ nξ λ ne Λ mξ Λ m ) e χz 2 sin(mπz), (6) whee, 8

9 and, Γ n = χ2 0 eχz sin(λ n z)dz (e χ ) 2 0 sin2 (λ n z)dz (62) Λ 2 m = χ2 4 + m2 π 2 Ã m = B mn = χ (e χ ) 3 2χ e χ 0 0 e χz ( χe χz + (χ+b)eχ B +B 0 sin2 (mπz)dz 2 cos(λ n z) sin(mπz)dz 0 sin2 (mπz)dz ) e χz 2 sin(mπz)dz (63) Composite solution In ode to use the Van Dyke s matching pinciple, the oute solution fo the velocity and the tempeatue distibution is expessed in tems of the inne vaiable including tems of the fist ode in, (Θ o ) i = χz + ϑ w + Na( 2ξ)Θ i 0 + O(Na 2 ) (64) (W o ) i = ( e χz ) e χ + Na( 3ξ)W0 i + O(Na 2 ) (65) The Van Dyke s matching pinciple is expessed using the fomulas, W c = W o + W i (W o ) i, Θ c = Θ o + Θ i (Θ o ) i (66) The composite solution fo tempeatue and velocity distibution is then given by, ( Θ c = χz + ϑ w + Na 2 (e χ ) 2 2Na n= [ e χz + z (χ + ]) B)eχ B + B Γ n e λ n( ) λ 3 n sin(λ n z) + O(Na 2 ) (67) W c = ( e χz ) e χ + NaW o + Na +Na m= n= Γ n Bmn λ 2 n(λ 2 m λ 2 n) 2Ãme m= ( ( ) λn e ( ) Λm Λ 2 m e χz 2 λ ne Λ ( ) m Λ m sin(mπz) ) e χz 2 sin mπz (68) 9

10 4 Numeical solution The domain Ω fo numeical computation is 0 < z < and 0 < <, shown in Figue. In ode to apply the finite element method, we need to ewite the two PDEs in vaiational foms. We multiply the continuity equation () by V (, z), cf. [22, 24, 25, 26], the test function with bounday conditions specified in Figue. Then we apply the integation by pats to obtain ( W V W V + 2 Ω + W V ) 2 2 ddz 2 W V dz = ( = Θ ( W Θ W 2 W )) V ddz. (69) Ω We do the same fo the second equation (2), with a test function V (, z), but of diffeent bounday conditions shown in Figue. ( Θ V Θ V ) + 2 ddz (Bϑ a BΘ)V d Ω z= ( ( W ) 2 ( W = Na e Θ + 2 W ) ) 2 V ddz. (70) Ω z (fo W equation (69)) V = 0 z (fo Θ equation (70)) V fee V = 0 V fee V fee V fee V = 0 V = 0 Figue : Bounday conditions fo the test functions V (fo W and Θ in (69) (70)). To obtain homogeneous bounday conditions fo the vaiational poblems (69) (70), we use the following decompositions W = W b + W 0, W b = z, (7) Θ = Θ b + Θ 0, Θ b B = ϑ w + z + B (ϑ a ϑ w ). (72) We seek solutions W 0 and Θ 0 instead, which have homogeneous bounday conditions, also depicted in Figue 2, W 0 W 0 =0,z=0,z= = 0, = W 0 =, Θ 0 Θ 0 z=0 = 0, Θ 0 = 0, =0,= = BΘ 0. z= That is, we will find finite element solutions Wh 0 and Θ0 h whee h stands fo the gid size. To discetize (69) and (70), due to the special domain of the unit squae, one may use spectal methods (cf. [22]) o tenso poduct methods (cf. [23]) to get a high ode appoximation. 0

11 z W b = z W 0 = 0 W b = 0 W b = z W b W b = 0 W 0 = 0 W 0 W 0 = 0 z W b = 0 Θ b + BΘb = Bϑ a W 0 = 0 z Θ 0 + BΘ0 = 0 Θ b = 0 Θ b = 0 Θ 0 = 0 Θ 0 = 0 Θ b = ϑ w Θ 0 = 0 Figue 2: Bounday conditions fo W 0 and Θ 0 in (7) (72). To handle the nonlineaity of the coupled system, and to handle possible iegula domains in futue, we use Q k finite elements, continuous and piecewise polynomials of sepaate degee k o less, on unifom gids K h = {K K = [ i h, i ] [z j h, z j ], i, j =,..., /h} of Ω: Q h := V C(Ω) V K = 0 i,j k a ij i z j, K K h H (Ω). We use the following notations fo the discete spaces with homogeneous bounday conditions: Q h,w := Q h {V = V (, z) C(Ω) V (0, z) = V (, 0) = V (, ) = 0}, (73) Q h,θ := Q h {V = V (, z) C(Ω) V (, 0) = 0}. (74) The finite element discetizations of (69) (70) ead: Find (W 0 h, Θ0 h ) Q h,w Q h,θ such that A W (W 0 h, V ) = F W,Θ(V ) A W (W b, V ) V Q h,w, (75) A Θ (Θ 0 h, V ) = G W,Θ(V ) A Θ (Θ b, V ) + Bc z (ϑ a, V ), V Q h,θ. (76)

12 whee the bilinea foms and functionals ae defined by A W (U, V ) = a(u, V ) + 2 ( U, V ) 2 c (U, V ), (77) A Θ (U, V ) = a(u, V ) + Bc z (U, V ), (78) F W,Θ (V ) = ( Θ ( W Θ W 2 W ), V ), (79) ( [ ( W ) 2 ( W G W,Θ (V ) = Na e Θ + 2 W ) ] ) 2, V. (80) ( U V U V ) a(u, V ) = + 2 ddz, (8) Ω (U, V ) = UV ddz, (82) Ω c (U, V ) = UV dz, (83) =,0 z c z (U, V ) = UV dz. (84) z=,0 We solve the nonlinea system of equations (75) (76) numeically by a staightfowad Seidel iteation. That is, given initially some guesses (both zeo in computation) of Wh 0 and Θ 0 h, we geneate the ight hand side of (75) and use the conjugate gadient method to solve (75) to get a new Wh 0. Then the new W h 0 and the old Θ0 h would be used to geneate the ight hand side vecto in (76). We solve (76) again by the conjugate gadient method to get a new Θ 0 h. The next lemma shows that the two linea systems at each step descibed above ae uniquely solvable, because both the coefficient matices ae symmetic and positive definite. Lemma 4. Fo any V Q h,θ Q h,w and V 0, a(v, V ) > 0. (85) Fo any V Q h,w and V 0, Fo any V Q h,θ, V 0, and B 0, A W (V, V ) > 0. (86) A Θ (V, V ) > 0. (87) Poof. (85) and (86) ae shown in [27]. (87) is a coollay of (85), noting the sign of B is positive. Algoithm 4. The coupled nonlinea system (75) (76) is solved by the Seidel iteation with the given initial guess Wh,0 0 = 0 and W Θ,0 0 = 0. Fo j =, 2,..., W 0 h,j = W 0 h,j + e W, whee e W solves the equation and A W (e W, V ) = F Wj,Θ j (V ) A W (W b, V ) A W (W 0 h,j, V ) V Q h,w, (88) 2

13 Θ 0 h,j = Θ0 h,j + e Θ, whee e Θ solves the equation A Θ (e Θ, V ) = G Wj,Θ j (V ) A Θ (Θ b, V ) + Bc z (ϑ a, V ) A Θ (Θ 0 h,j, V ) V Q h,θ. (89) Hee W j = W b + W 0 h,j and Θ j = Θ b + Θ 0 h,j fo j = 0,, 2,.... A typical pai of solutions (W h, Θ h ) is shown in Figue 3. z Figue 3: Solutions W and Θ fo () and (2) when =.0, Na =, ϑ w =.5, ϑ a = B = 0.. At z=0.5, fo χ=0, Na=0., B= (a) Numeical solution =0.25, (b) Asymptotic solution =0.25, (c) Numeical solution =0., (d) Asymptotic solution =0. (e) Exact solution(=0) Θ Θ Figue 4: Solutions obtained by () (2), (29) (30), and (75) (76). 3

14 8 x At =0.5, fo χ=0, Na=0., B=0 (a) Numeical solution =0.25, (b) Asymptotic solution =0.25, (c) Numeical solution =0., (d) Asymptotic solution =0. (e) Exact solution(=0) 0 Θ Θ z Figue 5: Solutions obtained by () (2), (29) (30), and (75) (76). x At z=0.5, fo χ=0, Na=0., B=0 2 W W (a) Numeical solution =0. (b) Asymptotic solution =0. (c) Exact solution(=0) Figue 6: Solutions obtained by () (2), (29) (30), and (75) (76). 4

15 0.08 At =.5 fo =0., θ a = θ w =.5 B= Θ Θ (a) Numeical solution Na=0. (b) Asymptotic solution Na=0. (c) Numeical solution Na= (b) Asymptotic solution Na= z Figue 7: Solutions obtained by () (2), (29) (30), and (75) (76). 5

16 5 Discussion In this section we compae the analytical solutions obtained in section 3 with the finite element solution obtained in section 4. The exact solution given in section 3. equations ()-(2)is valid fo = 0, B = 0 and all values of Na, the petubation solution given in section 3.2 equations (29)-(30) is valid fo all values of and small Na while the solution found in section 3.3 equations (67)-(68) is valid only fo small and Na. The numeical solution on the othe hand is valid fo all paamete values. 3 x 0 5 At =0.5 fo Na=0., θ a = θ w =.5 B=00 =0. 2 W W (a) Numeical solution (b) Asymptotic solution z Figue 8: Solutions obtained by () (2), (29) (30), and (75) (76). The plots in Figs 4-8 depict the deviation of the tempeatue and velocity fom the isothemal solution. We plot Θ Θ 0 and W W 0 whee Θ 0 and W 0 ae the isothemal solutions. Figs. 4 and 5 show the deviation of tempeatue at z = 0.5 and = 0.5 espectively fo Na = 0., B = 0 and selected values of. The case B = 0 coesponds to insulated bounday condition on the uppe plate. Fo = 0. all thee solutions agee vey well except nea the fee suface =. The eo in the exact solution fo = 0 aises because it does not satisfy the bounday condition at the fee suface. The eo between the numeical and asymptotic solutions is othewise vey small. Figue 6 shows the deviation of the velocity fo same values of the paametes. The ageement among all thee solutions is again vey good. In Figue 7, the deviation in tempeatue is shown fo = 0,, B =, θ w =.0, θ a =.5 fo two values of Na. While the ageement between numeical and asymptotic solutions is excellent fo Na = 0., thee is a small discepancy fo the case Na =.0. This is actually quite good since the petubation expansion was tuncated at ode O(Na). Figue 8 shows the deviation of the velocity fo a lage value of the Biot numbe B. In this limit the two plates ae nealy isothemal and we see a qualitatively diffeence fom the solution fo B = O(). Specifically, the pofile is symmetic about the mid-plane z = 0.5 6

17 0.25 (a) Numeical solution (=0., B=0.0) (a) Numeical solution (=, B=0.0) (b) Exact solution(=0) (b) Seies solution(=0) 0.2 τ Na Figue 9: Plot of the toque T vs Na. wheeas fo smalle values of B the pofile is asymmetic. Anothe qualitative diffeence is the location of the maximum tempeatue. On any fixed plane the maximum occus at the fee suface = when B = 0. As B inceases the location of the maximum moves away fom the fee suface. Fom these figues we also see that the deviation in of tempeatue velocity fom isothemal is ode O(Na) when Na is small. Lastly, in Figue 9 we plot the toque on the lowe stationay plate as a function of the Nahme numbe Na fo selected values of and B. Although the exact solution is valid only fo = 0 and B = 0 the ageement with the numeical solution fo = 0. and B = 0. is excellent. Thus, in applications in which the aspect atio is small the exact solution can be used to obtain vey accuate coections to the toque in viscometic applications. We also show the seies epesentation fo the toque equation (3) and the ageement is excellent fo Na < 2. 6 Summay Non-isothemal tosional flow with heat tansfe bounday condition at the uppe otating plate, isothemal bounday condition at the lowe stationay plate, and insulated bounday condition at the fluid/ai inteface has been analyzed. It is assumed that viscosity in an exponential function of tempeatue. We have obtained analytical solutions valid in the limit of small aspect atio and in the limit of small Nahme-Giffith numbe Na. The nonlinea coupled patial diffeential equations have also been solved numeically using the finite element method. Ou esults show that the asymptotic solutions agee vey well with the numeical solution. Fo small vales of Na the deviation of tempeatue and velocity fom the isothemal solution is small appoximately of ode O(Na). Futhemoe, we show that fo viscometic applications in which the aspect atio is typically less than 0., the exact solution obtained 7

18 fo = 0 and B = 0 gives vey accuate esults fo the non-isothemal coection to the toque fo small values of and the Biot numbe B. A Appendix A. Heat tansfe bounday condition In this section, the deivation of the heat tansfe bounday condition at the uppe otating plate is detailed. Following the convention adopted in Ozisik [29], k T ( ) z = h T Ta, at the suface S (90) whee, h is the heat tansfe coefficient and k is the themal conductivity of the otating plate. The suface S coesponds to the suface of the uppe otating plate at z =. Afte nomalization of vaiables and intoducing the thickness of the plate H, to pocue a meaningful paamete, the Biot numbe Bi = hh k. T = Bih H ( T + T a) (9) whee, h is the thickness of the gap between the two plates [28, 29]. In dimensionless fom this becomes whee, B Bi h H. Θ + BΘ = Bϑ a (92) A.2 The function F m () The function F m () appeaing in equation (30) satisfies the following odinay diffeential equation ϕ = 2 3 n= 2 F m + F m ( + m2 π 2 2 ) 2 F m = ϕ() [ ] [ mπ Γn ( cos(mπ) cos(λ n )) 2 I 0 ( λn ) ] m 2 π 2 λ 2 n λ 2 n I ( λ ) 2 n λ 2 42 n λ 4 n (93) The geneal solution to the above odinay diffeential equation is, ( mπ ) ( mπ ) ( mπ ) [ ϕ()k ( mπ F m () = C I + C 2 K + I ) ] d ( mπ ) [ ϕ()i ( mπ K ) ] d, (94) whee K 0 and K ae modified Bessel functions of the second kind. The evaluation of the integals ae shown next. 8

19 ϕ()k ( mπ ) ϕ()i ( mπ d = 2 n= mπ Γ n ( cos(mπ) cos(λ n )) λ n (m 2 π 2 λ 2 n) [ 4mπ 3 λ n (m 2 π 2 λ 2 n) 2 λ ni ( λ n )K 0( mπ ) + mπi 0( λ n )K ( mπ ) I ( λn ) λ n (m 2 π 2 λ 2 n) ) λ ni ( λ n )K ( mπ ) + mπi 0( λ n )K 0( mπ ) I ( λn ) + 4 mπ K 2( mπ ) m 2 π 2 K 3( mπ ) mπλ 2 K 2 ( mπ ] n ) d = 2 n= mπ Γ n ( cos(mπ) cos(λ n )) λ n (m 2 π 2 λ 2 n) [ 4mπ 3 λ n (m 2 π 2 λ 2 n) 2 λ ni 0 ( mπ )I ( λn ) mπi 0( λn )I ( mπ ) I ( λ n ) λ n (m 2 π 2 λ 2 n) λ ni ( mπ I ( λn ) mπi 0( λn )I 0) mπ ) I ( λ 4 n ) mπ I 2( mπ ) ] m 2 π 2 I 3( mπ ) 43 2 mπλ 2 n I 2 ( mπ ) (95) (96) The bounday conditions, F m = 0 at = 0, and F m Fm = 0 at = ae used to detemine the constants in the above equation. The bounday condition on the axis causes the constant of integation C 2 to be zeo. The othe bounday condition is used to detemine the constant of integation C. Howeve, because of the complexity of the natue of the solution as judged fom the above equations, the constant C is shown detemined in pinciple. Howeve, the actual fomulation is employed to geneate plots via CAS (Compute Algeba System). [ ϕ()k ( mπ C = ) ] d = K 0( mπ ) + K 2( mπ ) 2 mπ K ( mπ ) I 0 ( mπ ) + I 2( mπ ) 2 mπ I ( mπ ) [ ϕ()i ( mπ ) ] d = (97) Refeences [] R. Nahme, Ing. Ach. (940) [2] E. A. Keasley, The viscous heating coection fo viscometic flows, Tans Soc. Rheol. 6 (962) [3] R. B. Bid and R. Tuian, Viscous heating effects in a cone and plate viscomete, Chemical Engineeing Science,vol.7, (962) [4] R. M. Tuian and R. B. Bid, Viscous heating in the cone-and-plate viscomete-ii. Newtonian fluids with tempeatue-dependent viscosity and themal conductivity, Chem. Eng. Sci. 20 (965)

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