Effect of Slip Factor in Hydrodynamic Behaviors of basic Micro-Flows Problems

Transcription

1 ecent Adances in Manufactuing Engineeing Effect of Slip Facto in Hydodynamic Behaios of basic Mico-Flows Poblems K. Bataineh Mechanical Engineeing Dept. Jodan Uniesity of Science and Technology Ibid Jodan A. F. Khadawi Mechanical Engineeing Dept. Al-Balqa Applied Uniesity Al-Salt Jodan and M. A. Al-Nim Mechanical Engineeing Dept. Jodan Uniesity of Science and Technology Ibid Jodan Abstact: - This study inestigates the hydodynamic behaio on basic mico-flows poblems such as: micofalling film mico-otating cylinde ceeping flow aound micosphee and flow nea otating micosphee. ontinuum appoach with slip at the boundaies is adopted. Effect of slip factos is inestigated. It is found that the effect of elocity slip at the walls inceases with inceag the slip factos while they hae no significant effect fo alues less than.. Key-Wods: - falling film fully deeloped Incompessible mico-cylinde mico-sphee mico-flows hydodynamic behaio Newtonian. Intoduction Micochannels ae used in a aiety of deices incopoating gle-phase liquid flow. The ealy applications inoled micomachined deices such as micopumps micoales and micosensos. This was followed by a thust in the biological and life sciences with a need fo analyzing biological mateials such as poteins DNA cells embyos and chemical eagents. The field of micomixes futhe eceied attention with deelopments in micoeactos whee two chemical species ae mixed pio to intoducing them into a eaction chambe. The deelopments in the micoelectomechanical deices natually equie heat emoal systems that ae equally small. ooling of mios employed in high-powe lase systems inoles cooling systems that coe ey small footpints. Adances in biomedical and genetic engineeing equie contolled fluid tanspot and its pecise themal contol in passages of seeal micomete dimensions. A pope undestanding of fluid flow and heat tansfe in these micoscale systems is theefoe essential fo thei design and opeation. The deeloping micoscale themal and fluidic systems typically hae chaacteistic length of the ode of µ m and often opeate in gaseous enionments at standad conditions whee the molecula mean fee path is in the ode of nm. Fo these cases the Knudsen numbe which is defined as the atio of the mean fee path of gas molecules to the chaacteistic dimension of channel is in the ange fom to namely the elocity-slip and tempeatue-jump egime. Theefoe the micoscale themal and fluidic systems must take into account the effects of slip flow and tempeatue jump at wall. The analyses of lamina heat tansfe in slipflow egime wee fist undetaken by Spaow et al. [] and Inman [] fo tubes with unifom heat flux and a paallel plate channel o a cicula tube with unifom wall tempeatue ug continuum theoy subject to slip-elocity and tempeatue-jump bounday conditions. Thei woks show the Nusselt numbes decease in the pesence of slip. The slip flow poblem in micotubes has been widely conducted by inestigatos [ ]. Howee slip flow in micochannels has not been conducted as much as the slip flows in micotubes. Since the slip flow in micochannels equies a twodimensional appoach its solution is elatiely difficult compaed to that of the slip flow in micotubes. Some of the slip flow studies in micochannels ae summaized hee. Yu and Ameel [] studied slip flow heat tansfe in micochannels and found that heat tansfe inceases deceases o emains unchanged compaed to non-slip flow conditions depending on two dimensionless aiables that include effects of aefaction and the ISBN:

2 ecent Adances in Manufactuing Engineeing fluid/wall inteaction. Then thee has been an enomous inteest in eseach woks of micosystem oe the last decade [-6]. The objectie of this study is to inestigate the hydodynamic behaio on basic mico-flows poblems such as: fluid flow in mico-falling film otating mico-cylinde ceeping flow aound a micosphee and flow nea otating micosphee. The model that combines the continuum appoach with slip at the boundaies is adopted in this inestigation. Theoy onside the lamina incompessible flow of constant popety Newtonian fluid. This study is paticulaly concened with the hydodynamic behaio of mico-flow in the slip flow egime. Thee ae basically fou flow cases consideed in this study fluid flow in mico-falling film otating mico-cylinde ceeping flow aound a micosphee and flow nea otating micosphee these ae shown schematically in Fig. : I. Inclined suface We conside a liquid flows steadily down an inclined flat plate of length L and width h as shown in Fig. case. Fo small flow ates we expect that the iscous foces will peent continued acceleation of fluid down the wall so that u becomes independent of z in a shot distance. The flow is assumed to be two dimensional and fully deeloped. The flow being steady eliminates time aiations in any fluid popety. Being D flow implies no z components of elocity and no popety aiation in z diection. All tems of z components of Naie Stokes equations cancel. Afte applying fully deeloped assumption into continuity equation and ce no penetation of fluid though the plate y component of elocity must be zeo. Applying all assumptions mentioned aboe simplifies Naie -Stokes equations futhe to: p u ρ g x µ x y p ρg y y u at most is function of y only. Integate equation twice leads to: y u ρ g c y c µ The bounday conditions needs to ealuate the constant ae: Zeo shea stess at liquid fee suface u y yh Slip elocity at solid suface u u u y wall 5 y Applying the aboe bounday conditions the x- component of elocity becomes ρg y u h hy 6 µ Velocity applying no slip bounday condition is gien by ρg y u hy 7 µ II. Shape of suface of a otating liquid onside a liquid of constant density and iscosity placed in a cylindical containe of adius as shown in Fig. case. The containe is otating aound its own axis at an angula elocity. ylindical coodinates ae appopiate fo this poblem. At steady state both adial and axial elocity ae zeo. Naie -Stokes equations educe to: p - omponent ρ 8 d d omponent µ 9 d d p z-component ρg z Equations 8- assume the following bounday conditions: at at ω σ λ σ Whee Accoding to the bounday conditions gien in Eq. Eqs. 8- ae soled to gie ω Substitute Eq. and Eq. into Eq.8 and integate to gie: ω p ρ f z and p -ρgz f ISBN:

3 whee f and f ae abitay functions of integation ce p does not depend on the solution fo these become p gz ρ ω ρ The constant can be obtained by equiing that p p atm at and zz o gz - z o ρ ω ρ p p atm 5 The liquid ai inteface is o - z z g ω 6 III. eeping flow aound a sphee: We conside hee the flow of an incompessible Newtonian fluid about solid sphee of adius and diamete D as shown in Fig. case below. The fluid appoaches the fixed sphee etically upwad in the z diection with unifom elocity. Fo ey small eynolds numbe the diffeential equations fo steam function which ae equialent to Naie- Stokes equations in spheical coodinates ae: υ cos E E E E t 7 whee opeato E is gien by: E Fo steady ceeping flow the entie left side may be set equal to zeo and the equation fo axisymmetic flow becomes: ] [ 7 Subject to the following bounday conditions: _. at el slip at pentation no at wall 8 Now we looking fo a solution of the fom f 9 Substitute this into Eq. 7 yields: f d d d d This is equidimensional fouth ode equation assuming the following solution: F Fom the thid bounday condition must be zeo and must equal to Hence the steam function is The elocity components ae: cos Accoding to the bounday conditions gien in Eq. 8 Eqs. - ae soled to gie: cos 5 Velocity components fo no slip bounday conditions ae obtained by setting equal to zeo: cos 6 7 IV. Flow nea otating micosphee We conside a solid micosphee of adius otating at constant angula elocity in a lage body of quiescent fluid as shown in Fig. case. We ae inteested to find the elocity and pessue ecent Adances in Manufactuing Engineeing ISBN:

4 ecent Adances in Manufactuing Engineeing distibutions. Also the toque equied to maintain the motion. Since the eynolds numbe is small we use the ceep flow esion of the equation of motion. ontinuity and motion equations in spheical coodinates ae the goening equations fo this poblem. Since the solution is expected to be symmetic about z-axis thee is no dependence on the angle Φ. The pessue will be in the fom of pp. The component of ceeping flow equation of motion become p - omponent 8 p omponent 9 Φ - component The bounday conditions may be summaized as: at ω at at p po Ty a solution in the fom of: f Substitute into Eq. yields: d df f d d This is equidimensional equation assuming the solution in the fom: f 5 Applying the bounday conditions in Eq. to get: ω 6 The toque needed to maintain the otation of the sphee is ππ τ T z dd 7 Whee τ µ 8 τ µ ω µω 9 Substitute this into Eq. 7 and pefom double integation yields: 8πµω T z 9 esults and Discussion The solutions obtained in this study ae plotted fo each flow poblem unde consideation. The flow was teated as lamina two-dimensional steady incompessible flow with constant themophysical popeties and without heat souce/k. The elocity distibution at diffeent slip facto is shown in Figs. 7. The geneal tend in these figues can be summaized as: at small slip facto less than. the obtained elocity pofiles ae simila to the coesponding classical maco-flow pofiles. Figue shows effect of the slip facto on the elocity distibution of liquid flows steadily down an inclined flat plate. It is clea that the elocity slip at the walls inceases with inceag the slip facto. Inceag implies that the flow is not completely awae of the pesence of the wall due to the elatiely low numbe of collisions between the fluid molecules. As inceases the fluid molecules hae highe mean fee path length and as a esult the molecules tael longe distance without colliding with othe molecules. If a cetain goup of the molecules collide with the wall they will tael longe distance befoe colliding with othe molecules to decease thei momentum and communicate with them about the pesence of the wall. Effect of the slip facto on the elocity distibution of a liquid with constant density and iscosity placed in a otating cylindical containe of adius with angula elocity is shown in Fig.. It is obious that the slip elocity inceases as the slip facto inceases also the slip facto has insignificant effect fo alues less than.. Figues 5-6 show the effect of slip facto on the adial elocity distibution and the tangential elocity distibution of ceeping flow aound a ISBN:

5 ecent Adances in Manufactuing Engineeing sphee. Also it is clea that fom both figues the slip facto has insignificant effect fo alues less than.. The elocityφ component distibution aound a solid micosphee of adius otating at constant angula elocity is shown in Fig. 7. Also the elocity slip inceases as inceases. onclusion In this aticle the theoetically pedicting method fo hydodynamic behaio of the fully deeloped lamina mico-flows in the slip-flow egime is studied. It is found that the elocity slip at the walls inceases with inceag the slip facto while the slip facto has insignificant effect fo alues less.. efeences: [] E. M. Spaow S.H. Lin Lamina heat tansfe in tubes unde slip-flow conditions ASME J. Heat Tansfe 8 96 pp [] []. M. Inman Heat tansfe fo lamina slip flow of a aefied gas in a paallel plate channel o a cicula tube with unifom wall tempeatue NASA TN 96 D-. [].F. Baon X.Wang T.A. Ameel.O.Waington The Gaetz poblem extended to slip-flow Intenat. J. Heat Mass Tansfe pp [] T.A. Ameel.F. Baon X. Wang.O. Waington Lamina foced conection in a cicula tube with constant heat flux and slip flow Micoscale Themophys. Engg. 997 pp.. [5].F. Baon X. Wang.O. Waington T. Ameel Ealuation of the eigenalues fo the Gaetz poblem in slip-flow Intenat. ommun. Heat Mass Tansfe 996pp [6] F.E. Laode. Housiadas Y. Dosos Slip-flow heat tansfe in cicula tubes Intenat. J. Heat Mass Tansfe 5 pp [7] H. Zhao The numeical solution of gaseous slip flows in micotubes Intenat. ommun. Heat Mass Tansfe 8 pp [8] M. Vasudeiah K. Balamuugan Stokes slip flow in a cougated pipe Intenat. J. Engg. Sci [9].F. Baon X.Wang T.A. Ameel.O.Waington The Gaetz poblem extended to slip-flow Intenat. J. Heat Mass Tansfe [] Y.-P. Shih.-. Huang S.-Y. Tsay Extended Leeque solution fo lamina heat tansfe to powe-law fluids in pipes with wall slip Intenat. J. Heat Mass Tansfe [] S. Yu T.A. Ameel Slip-flow heat tansfe in ectangula micochannels Intenat. J. Heat Mass Tansfe pp. 5. [] M. A. Al-Nim and A. F. Khadawi Themal behaio of a stagnant gas confined in a hoizontal micochannel as descibed by the dual-phase-lag heat conduction model Int. J. Themophysics 5: 95. [] A. F. Khadawi A. Othman and M. Al-Nim Tansient fee conection fluid flow in a etical micochannel as descibed by the hypebolic heat conduction model Int. J. Themophysics 6: [] O. Haddad M. Abuzaid and M. Al-Nim Entopy geneation due to lamina incompessible foced conection flow though paallel-plates micochannel Entopy 65 pp.-6. [5] Haddad O. M. Al-Nim M. A. and Abuzaid M. M.: The effect of fequency of fluctuating diing foce on basic slip micoflows Acta Mechanica Vol. 79 5pp [6] Haddad O. M. Abuzaid M. M. and Al- Nim M. A.: Deeloping fee conection gas flow in a etical open-ended micochannel filled with poous media Numeical Heat Tansfe Pat A Vol [7] Haddad O. M. Al-Nim M. A. and Taamneh Y.: Hydodynamic and themal behaio of gas flow in mico channels ISBN:

6 ecent Adances in Manufactuing Engineeing filled with poous media J. Poous Media Vol. 956 pp. -. [8] Haddad O. M. Al-Nim M. A. and Abuzaid M. M.: Effect of peiodically oscillating diing foce on basic micoflows in poous media Jounal Poous Media Vol pp [6] Haddad O. M. Al-Nim M. A. and Abuzaid M. M The effect of fequency of fluctuating diing foce on basic gaseous mico-flows Acta Mechanica 79 5pp [9] Haddad O.M. Al-Nim M. A. and Al- Omay J. Sh.: Foced conection of gaseous slip-flow in poous micochannels unde local themal nonequilibium conditions Tanspot in Poous Media Vol pp. 5-7 [] Haddad O.M. Al-Nim M. A. and Sai M.: Foced conection gaseous slip flow in cicula poous mico-channels Tanspot in Poous Media Vol. 7 7 pp [] Al-Jaah J. Khadawi A. and Al-Nim M. A.: Film condensation on a etical micochannel Int. ommunication in Heat and Mass Tansfe Vol [] pp Haddad O. M. Al-Nim M. A. and Shatnawi G. Fee conection gas flow in poous mico-channels Accepted fo pesentation in the 6th IASME/WSEAS Intenational onfeence on Heat Tansfe Themal Engineeing and Enionment HTE'8 hodes Island Geece August - 8. [] Kiwan S. and Al-Nim M. A.: Flow and Heat Tansfe oe a Stetched Mico- Suface Accepted fo publication in ASME J Heat Tansfe 8. [] Al-Nim M. A. Maqapleh A. M. Khadawi A. F. and Ammouah S. A.: Fully deeloped themal behaios fo paallel flow micochannel heat exchange Accepted fo publication in Intenational ommunications in Heat and Mass Tansfe 9. [5] Bataineh K. and Al-Nim M. A.: -D Naie Stokes Simulations of Micoscale Viscous Pump with Slip Flow Accepted fo publication in ASME Jounal of Fluids Engineeing 9. ISBN:

7 ecent Adances in Manufactuing Engineeing Figue : elocity distibution fo seeal alues of with fixed to unity Fig. schematic diagam fo poblems unde consideation Figue : shape of the fluid Figue : elocity distibutions fo aious alues of fo fixed h Figue 5 adial elocity distibution fo ISBN:

8 ecent Adances in Manufactuing Engineeing no slip V/V h / Figue 6 Tangential elocity distibution fo.9.8 /ω /ω no slip Figue7 Φ elocity components distibution ISBN: