A similarity approach to boundary layer equations of a non-Newtonian fluid: Carreau-Yasuda model

Yıl 2021, Cilt: 10 Sayı: 2, 791 - 799, 27.07.2021

Yiğit Aksoy Hikmet Sümer Kıvanç Samra

https://doi.org/10.28948/ngumuh.840284

Öz

The present study considers a non-Newtonian flow over a horizontally immersed flat plate kept at a different temperature relative to the fluid. An inviscid free stream with uniform velocity induces the flow over the plate where an incompressible boundary layer viscously occurs. It is stipulated that the fluid obeys the Carreau-Yasuda constitutive equation. Analytical investigations begin with the derivation of momentum and energy equations followed by boundary layer simplifications. Scaling symmetries are subsequently calculated to define similarity variables to transform boundary layer equations into ordinary differential forms. Later, solutions of the governing equations are pursued by a numerical scheme based on finite differences. Thanks to those solutions, the effects of significant non-dimensional parameters, such as Deborah and Prandtl numbers, on both momentum and thermal boundary layers are examined throughout the figures. The Nusselt number's variation with non-dimensional numbers is also questioned for the study's heat transfer part.

Anahtar Kelimeler

Carreau-Yasuda, non-Newtonian Fluid, Boundary Layer Flow, Similarity Solutions

Bir non-Newtonyen akışkanının sınır tabakası denklemleri için benzerlik yaklaşımı: Carreau-Yasuda modeli

Öz

Bu çalışmada, akışkana göre farklı bir sıcaklıkta ve yatay olarak yerleştirilmiş düz bir plaka üzerinde Newtonyan olmayan bir akış göz önüne alınmıştır. Sıkıştırılamaz, kararlı ve düzgün hıza sahip viskoz olmayan bir serbest akım plaka üzerinde viskoz bir sınır tabakası akışına neden olmaktadır. Newtonyen olmayan akışın Carreau-Yasuda akışkan modeline uyması öngörülmüştür. Analitik yaklaşım, momentum ve enerji denklemlerinin türetilmesi ve ardından sınır tabakası basitleştirmeleri ile başlar. Denklemlerin ölçekleme simetrileri kullanılarak hesaplanan benzerlik değişkenleri vasıtası ile kısmi diferansiyel denklem formunda olan sınır tabakası denklemleri adi forma indirgenmiştir. Daha sonra, Söz konusu denklemlerin sayısal çözümleri sonlu farklar algoritmasına dayanan sayısal bir çözümleyici ile bulunmuştur. Bu çözümler sayesinde, Deborah ve Prandtl sayıları gibi önemli boyutsuz parametrelerin hem momentum hem de termal sınır tabakası kalınlıkları üzerindeki etkileri grafikler üzerinden incelenmiştir. Ayrıca Nusselt sayısının boyutsuz sayılara göre değişimi de çalışmanın ısı transferi kısmı için araştırılmıştır.

Anahtar Kelimeler

Newtonyen olmayan akışkan, Carreau-Yasuda, sınır tabakası akışı, benzerlik dönüşümleri

Kaynakça

• F. A. Morrison, Understanding Rheology. Oxford University Press, New York ABD, 2001.
• H. Ozoe and Stuart W. Churchill, Hydrodynamic stability and natural convection in Ostwald‐de Waele and Ellis fluids: The development of a numerical solution, AIChE Journal, 18(6), 1196-1207, 1972. https://doi.org/10.1002/aic.690180617
• M. M. Cross, Rheology of non-newtonian fluids: a new flow equation for pseudoplastic systems, Journal of Colloid Science, 20(5), 417-437, 1965. https://doi.org/ 10.1016/0095-8522(65)90022-X
• A. M. Siddiqui, M. Ahmed and Q. K. Ghori, Couette and Poiseuille flows for non-Newtonian fluids, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), 15-26,2006. https://doi.org/ 10.1515/IJNSNS.2006.7.1.15
• K. Khellaf and G. Lauriat, Numerical study of heat transfer in a non-Newtonian Carreau-fluid between rotating concentric vertical cylinders, Journal of non-Newtonian Fluid Mechanic, 89, 45-61, 2000. https://doi.org/10.1016/S0377-020257(99)0030-0
• I. Lashgari, J. O. Pralits, F. Giannetti and L. Brandt, First instability of the flow of shear-thinning and shear thickening fluids past a cylinder, Journal of Fluid Mechanics, 701, 201-227, 2012. https://doi.org/ 10.1017/jfm.2012.151
• F. M. Abbasi, T. Hayat and A. Alsaedi, Numerical analysis for MHD peristaltic transport of Carreau–Yasuda fluid in a curved channel with Hall effects, Journal of Magnetism and Magnetic Materials, 382, 104-110, 2015. https://doi.org/10.1016/j.jmmm.2015. 01.040
• J. Boyd, M. J. Buick and S. Green, Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flow using the lattice Boltzmann method, Physics of Fluids, 19,093103,2007. https://doi.org/10.1063/1.2772250
• K. V. S. N. Raju, D. Krishna, G. Rama Devi, P. J. Reddy and M. Yaseen, Assessment of applicability of Carreau, Ellis, and Cross models to the viscosity data of resin solutions, Journal of Applied Polymer Science, 48, 2101-2112, 1993. https://doi.org/10.1002/app. 1993.070481205
• J. Koszkul and J. Nabialek, Viscosity models in simulation of the filling stage of the injection molding process, Journal of Materials Processing Technology, 157-158, 183-187, 2004. https://doi.org/10.1016/ j.jmatprotec.2004.09.027
• H. Schlichting, Boundary Layer Theory McGraw-Hill. New York, ABD, 1979.
• A. Pantokratoras, Non-similar Blasius and Sakiadis flow of a non-Newtonian Carreau fluid, Journal of the Taiwan Institute of Chemical Engineers, 56, 1-5, 2015. https://doi.org/10.1016/j.jtice.2015.03.021
• M. Khan and A. Hashim, Boundary layer flow and heat transfer to Carreau fluid over a nonlinear stretching sheet, AIP Advances, 5(10), 107203, 2015. https://doi.org/10.1063/1.4932627
• G. W. Bluman and S. Kumei, Symmetries and Differential Equations. Springer Science & Business Media, 2013.
• Y. Aksoy, T. Hayat and M. Pakdemirli, Boundary layer theory and symmetry analysis of a Williamson fluid, Zeitschrift für Naturforschung, 67(6-7), 363-368, 2012. https://doi.org/10.5560/zna.2012-0028
• G. Sarı, M. Pakdemirli, T. Hayat and Y. Aksoy, Boundary layer equations and Lie group analysis of a Sisko Fluid, Journal of Applied Mathematics, 2012, 259608, 2012. https://doi:10.1155/2012/259608
• T. Hayat, M. Pakdemirli and Y. Aksoy, Similarity solutions for boundary layer equations of a Powel-eyring fluid, Mathematical and Computational Applications, 18(1), 62-70, 2013. https://doi.org/ 10.3390/mca18010062
• M. Yürüsoy and M. Pakdemirli, Group classification of a non-Newtonian fluid model using classical approach and equivalence transformations, International Journal of Non-Linear Mechanics, 34(2), 341-346, 1999. https://doi.org/10.1016/S0020-7462(98)00037-7