Similarity Solutions of a non-Newtonian Fluid’s Momentum and Thermal Boundary Layers: Cross Fluid Model

Year 2022, , 222 - 239, 28.02.2022

Hikmet Sümer , Yiğit Aksoy

https://doi.org/10.35414/akufemubid.1028006

Abstract

The steady, incompressible and laminer flow of a non-Newtonian fluid that fits the Cross-fluid model over a flat plate is investigated. Dimensionless momentum and energy equations in partial differential form are derived to examine the variation of fluid velocity and temperature. The equations are simplified by the boundary layer theory based on the assumption that the change occurs in a narrow region, then scaling symmetries are calculated. By means of symmetries, equations in a partial form are reduced to an ordinary form by computing similarity variables and functions. The sbvp2.0 package developed for the Matlab environment based on collocation methods was used for the numerical solutions of the equations. In the light of analytical approach and solutions, the heat transfer is investigated by the Nusselt number. The study reveals that increases in Weissenberg number and power-law index, as non-Newtonian properties, are in charge of the thinner boundary layers, thus causing less friction and effective convection. As a result of numerical parts of the study, sbvp2.0 package is recomended for stiff equations with high nonlinearity, especially arising from boundary layer flows.

Keywords

Non-Newtonian fluid, Boundary layer flow, Lie group symmetries, Numerical solutions, Heat transfer

Bir Newtonyen Olmayan Akışkanın Momentum ve Isıl Sınır Tabakalarının Benzerlik Çözümleri: Cross Akışkan Modeli

Abstract

Bu çalışma kapsamında Newtonyen olmayan Cross akışkanının sabit bir plaka üzerinde sıkıştırılamaz laminer akışı incelenmiştir. Kısmi diferansiyel denklem formundaki boyutsuz momentum ve enerji denklemleri çözümlenerek akışkanın hızı ve sıcaklık değişimleri incelenmiştir. Bu denklemler akışkan hız ve sıcaklık değişiminin dar bir bölgede gerçekleştiği varsayımına dayanan sınır tabakası teorisi ile sadeleştirilmiştir. Sınır tabakası denklemlerinin simetrileri ölçekleme dönüşüm formülleri ile tespit edilip, bu simetriler yardımıyla benzerlik değişkenleri ve fonksiyonlar kullanılarak, kısmi diferansiyel denklemlerin eşdeğer adi diferansiyel denklemleri bulunmuştur. Denklemlerin sayısal çözümleri için, sıralama noktalarını kullanarak denklemlerin nümerik çözümlerini bulmayı sağlayan Matlab ortamı için geliştirilen sbvp2.0 paketi kullanılmıştır. Analitik yaklaşım ve çözümler ışığıyla akışkanın ısı transferi Nusselt sayısı ile incelenmiştir. Artan Weissenberg sayısı ve power-law indeksi ile sınır tabakalarının kalınlaştığı ve bu sayede daha az sürtünme ve etkili konveksiyona sebep olduğu çalışmadan bulunmuştur. Çalışmanın sayısal kısmının sonucu olarak sbvp2.0 paketi yüksek doğrusal olmayan davranışa sahip özellikle sınır tabakası akışlarından ortaya çıkmış denklemler için önerilmektedir.

Keywords

Newtonyen olmayan, Sınır tabakası, Lie Grup Simetrileri, Nümerik çözümler, Isı transferi

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