Free convection flow of Couple Stress fluid between parallel Disks

Transcription

1 International Conference on Fluid Dynaics and Therodynaics Technologies (FDTT ) IPCSIT vol.33() () IACSIT Press, Singapore Free convection flow of Couple Stress fluid between parallel Disks D. Srinivasacharya * and K. Kaladhar Departent of Matheatics, National Institute of Technology, Warangal-564, Andhra Pradesh, India *dsrinivasacharya@yahoo.co, dsc@nitw.ac.in Abstract. The steady incopressible couple stress fluid flow between parallel disks is considered. The lower and upper disks rotate with the sae angular velocity Ω. Due to the rotating frae of reference, Coriolis and centrifugal forces appear explicitly in the oentu equation. The governing non-linear partial differential equations are transfored by a siilarity transforation into a syste of ordinary differential equations. The resulting equations are solved using Hootopy Analysis Method (HAM). The nondiensional velocity and teperature profiles are displayed graphically for different values of the Prandtl nuber and couple stress fluid paraeter. Keywords: Natural convection, Couple stress fluid, HAM.. Introduction Rotating-disk flow has long been an iportant topic in fluid dynaic research for the interests in practical as well as acadeic senses. Karan [] introduced an ingenious siilarity transforation to study the axisyetric flow induced by a single rotating disk. Batchelor [] showed that this transforation can be used even when the fluid is confined between two parallel disks rotating about a coon axis at different speeds. Since then, several researchers considered the flow between two disks rotating about a coon axis with the sae angular velocity or different angular velocities. Recently, Batista [3] obtained analytical solution of the Navier Stokes equations for the case of the steady flow of an incopressible fluid between two uniforly co-rotating disks. On the other hand, flow and Heat transfer associated with a rotating disk syste has attracted the attention of researchers for the past few decades due to its nuerous industrial and engineering applications of rotating achinery, lubrication, coputer storage devices and crystal growth processes. Recently, Jiji and Ganatos [4] considered steady lainar flow and heat transfer generated by two infinite parallel disks. Flow of non-newtonian fluids between rotating disks has also drawn attention in view of its applications in engineering practice. Different odels have been proposed to explain the behavior of non-newtonian fluids. Aong these, couple stress fluids introduced by Stokes [5] have distinct features, such as the presence of couple stresses, body couples and non-syetric stress tensor. The ain feature of couple stresses is to introduce a size dependent effect. Classical continuu echanics neglects the size effect of aterial particles within the continua. This is consistent with ignoring the rotational interaction aong particles, which results in syetry of the force-stress tensor. The study of couple-stress fluids has applications in a nuber of processes that occur in industry such as the extrusion of polyer fluids, solidification of liquid crystals, cooling of etallic plate in a bath, and colloidal solutions etc. A review of couple stress (polar) fluid dynaics was reported by Stokes [6]. Recently, Srinivasacharya and Kaladhar [7] presented analytical solution of cheically reacting free convective couple stress fluid in an annulus with Soret and Dufour effects. 3

2 The hootopy analysis ethod [8] was first proposed by Liao in 99, is one of the ost efficient ethods in solving different types of nonlinear equations such as coupled, decoupled, hoogeneous and non- hoogeneous. Also, HAM provides us a great freedo to choose different base functions to express solutions of a nonlinear proble [9]. The application of the Hootopy Analysis Method (HAM) in engineering probles is highly considered by scientists, because HAM provides us with a convenient way to control the convergence of approxiation series, which is a fundaental qualitative difference in analysis between HAM and other ethods. Recent developents of HAM, like convergence of HAM solution, Optiality of convergence control paraeter discussed by Srinivasacharya and Kaladhar [] for couple stress fluids. In the present work the free convection in couple stress fluid with heat transfer flow between two infinite rotating disks is studied. The Hootopy Analysis ethod is eployed to solve the governing nonlinear equations. Convergence of the derived series solution is analyzed. The behavior of eerging flow paraeters on the velocity and teperature is discussed.. Forulation of the Proble Consider the steady axisyetric flow of a incopressible couple stress fluid between two parallel disks separated by a distance d. Choose the cylindrical coordinates (R, ϕ, Z) with the origin at the center of the lower disk. The lower disk Z = is aintained at unifor constant teperature T and concentration C, while the upper disk Z = d at unifor constant teperature T and concentration C. Assue that the lower and upper disks are rotating with the sae angular velocity Ω. With respect to the rotating frae of reference, Coriolis and centrifugal forces appear explicitly in the oentu equation. The fluid has constant physical properties and the Boussinesq approxiation is invoked for study of the centrifugal buoyancy effects. In the present study, the wall condition at lower disk is used as the reference state, at which the fluid confined by the disks lies at the teperature T r = T and rotates with the reference frae as a solid body, therefore, U = V = W = and P r /ρ r = Ω Ω R + g. The velocity coponents in the R, ϕ and Z directions are U, V and W, respectively. Considering the lainar density teperature relation ρ = ρ r [ β(tt r )], the governing equations can be depicted as: q = () q. ) q+ Ω [ β( T Tr )]( ez q) + Ω Rβ( T Tr ) er gβ ( T Tr ) ez = P */ ρr ν q η / ρr q () ( ( q. ) T = α T (3) in which P* = P P r is the pressure departure fro the reference condition, Ω = Ω ē z, R = R ē r, and ē z and ē r are the unit vectors in the axial and radial directions, respectively. In the present analysis, the following diensionless variables are used, Z = ηd, U = RΩF(, V = RΩ, W = νω H (, T T = ΔT θ where ΔT = T T is the characteristic teperature difference. G, H, and θ are the tangential, axial velocities and the teperature function, respectively. The transforation is essentially of von Karan type but with an additional treatent to the teperature function in the energy equation. The governing equations ()-(3) can thus be cast into the following diensionless for: vi iv 3/ / 3/ S H H + 4Re [( + G) G B( Gθ + G θ )] + Re H H B Re θ = (4) iv / S G + Re [ HG H G H + BH θ ] G = (5) / θ Pr Re H θ = (6) in which the continuity equation H = Re / F (7) has been introduced to siplify the syste by eliinating the radial velocity function F(.Here Pr = ν/α is the Prandtl nuber,. Re = d Ω/ ν is the Reynolds nuber, B = β ΔT is the theral Rossby nuber and 4

3 S = η /μd is the Couple stress paraeter, The effects of couple-stress are significant for large values of S (= l/d), where l = η /μ is the aterial constant. The superscript ( ) denotes differentiation with respect to η, The boundary conditions are H() =, H () =, ) =, H() =, H () =, ) =, θ () =, θ () = (8a) H () =, H () =, ) =, ) = (8b) The boundary condition (8b) iplies that the couple stresses are zero at the Disc surfaces. 3. The HAM solution of the proble For HAM solutions, we choose the initial approxiations of H(, H( and θ( as follows: H η ) =, ( =, θ ( = η (9) ( G and choose the auxiliary linear operators: L = / η, L = / η and L = / η 3 () such that L ( c c η + c η + c η + c η + c η ) =, L ( c + c η + c η + c 3 η ) = L ( c + c η ) = () , 3 where c i (i =,,..., ) are constants. Introducing non-zero auxiliary paraeters h, h and h 3 we develop the zeroth-order deforation probles as follow: ( L [ H ( λ ; H ( λ) ] = ph N [ H ( λ; ] () ( L [ λ ; G ( λ) ] = ph N [ λ; ] (3) ( L [ θ ( λ; θ ( λ) ] = ph N [ θ ( λ; ] (4) subject to the boundary conditions H (; = H (; = ; = H (; = G (; =, θ (; = H (; = H (; = ; = H (; = G (; =, θ (; = where p [, ] is the ebedding paraeter and the non-linear operators N, N and N 3 are defined as: vi iv N [H( η;, η;, θ ( η; ] = S H H + 4 Re [( + G) G B( Gθ + G θ )] + Re H (6) 3 / B Re θ iv / N [H( η ;, η;, θ ( η;] = S G + Re [ HG H G H + BH θ ] G (7) / N 3 [H( η;, η;, θ ( η;] = θ Pr Re H θ (8) For p =, we have the initial guess approxiations H ( η ;) = H(, η;) =, θ ( η;) = θ( (9) When p =, equations () - (4) are sae as (4) - (6) respectively, therefore at p = we get the final solutions H ( η ;) = H (, η;) =, θ ( η;) = θ ( () Hence the process of giving an increent to p fro to is the process of H(η; varying continuously fro the initial guess H ( to the final solution H( (siilar for η, and θ(η,). This kind of continuous variation is called deforation in topology so that we call syste Eqs. () - (5), the zerothorder deforation equation. Next, the th -order deforation equations follow as H G θ L [ H ( χh ( ] = h R, L[ G ( χg( ] = hr, L3[ θ( χθ ( ] = h3r () with the boundary conditions H () = H () = ) = H () = G () = H () = H () = ) = H () = G () = () θ() = θ() = where H vi iv 3/ / R ( η ) = S H H + 4Re G + G ngn B ( Gnθ n + G nθ n ).5Bθ + Re H nh (3) n n= n= n= 3 / / (5) 5

4 R G iv / ( η ) = S G + Re ( GnH n H ngn + BH nθ n ) H G (4) n= R θ ( = θ Pr / Re H θ n n n = and, for being integer χ = for ; χ = for > (6) The initial guess approxiations H (),G ( and θ (, the linear operators L, L and L 3 and the auxiliary paraeters h, h and h 3 are assued to be selected such that equations () - (5) have solution at each point p [, ] and also with the help of Taylors series and due to eq. (9) H(η;,η; and θ(η; can be expressed as H ( η; = H ( + H ( p, η; = G ( + G ( p, θ ( η; = θ( + θ( = = = in which h, h and h 3 are chosen in such a way that the series (7) are convergent at p =. Therefore we have fro () that ( + H (, η; = G ( + G (, θ ( η; = θ( + θ( = = = H ( η ; = H (8) for which we presue that the initial guesses to H, G and θ the auxiliary linear operators L and the non-zero auxiliary paraeters h, h and h 3 are so properly selected that the deforation H(η;, η; and θ(η; are sooth enough and their th -order derivatives with respect to p in equations (8) exist and are given respectively by H ( η ) =! H ( η; p p=, G ( =! η; p p= θ ( =! θ ( η; It is clear that the convergence of Taylor series at p = is a prior assuption, whose justification is provided via a theore, so that the syste in (8) holds true. The forulae in (8) provide us with a direct relationship between the initial guesses and the exact solutions. Moreover, a special ephasize should be placed here that the th -order deforation syste ()-(4) is a linear differential equation syste with the auxiliary linear operators L whose fundaental solution is known. 4. Results and Discussion The expressions for H,G and θ contain the auxiliary paraeters h, h and h 3. As pointed out by Liao (3), the convergence and the rate of approxiation for the HAM solution strongly depend on the values of auxiliary paraeter h. For this purpose, h-curves are plotted by choosing h, h and h 3 in such a anner that the solutions (9)-() ensure convergence [9]. Here to see the adissible values of h, h and h 3, the h- curves are plotted for 5 th -order of approxiation in Figs.()-(4) by taking the values of the paraeters Re = 5, Pr =., S = and B =.5.. It is clearly noted fro Fig. that the range for the adissible values of h is.35 < h < -.6. Fro Fig.3, it can be seen that the h-curve has a parallel line segent that corresponds to a region.5 < h <.6. Fig.4 depicts that the adissible value of h 3 are.5< h 3 <.6. A wide valid zone is evident in these figures ensuring convergence of the series. The average residual errors are calculated at different order of approxiations () and found that they are iniu at h = -.3, h = - and h 3 = -.9 respectively. Therefore, the optiu values of convergence control paraeters are taken as h = -.3, h = -, h 3 = -.9. p p p= (5) (7) Fig. : h curve for H( Fig. 3: h curve for Fig. 4: h curve for θ( The solutions for F(,, H( and θ( have been coputed and shown graphically in Figs. 5 to. The effects of couple stress fluid paraeter (S) and Prandtl nuber have been discussed. To study the effects of S and Pr, coputations were carried out by taking B =.5. 6

5 Figures 5 to 8 indicate the effect of the Couple stress fluid paraeter S on F(,, H( and θ(. As the Couple stress fluid paraeter S increases, the radial velocity F(η, the tangential velocity and axial velocity H( decrease. It is also clear that the teperature θ( decreases with an increase in S. It can be noted that the velocity in case of couple stress fluid is less than that of a Newtonian fluid case. Thus, the presence of couple stresses in the fluid decreases the velocity and teperature Fig. 5: Effect of S on F at Pr =. Fig. 6: Effect of S on G at Pr =. Fig. 7: Effect of S on H at Pr =. Fig. 8: Effect of S on θ at Pr =. Fig. 9: Prandtl nuber Pr Effect on F at S=.5 Fig. : Prandtl nuber Pr Effect on F at S=.5 Fig. : Prandtl nuber Pr Effect on F at S=.5 Fig. : Prandtl nuber Pr Effect on θ at S=.5 In Figure 9 to with B =.5, the Prandtl nuber effect significantly alters the flow fields. For large Prandtl nuber, Pr =, the teperature function changes abruptly in the thin theral boundary layer but reain unifor in large portion of the wheel space. As Pr decreases fro to., the theral diffusion is getting ore and ore iportant and, then, the teperature variation appears notably in the whole doain rather than confined in a narrow region. For sall Pr, the teperature gradient near the disk, i.e. Z =, is alleviated. Due to coupling of the Coriolis induced buoyancy in circuferential fluid otion and the Prandtl nuber effects, salient Pr-dependence of the circuferential velocity is presented. 5. Conclusions In this paper, the fully developed couple stress fluid flow between two parallel Discs has been studied. The governing equations are expressed in the non-diensional for and are solved by using HAM. The features of flow characteristics are analyzed by plotting graphs and discussed in detail. The ain findings are suarized as follows: The presence of couple stresses in the fluid decreases the velocity and teperature. Increase in the Prandtl nuber leads to the increase in teperature and decrease in velocity. 6. References [] T. von Karan. Lainar und turbulente Reibung. ZAMM. 9, : [] G. K. Batchelor. Note on a class of solutions of the Navier-Stokes equations representing steady rotationallysyetric flow. The Quart. J. Mech. App. Maths. 95, 9 4. [3] M. Batista. Steady flow of incopressible fluid between two co-rotating disks. App. Math. Model., 35: [4] Latif M. Jiji, and P. Ganatos. Microscale flow and heat transfer between rotating disks. Int. J. of Heat and Fluid Flow., 3: 7-7 [5] V.K. Stokes. Couple Stresses in fluid. The phy of fluids. 966: [6] V.K. Stokes. Theories of Fluids with Microstructure: An Introduction. Springer Verlag, New York, 984. [7] D. Srinivasacharya, and K. Kaladhar. Analytical solution for cheically reacting free convective couple stress fluid in an annulus with Soret and Dufour effects. Int. Re. Mech Engineering., 5(5): [8] S.J. Liao. Beyond perturbation. Introduction to hootopy analysis ethod. Chapan and Hall/CRC Press, and Boca Raton, 3. [9] S.J. Liao. On the hootopy analysis ethod for nonlinear probles. Appl Math Coput. 4, 47(): [] D. Srinivasacharya, and K. Kaladhar. Mixed convection flow of couple stress fluid between parallel vertical 7

6 plates with Hall and Ion-slip effects. Co. Nonlinear Sci. and Nu, Si. In Press, doi:.6/ j.cnsns