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

Year 2022, Volume: 22 Issue: 1, 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

References

• Aksoy, Y., Hayat, T., Pakdemirli, M., 2012, Boundary layer theory and symmetry analysis of a Williamson fluid, Zeitschrift für Naturforschung A, 67a(6-7), 363-368.
• Aksoy, Y., Pakdemirli M., Khalique, C. M., 2007, Boundary layer equations and stretching sheet solutions for the modified second grade fluid, International Journal of Engineering Science, 45(10), 829-841.
• Bird, R. B., 1976, Useful Non-Newtonian models, Annual Review of Fluid Mechanics, 8, 13-34.
• Bluman, G. W., Kumei, S., 2013, Symmetries and differential equations. Springer Science & Business Media, NY, 413.
• Cross,M. M., 1965, Rheology of Non-Newtonian Fluids: A new Flow Equation For Pseudoplastic Systems, Journal of Colloid Science, 20(5), 417-437.
• Değer, G., Pakdemirli, M., Aksoy, Y., 2011, Symmetry Analysis of Boundary Layer Equations of an Upper Convected Maxwell Fluid with Magnetohydrodynamic Flow, Zeitschrift fur Naturforschung A-Journal of Physical Sciences, 66(5), 321-328.
• Fallahpour,M., Mckee,S., Weinmüller, E. B., 2018, Numerical simulation of flow in smectic liquid crystals, Applied Numerical Mathematics, 132, 154-162.
• Galindo-Rosales, F. J., Rubio-Hernandez, F. J., Sevilla, A., 2011, An apparent viscosity function for shear thickening fluids, Journal of non-Newtonian Mechanics, 166(5-6), 321-325.
• Hayat, T., Pakdemirli, M., Aksoy, Y., 2013, Similarity solutions for boundary layer equations of a Powel-Eyring fluid, Mathematical and Computational Applications, 18(1), 62-70.
• Iftikhar, N., Riaz, M. B., Awrejcewicz, J., Akgül, A., 2021, Effect of magnetic field with parabolic motion on fractional second grade fluid, Fractal and Fractional, 5(4), 163.
• Javaid, M., Tahir, M., Imran, M., Baleanu, D., Akgül, A., Imran, M. A., 2022, Unsteady flow of fractional Burger’s fluid in a rotating annulus region with power-law kernel, Alexandria Engineering Journal, 61(1), 1727.
• Javed, F., Riaz, M. B., Iftikhar, N., Awrejcewicz, J., Akgül, A., 2021, Heat and mass transfer impact on differential type nanofluid with carbon nanotubes: a study of fractional order system, Fractal and Fractional, 5(4), 231.
• Kitzhofer, G., Koch, O., Lima, P., Weinmüller, E., 2007, Efficient numerical solution of the density profile equation in hydrodynamics, Journal of Scientific Computing, 32(3), 411-424.
• Morrison, F. A., 2001, Understanding Rheology. Oxford University Press, New York ABD , 446.
• NA, T. Y., 1994, Boundary layer flow of Reiner-Philippoff fluids, International Journal of Non-Linear Mechanics, 29(6), 871-877.
• Pakdemirli,M., Aksoy Y., Yürüsoy, M., Khalique M., 2008, Symmetries of boundary layer equations of power-law fluids of second grade, Acta Mechanica Sinica, 24(6), 661-670.
• Pakdemirli,M., HAYAT, T., Aksoy Y., 2013, Group-theoretic approach to boundary layer equations of an oldroy-B fluid, Zeitschrift für Naturforschung A, 68(12), 785-790.
• Raju,K. V. S. N., Krishna,D., Rama Devi, G., . Reddy, P. J., Yaseen, M., 1993, Assessment of applicability of Carreau, Ellis, and Cross models to the viscosity data of resin solutions, Journal of Applied Polymer Science, (48), 2101-2112.
• Riaz, M. B., Rehman, A., Awrejcewicz, J., Akgül, A., 2021, Power-law kernel analysis of mhd Maxwell fluid with ramped boundary conditions: transport phenomena solutions based on special functions, Fractal and Fractionals, 5(4), 248.
• Riaz, M. B., Abro, K. A., Abualnaja, K. M., Akgül, A., Rehman A., Abbas, M., Hamed, Y. S., 2021, Exact solutions involving special functions for steady convective flow of magnetohydrodynamic second grade fluid with ramped conditions, Advances in Difference Equations, 2021(1), 1-14.
• Sarı, G., Pakdemirli, M., Hayat, T., Aksoy, Y.,2012, Boundary layer equations and Lie group analysis of a Sisko fluid, Journal of Applied Mathematics, doi: 10.1155/2012/259608.
• Schlichting, H., Gersten K., 2017, Boundary-Layer Theory. Berlin, Springer, Heidelberg, Berlin, 805.
• Sisko, A. W., 1958, The flow of lubricating greases, Industrial & Engineering Chemistry, 50(12), 1789-1792.
• Sunthrayuth, P., Alderremy, A., Aly, S., Shah, R., Akgül, A., 2021, Exact analysis of electro-osmotic flow of Walters’-B fluid with non-singular kernel, Pramana, 95(4), 1-10.
• Wan Nik, W.B., Ani, F. N., Masjuki, H. H., Eng Giap S. G., 2005, Rheology of bio-edible oils according to several rheological models and its potential as hydraulic fluid, Industrial Crops and Products, 22(3), 249-255.
• Weinmuller, E. B., 1986, Collocation for singular boundary value problems of second order, SIAM journal on numerical analysis, 23(5), 1062-1095.
• Williamson, R. V., 1929, The Flow of Pseudoplastic Materials, Industrial & Engineering Chemistry, 21(11), 1108-1111.
• Wurm, S., At 2016, BVPsuite 2.0 a new version of a collocation code for singular BVPs in ODEs EVPs and DAEs, Ph.D. dissertation, TU, Wien, 110.
• Yasuda,K., Armstrong,R. C., Cohen, R. E., 1981, Shear flow properties of concentrated solutions of linear and star branched polystyrenes, Rheologica Acta, 20, 163-178.