Boundary layer problem on conveyor belt. Gabriella Bognár University of Miskolc 3515 Miskolc-Egyetemváros, Hungary

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1 Boudary layer problem o coveyor belt Gabriella Bogár Uiversity of Miskolc 355 Miskolc-Egyetemváros, Hugary matvbg@ui-miskolc.hu Abstract: A techologically importat source of the boudary layer pheomeo is the flow over a movig coveyor belt or the flow o a cotiuous movig surface. This article makes a theoretical aalysis of the boudary layer flow o a flat surface movig i a otherwise quiescet o-newtoia fluid medium. A special emphasis is give to the formulatio of boudary layer equatios, which provide similarity solutios. Itroductio Automated bulk optical processig equipmet ca perform a variety of tasks such as, for example, ispectig or sortig bulk articles icludig raw or processed fruit, vegetables, wood chips, recycled plastics ad other similar products. The articles may be characterized accordig to size, color, shape or other qualities. Moder bulk optical processig equipmet ca rapidly separate very large quatities of articles ito umerous categories. Such equipmet typically icludes a coveyor system that moves the articles past a ispectio statio. Rapid ispectio or sortig of large quatities of articles typically requires high-speed coveyor systems such as, for example, coveyor belts with widths of 3-4 ft (-.3 m) ad that carry articles at speeds of over 0 ft/sec (3 m/sec). A problem with coveyor systems drive at such speeds is that may articles are relatively ustable o the belts ad ted to roll, tumble, bouce ad collide with each other. Ustable articles carried by a high-speed coveyor system are difficult to ispect, sort or otherwise process for at least two reasos. The problems of the boudary layer over a cotiuous surface movig i a otherwise quiescet fluid medium have attracted cosiderable attetio ([]-[7]). The iterest has bee devoted to the problem of how to predict the trasfer behavior of o-newtoia fluids. The mai reaso for this is probably that fluids (such as pulps, slurries, molte plastics, ad emulsios), which do ot obey the Newtoia postulate that the tress tesor is directly proportioal to the deformatio tesor, are produced idustrially i icreasig quatities. Several models have bee suggested for o-newtoia fluids. I this paper we use the powerlaw rheological model. The flow of a ielastic power law fluid over a sheet has bee ivestigated. I the absece of a exact aalytical solutio umerical solutios for differet power idex will be preseted. Boudary layer goverig equatios Cosider the two-dimesioal steady flow of a o-newtoia fluid of desity ρ modeled by a power law fluid due to Ostwald-de Waele over a flat plate movig cotiuously with a velocity U w i a otherwise quiescet fluid medium. The assumptios are the followigs: (i) the absece of body force ad pressure gradiet (ii) X ad Y axes are take alog ad perpedicular to the plate (iii) U ad V are velocity compoets parallel ad ormal to the plate

2 The cotiuity ad mometum equatios after makig the ecessary boudary layer approximatios are V + = 0, () X U X + V τ xy =, (2) ρ where ( U, V ) are the compoets of the velocity at ay poit ad τ xy U = γ is the shear compoet of the stress tesor τ applyig the power-law model. γ ad are the material costats, is the rate of strai ad ν = γ is the kiematic viscosity. The costat is kow as the power-law idex of the fluid. Boudary coditios ca be formulated such as U = U, = 0, = 0 = 0 w V U. Y Y = 0 Y =+ Let us itroduce the followig dimesioless variables as follows: X L 2 Y = γ, x =, y ( Uw L / ) + L U U w V v w / + Uw 2 u =, ( U L ) = γ, 2 ( U w L /γ ) NRe =. The boudary coditios of the flow ca be expressed as Similarity solutio approach u ( x, 0) =, v( x,0) = 0, lim u( x, y) = 0 y. (3) The stream fuctio ψ is defied by the u = ψ, v =,

3 ψ ψ ψ ψ = y (4) ad the cotiuity equatio is automatically satisfied. The boudary coditios ca be formulated for ψ as ( x, 0) =, ( x,0) = 0, lim = y ( x,0) 0. I (4) applyig the similarity variables ψ ( x, y) = x + f ( η) ad /( + ) = y / x η, oe ca get the oliear ordiary differetial equatio ' f '' f '' + f f '' = 0 (5) with boudary coditios f ( 0) = 0, f '(0) =, lim f '( η) = 0. (6) η Remark: (i) Let us ote that from (5) it follows ''( η) < 0 0 ad lim f ''( η) = 0. η (ii) Equatio (5) is the same as the Blasius equatio for =, but the boudary coditios differ from the usual coditios applied i case of Blasius problem. f i (,+ ) Numerical results

4 Figure. Dimesioless velocity profiles for =0.3, 0.7, Equatio (5) subject to coditios (6) must be solved. The shootig method is applied by usig the stadard fourth order Ruge-Kutta method for the determiatio of f ' which provides the velocity compoet u. The calculatios were doe as follows: -for three various parameters of ( = 0.3, 0.7, ) -umerical itegratio restricted to the fiite dimesios ( η max = 5) -take care for the coditio lim f '( η) = 0 η η max Figure exhibits f ' ad also the velocity profile for differet power-law expoets. Table 2 shows the values of f ''(0) for the three values of. f ''(0) Table 2. O the base of Table 2 we ote that the ski frictio parameter σ = ( f ''(0)) decreases with a icrease of ad it gives the followig coclusios: (i) by a small fluid exert a greater shear stress o the plate,

5 (ii) the boudary thickess teds to icrease with decreasig power law idex. Refereces: [] Magyari E., Keller B., Exact solutios for self-similar boudary layer ows iduced by permeable stretchig walls, Eur. J. Mech. B Fluids 9 o. (2000) [2] Guedda M., Nouiqueess of solutios to differetial equatios for a boudary-layer approximatios i porous media, C. R. Mecaique 300 (2002) [3] Magyari E., Keller B., Pop I., The missig" self-similar free covectio boudary layer flow over a vertical permeable surface i a porous medium, Trasp. Porous Media 46 o. (2002) [4] Liao S-J., -Pop I., Explicit aalytic solutio for similarity boudary layer equatios, Iteratioal Joural of Heat ad Mass Trasfer 47 (2004) [5] Brighi B., Hoerel J-D., Asymptotic behavior of the ubouded solutios of some boudary layer equatios, arxiv:math/052345v (2005) [6] Liao S-J., Aewbrach of solutios of boudary-layer flows over a permeable stretchig plate, Iteratioal Joural of No-Liear Mechaics 42 (2007) [7] Fag T., Liag W., Lee C-f F., A ew solutio brach for the Blasius equatio_a shrikig sheet problem, Computers ad Mathematics with Applicatios 56 (2008) [8] Gaifulli A.M., Zubtsov A.B., Flow past a plate with a movig surface, Fluid Dyamics, 44 No. 4 (2009)

A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD

IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: 39-63 pissn: 3-7308 A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish