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1、LU国超仅lUXtt 口http:/The con stitutive equati on for en ergy boun dary layer inpower law non-Newt onian fluids12Liancun Zheng , Xinxin Zhang1Departme nt of Mathematics and Mecha nics, Un iversity of Scie nee and Tech no logy Beiji ng,Beijing 100083, China, e-mail: lia ncun zhe ngs in 2Mecha ni cal Engi
2、n eeri ng School, Uni versity of Scie nce and Tech no logy Beiji ng,Beijing 100083, China, e-mail: Abstract: A new energy boundary layer equation model for power law non-Newtonian fluids is established first time by assuming that the thermal diffusivitya is characterized as a power law function ofte
3、mperature gradie nt. The Pran dtl nu mber is characterized by a relati on ship of velocity gradie nt, temperature gradient, and the power law index. Furthermore, a new similarity number are derived by supposing that the heat boundary layer equation existing similarity solution.Keywords: Power law fl
4、uids, heat tran sfer, similarity soluti on, non li near boun dary value problem.AMS Subject Classificatio n:34B15, 76D101. In troducti onRecen tly, con siderable atte ntio n has bee n devoted to the problem of how to predict the drag force behavior of non-Newtonian fluids. The main reason for this i
5、s probably that fluids(such as molten plastics, pulps, slurries, emulsi on s), which do not obey the Newt onian postulate that the stress ten sor is directly proportional to the deformation tensor, are produced industrially in increasing quantities, and are therefore in some cases just as likely to
6、be pumped in a pla nt as the more com mon Newt onian fluids. Un dersta nding the nature of this force by mathematical modeling with a view to predicting the drag forces and the associated behavior of fluid flow has bee n the focus of con siderable research work. In additi on, the mathematical model
7、considered in the present paper has significance in studying many problems of.1-3, 6-16engin eeri ng2. Boun dary Layer Gover ning Equati onsWhen a fluid flows past a solid body at high Reyno Ids nu mber R, a thin viscous boun dary layer is known to form at least along the forward portion of the soli
8、d surface. Historically, the boundary layer flow past a flat plate was first example considered by Blasius to illustrate the application of Prandtl layer theory. Schowalterapplied the boun dary layer theory to power law pseu-doplastic fluids anddeveloped the two-dime nsional and three dime nsional b
9、oun dary layer equati ons for the mome ntum3tran sfer. Acrivos and Shahcon sidered the mome ntum and heat tran sfer for a non-Newt onian fluids past#BMSiaxEghttp:/m risu t: rrt T-? LU Bl AH MICarbitrary exter nal surfaces. Followi ng the discussi on by Schowalter and Acrivos, the similarity equati o
10、n of mome ntum boun dary layer has bee n known asn-1(|f(训f(n)+f(n f(n = oEqs.(1) has been used to describe the momentum transfer in power law fluids boundary layer for more tha n 40 years2-20. However, the similarity equati on for thermal boun dary layer has not bee n established up to now. This pap
11、er investigates the applicability of boundary layer theory for the flow of power law fluids. A special emphasis is given to the formulation of boundary layer equations, which provide similarity soluti ons.Consider a semi-infinite plate aligned with a uniform power law flow of constant speedU m at un
12、iform wall temperature. The lam inar boun dary layer equati ons express ing con servati on of mass, mome ntum and en ergy should be writte n as follows:?V+ = 0? Y?U? X+ V?U =丄土?Y p ?Y2Z + v?T? X ?Y? ?TNWwhere the X and Y axes are taken along and perpendicular to the plate, U and V are the velocity?U
13、 comp onents parallel and no rmal to the plate,v = Y?Yn- 1(丫 = K / p) is the kinematic viscosity, thethermal diffusivity a may be defi ned as?T?Yn-1withYand 3 as positive constant. The casen = 1 corresp onds to a Newt onian fluid and the case0 n 1 describes the dilatant fluid. Theappropriate boun da
14、ry con diti ons are:Y=0 = 0, V y=0 = 0, UT Y=0 = Tw ,Y = +sY = +0=T=U0000#BMSiaxEghttp:/#BMSiaxEghttp:/3. Non li near Boun dary Value Problem.The dimensionless variables, the stream function书(x, y) , the similarity variable n and thedimensionless temperature functionw(叶)are introduced as 12-14, we a
15、rrive at the nonlinear boundaryvalue problems of the form:n-1(f(nf(n)+f(n f(n = (8)f (0) = 0, f(0) =0, f(n n=+o =1n- 1(9)(10)(w(nw(n)+NZh f(nw(n = ow(0) = 0, W(n n = +a =1Eqs.(7)_(10) are the similarity equations for both momentum and thermal boundary layer innon-Newtonian fluids. It is clearly that whenn = 1, Eqs.(7)-(10) reduce to the
