Engineering, 00,, 705-709 doi:0.436/eng.00.909 Published Online September 00 (http://www.sirp.org/journl/eng) Some Aspets of Non-Orthogonl Stgntion-Point Flow towrds Strething Surfe Abstrt Mothr Rez, Andi Snkr Gupt Deprtment of Mthemtis, Ntionl Institute of Siene & Tehnology, Berhmpur, Indi Deprtment of Mthemtis, Indin Institute of Tehnology, Khrgpur, Indi E-mil: rez@nist.edu Reeived My, 00; revised July, 00; epted August 3, 00 The problem of stedy two-dimensionl oblique stgntion-point flow of n inompressible visous fluid towrds strething surfe is reexmined. Here the surfe is strethed with veloity proportionl to the distne from fixed point. Previous studies on this problem re reviewed nd the errors in the boundry onditions t infinity re retified. It is found tht for very smll vlue of sher in the free strem, the flow hs boundry lyer struture when, where x nd x re the free strem stgntion-point veloity nd the strething veloity of the sheet, respetively, x being the distne long the surfe from the stgntion-point. On the other hnd, the flow hs n inverted boundry lyer struture when. It is lso observed tht for given vlues of nd free strem sher, the horizontl veloity t point dereses with inrese in the pressure grdient prmeter. Keywords: Oblique Stgntion-Point Flow, Strething Surfe. Introdution The study of the flow of n inompressible visous fluid over strething surfe hs importnt bering on severl tehnologil nd industril proesses. Problems suh s the extrusion of polymers in melt-spinning, glss blowing, spinning of fibers severl metllurgil s well s metl-working proesses involve ertin spets of flow over strething sheets. Crne [] obtined similrity solution in losed nlytil form for stedy twodimensionl flow of n inompressible visous fluid used solely by the strething of n elsti sheet whih moves in its own plne with veloity vrying linerly with distne from fixed point. Chim [] investigted stedy two-dimensionl orthogonl nd oblique stgntion-point flow of n inompressible visous fluid towrds strething surfe in the se when the prmeter b representing the rtio of the strin rte of the stgntion-point flow to tht of the strething surfe is equl to unity. By removing this highly restritive ssumption ( b ), Mhptr nd Gupt [3] nlyzed the stedy two-dimensionl orthogonl stgntion-point flow of n inompressible visous fluid to-wrds strething surfe in the generl se b. They observed tht the struture of the boundry lyer depends ruilly on the vlue of b. Rez nd Gupt [4] generlized the problem of n oblique stgntion-point flow over strething surfe by Chim [] to inlude surfe strin rte different from tht of the stgntion flow. But sine the displement thikness rising out of the boundry lyer on the surfe ws ignored in their boundry ondition t infinity, the nlysis in [4] is of doubt full vlidity. This ws retified in pper by Lok, Amin nd Pop [5]. However, these uthors [5] did not tke into ount the pressure grdient prmeter in the boundry ondition t infinity. This is serious omission sine the pressure grdient prmeter is linked to the free strem sher in the oblique stgntion-point flow (Drzin nd Riley [6]). Hene the results of the pper in [5] re lso of doubtful vlidity. It is noted tht plnr oblique stgntion-point flow of n inompressible visous flow of n inompressible visous fluid towrds ffixed rigid surfe ws first studied by Sturt [7]. This problem ws lter independently investigted by Tmd [8] nd Dorrepl [9]. The nlogue of the plnr oblique stgntion-point flow to stgntion flow obliquely impinging on rigid irulr ylinder ws disussed by Weidmn nd Putkrdze [0]. Ext similrity solutions for impingement of two visous immisible oblique stgntion flows forming flt Copyright 00 SiRes.
706 M. REZA ET AL. interfe ws given by Tilley nd Weidmn []. On the other hnd het trnsfer in oblique stgntion-point flow of n inompressible visous fluid towrds strething surfe ws investigted by Mhptr, Dholey nd Gupt []. Further oblique stgntion-point flow of visoelsti fluid towrds strething surfe ws studied by Mhptr, Dholey nd Gupt [3]. The objetive of the present pper is to retify the errors in [4] nd [5] nd give orret solution to the bove problem. It is worth pointing out tht n oblique stgntion-point flow ours when seprted visous flow retthes to surfe.. Flow Anlysis Consider the stedy two-dimensionl flow ner stgntion point when n inompressible visous fluid impinges obliquely on n elsti surfe oiniding with the plne y 0, the flow being onfined to y 0. Two equl nd opposite fores re pplied long the x-xis so tht the surfe is strethed keeping the origin fixed, s shown in Figure. The veloity omponents in the invisid free strem long the x nd y diretions re U x b( y ), V ( y ), () 0 0 respetively, where nd b re onstnts. Further is the displement thikness rising out of the boundry lyer on the strething surfe nd is the prmeter whih ontrols the horizontl pressure grdient tht produes the sher flow. Note tht the whole flow field given by () my be viewed s being omposed of n orthogonl stgntion-point flow ombined with horizontl sher flow. The orresponding strem funtion for the bove veloity distribution is 0 x( y ) b( y ) () There ppers boundry lyer on the surfe t high Reynolds number. At the strething surfe, the no-slip ondition gives u x, v 0 t y 0, (3) where is positive onstnt nd u nd v re the veloity omponents long x nd y diretions, respetively. In Rez & Gupt [4], strem funtion in the boundry lyer ws ssumed in the form F( ) W( ), (4) where is the kinemti visosity nd x, y, (5) This gives the dimensionless veloity omponents from (4) nd (5) s U F( ) W( ), V F( ), (6) where U u nd V v. Using (6) in the Nvier-Stokes equtions it ws shown in [4] tht F( ) nd W ( ) stisfy the following equtions (7) F FF F, F W FW W, (8) where nd re onstnts. From (6), no-slip onditions (3) beome F(0) 0, F(0), (9) W(0) 0, W(0) 0. (0) Further from () nd (6), the boundry ondition for F( ) nd W ( ) t infinity re F( ) ; F( ) ( d ) s, () b W( ) ( d) s, () where d / is the dimensionless displement thikness prmeter nd d / is the dimensionless pressure grdient prmeter linked to the free strem sher flow. Rez nd Gupt [4] ignored both the onstnts δ nd δ in (). While pointing out tht δ should be tken into ount (s mentioned in the Introdution), Lok, Amin & Pop [5] retified this error in [4]. However, these uthors in [5] lost sight of the onstnt δ in () nd onsequently rrived t governing equtions for the veloity distribution one of whih is inorret. Hene their nlysis is of doubtful vlidity. Using the boundry onditions () nd () in (7) nd (8), we get Figure. A sketh of the physil problem. Copyright 00 SiRes.
M. REZA ET AL. 707 b, ( d d). (3) Thus the governing equtions for F(η) nd W(η) beome, (4) F FF F b FW FW W ( d d). (5) Note tht Eqution (5) derived by Lok et l. [5] does not inlude d. Further the boundry ondition () in [5] is lso erroneous due to the bsene of d. Substitution of (4) nd (5) in the x nd y momentum equtions followed by integrtion gives the pressure distribution p( xy, ) in the flow s pxy (, ) (6) b F F ( d d) onstnt. whih n be found one F( ) is known. Equtions (4) nd (5) subjet to the boundry onditions (9)-() re solved numerilly by finite differene method using Thoms lgorithm (Flether [4]). tht for, displement thikness is pproximtely zero (numerilly). This is due to ft tht when, the strething veloity of the plte is preisely equl to the irrottionl strining veloity. From physil point of view, the bsene of boundry lyer in this se rises from the ft tht lthough the flow is not fritionless in strit sense, the frition is uniformly distributed nd does not therefore ffet the motion. Sturt [7] nd Tmd [8] showed tht the vlue of the dimensionl displement thikness is 0.6479 for oblique stgntion point flow over rigid plte. This result n be ompred with tht of our problem by onsidering = 0 in the boundry ondition (3) whih gives F(0) = 0 nd F0 0. We hve found tht the vlue of the displement thikness is d 0.64788. It my be noted tht in both the studies of Sturt [7] nd Tmd [8], the pressure grdient prmeter 0. 3. Results nd Disussion Figure shows the vrition of U (, ) with η t fixed vlue of ξ(= 0.5) for severl vlues of when the pressure grdient prmeter d 0.5 nd b/.0. It n be seen tht t given vlue of, U inreses with inrese in. Further when b/ is very smll nd equl to 0.05, sy, the veloity profile t fixed vlue of 0.5 for severl vlues of with d 0.5 shows boundry lyer struture (see Figure 3) nd the thikness of the boundry lyer dereses with inrese in. From physil point of view, this stems from the ft tht inrese of strining motion in the free strem (e.g., inrese in for fixed vlue of ) leds to inrese in elertion of the free strem. This results in thinning of boundry lyer. Figure 3 shows tht when the free strem sher is negligible ( b/ 0.05 for given vlue of ), the flow hs boundry lyer struture beuse in this se strining motion domintes over the sher. However, this boundry lyer struture is ffeted to gret extent in the presene of onsiderble sher in the free strem (see Figure ). The dimensionless displement thikness d is omputed for different vlues of from the solution of Eqution (4) subjet to the boundry onditions (9) nd () nd shown in the bove Tble. It my be notied Figure. Vrition of U (, ) with t 0.5 for severl vlues of when d 0.5 nd b/ =.0. Figure 3. Vrition of U (, ) with t 0.5 for severl vlues of when d 0.5 nd b/ = 0.05. Copyright 00 SiRes.
708 M. REZA ET AL. Tble. Vlues of the displement thikness d for severl vlues of /. 3.5.0 d 0.3578 0.0890 0.548889 7 9.36050 with t fixed lotion ξ(= 0.5) for severl vlues of the pressure grdient prmeter d when 3 nd b/.0. It my be seen tht t given vlue of η, the horizontl veloity U dereses with inrese in d. The stremline ptterns for the oblique stgntionpoint flow re shown in Figures 5() nd 5(b) for very smll vlue of the free strem sher b/ 0.05 nd d 0.4 in two ses ) 0., ) 5.0. It n be seen tht for <, the strem lines re slightly tilted towrds the left. but when is lrge (= 5), the flow lmost resembles tht of n orthogonl stgntion-point flow s long s the free strem sher is very smll (see Figure 5(b)). For moderte vlue of free Figure 4 displys the vrition of U, strem sher ( b/ ), the disposition of the stremlines is shown in Figures 6() nd 6(b) for = 0., nd =.0, respetively. It is observed from Figures 5() nd 6() tht for given vlue of (= 0.), with inrese in the free strem sher, the stremlines beome more tilted towrds the left. We lso find tht with inrese in the strining motion in the free strem, the stremlines re less nd less tilted to the left. This is plusible on physil grounds beuse with inrese in for given vlue of b/, the flow tends to resemble n orthogonl stgntion-point flow. () when = 0. 4. Summry An ext solution of the Nvier-Stokes equtions is given whih represents stedy two-dimensionl oblique (b) when = 5.0 Figure 5. Stremline ptterns for b/ = 0.05 nd d = 0.4 () when = 0. (b) when = 5.0. Figure 4. Vrition of U (, ) with t 0.5 for severl vlues d when 3.0 nd b/ =.0. stgntion-point flow of n inompressible visous fluid towrds surfe strethed with veloity proportionl to the distne from fixed point. It is shown tht when the free strem sher is negligible, the flow hs boundry lyer behviour when the strething veloity is less thn the free strem veloity ( ), nd it hs n inverted boundry lyer struture when just the reverse is true ( ). It is found tht the obliquity of the flow towrds the surfe inreses with inrese in b/. This is onsistent with the ft tht inrese in b/ (for fixed vlue of ) results in inrese in the shering motion whih in turn leds to inresed obliquity of the flow towrds the surfe. Copyright 00 SiRes.
M. REZA ET AL. 6. Referenes 709 () when = 0.. (b) when =.0 Figure 6. Stremline ptterns for b/ =.0 nd d = 0.4 () when = 0.; (b) when =.0. 5. Aknowledgements One of the uthors (A. S. G) knowledges the finnil ssistne of Indin Ntionl Siene Ademy, New Delhi for rrying out this work. Authors would lso like to knowledge the use of the filities nd tehnil ssistne of the Center of Theoretil Studies t Indin Institute of Tehnology, Khrgpur. [] L. J. Crne, Flow Pst Strething Plte, Zeitshrift für ngewndte Mthemtik und Physik, Vol., 970, pp. 645-657. [] T. C. Chim, Stgntion-Point Flow towrds Strething Plte, Journl of Physil Soiety of Jpn, Vol. 63, No. 6, 994, pp. 443-444. [3] T. R. Mhptr nd A. S. Gupt, Het Trnsfer in Stgntion-Point Flow towrds Strething Sheet, Het nd Mss Trnsfer, Vol. 38, No. 6, 00, pp. 57-5. [4] M. Rez nd A. S. Gupt, Stedy Two-Dimensionl Oblique Stgntion Point Flow towrds Strething Surfe, Fluid Dynmis Reserh, Vol. 37, No. 5, 005, pp. 334-340. [5] Y. Y. Lok, N. Amin nd I. Pop, Non-Orthogonl Stgntion Point towrds Strething Sheet, Interntionl Journl of Non-Liner Mehnis, Vol. 4, No. 4, 006, pp. 6-67. [6] P. G. Drzin nd N. Riley, The Nvier-Stokes Equtions: A Clssifition of Flows nd Ext Solutions, Cmbridge University Press, Cmbridge, 006. [7] J. T. Sturt, The Visous Flow ner Stgntion Point when Externl Flow hs Uniform Vortiity, Journl of the Aero/Spe Sienes, Vol. 6, 959, pp. 4-5. [8] K. Tmd, Two-Dimensionl Stgntion-Point Flow Impinging Obliquely on Plne Wll, Journl of Physil Soiety of Jpn, Vol. 46, No., 979, pp. 30-3. [9] J. M. Dorrepl, An Ext Solution of the Nvier-Stokes Eqution whih Desribes Non-Orthogonl Stgntion- Point Flow in Two Dimensions, Journl of Fluid Mehnis, Vol. 63, 986, pp. 4-47. [0] D. Weidmn nd V. Putkrdzeb, Axisymmetri Stgntion Flow Obliquely Impinging on Cirulr Cylinder, Europen Journl of Mehnis - B/Fluids, Vol., No., 003, pp. 3-3. [] B. S. Tilley, P. D. Weidmn, Oblique Two-Fluid Stgntion-Point Flow, Europen Journl of Mehnis - B/Fluids, Vol. 7, No., 998, pp. 05-7. [] T. R. Mhptr, S. Dholey nd A. S. Gupt, Het Trnsfer in Oblique Stgntion-Point Flow of n Inompressible Visous Fluid towrds Strething Surfe, Het nd Mss Trnsfer, Vol. 43, No. 8, 007, pp. 767-773. [3] T. R. Mhptr, S. Dholey nd A. S. Gupt, Oblique Stgntion-Point flow of n Inompressible Viso-Elsti Fluid towrds Strething Surfe, Interntionl Journl of Non-Liner Mehnis, Vol. 4, No. 3, 007, pp. 484-499. [4] C. A. J. Flether, Computtionl Tehniques for Fluid Dynmis, Vol., Springer-Verlg, Berlin, 988. Copyright 00 SiRes.