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

Abstract

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.

Keywords

• 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

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