A New Approach for Solving Heat and Mass Transfer Equations of Viscoelastic Nanofluids using Artificial Optimization Method
Authors
Abstract
The behavior of many types of fluids can be simulated using differential equations. There are many approaches to solve differential equations, including analytical and numerical methods. However, solving an ill-posed high-order differential equation is still a major challenge. Generally, the governing differential equations of a viscoelastic nanofluid are ill-posed; hence, their solution is a challenging task. In addition, the presence of very tiny nanoparticles (lower than 100 nm) induces new heat and mass transfer mechanisms which can increase the complexity of the behavior of the viscoelastic nanofluids. Therefore, creating or developing new analytical or semi-analytical approaches to solve the governing equations of these types of nanofluids is highly demanded. In the present study, by using a new idea and utilizing an optimization approach, a new solution approach has been presented to solve the governing equations of viscoelastic nanofluids. By using the optimization method, a basic initial guess was changed toward an optimized solution satisfying all boundary conditions and the governing equations. The results indicate the robustness and accuracy of the presented method in dealing with the high-order ill-posed governing differential equations of viscoelastic nanofluids.
Keywords
1. Barnes, H. A., A Handbook of Elementary Rheology, Institute of Non-Newtonian Fluid Mechanics: University of Wales, 2000.
2. Sheu, L. J., Chiou, H. S., Weng, W. T., and Lee, S. R., “The Onset of Convection in a Viscoelastic Nanofluid Layer”, International Conference on Electronic & Mechanical Engineering and Information Technology, Vol. 2011, pp. 2044-2047, 2011.
3. Chhabra, R. P., and Richardson, J. F., Non-Newtonian Flow in the Process Industries, Oxford: Butterworth-Heinemann, 1999.
4. Kevorkian, J., and Cole, J. D., Perturbation Methods in Applied Mathematics, New York: Springer-Verlag, 1981.
5. Liao, S., “Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems”, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
6. Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, 1994.
7. Noghrehabadi, A., Ghalambaz, M., and Ghanbarzadeh, A., “Heat Transfer of Magnetohydrodynamic Viscous Nanofluids Over an Isothermal Stretching Sheet”, Journal of Thermophysics and Heat Transfer, Vol. 26, pp. 686-689, 2012.
8. Noghrehabadi, A., Pourrajab, R., and Ghalambaz, M., “Effect of Partial Slip Boundary Condition on the Flow and Heat Transfer of Nanofluids Past Stretching Sheet Prescribed Constant Wall Temperature”, International Journal of Thermal Sciences, Vol. 54, pp. 253-261, 2012.
9. Noghrehabadi, A., Pourrajab, R., and Ghalambaz, M., “Flow and Heat Transfer of Nanofluids Over Stretching Sheet Taking Into Account Partial Slip and Thermal Convective Boundary Conditions”, Heat and Mass Transfer, Vol. 49, pp. 1357-1366, 2013.
10. Noghrehabadi, A., Izadpanahi, E., and Ghalambaz, M., “Analyze of Fluid Flow and Heat Transfer of Nanofluids Over a Stretching Sheet Near the Extrusion Slit, Computers & Fluids, Vol. 100, pp. 227-236, 2014.
11. Sadeghy, K., and Sharifi, M., “Local Similarity Solution for the Flow of a Second-grade Viscoelastic Fluid Above a Moving Plate”, International Journal of Non-Linear Mechanics, Vol. 39, pp. 1265-1273, 2004.
12. Cortell, R., “Analysing Flow and Heat Transfer of a Viscoelastic Fluid over a Semi-infinite Horizontal Moving Flat Plate”, International Journal of Non-Linear Mechanics, Vol. 43, pp. 772-778, 2008.
13. Munawar, S., Mehmood, A., and Ali, A., “Comment on “Analysing Flow and Heat Transfer of a Viscoelastic Fluid over a Semi-infinite Horizontal Moving Flat Plate”, IJNLM, 43 (2008) 772”, International Journal of Non-Linear Mechanics,Vol. 46, pp. 1280-1282, 2011.
14. Madani Tonekaboni, S. A., Abkar, R., and Khoeilar, R., “On the Study of Viscoelastic Walters' B Fluid in Boundary Layer Flows”, Mathematical Problems in Engineering, Vol. 2012, pp. 1-18, 2012.
15. Choi, S. U. S., and Eastman, J. A., “Enhancing Thermal Conductivity of Fluids with Nanoparticles”, American Society of Mechanical Engineers, pp. 99-105, 1995.
16. Das, S. K., Choi, S. U. S., Yu, W., and Pradeep, T., Nanofluids: Science and Technology, John Wiley & Sons Inc, 2007.
17. Buongiorno, J., 2005. “Convective Transport in Nanofluids”, Journal of Heat Transfer, Vol. 128, pp. 240-250, 2005.
18. Cavuto, D. J., “An Exploration and Development of Current Artificial Neural Network Theory and Applications with Emphasis on Artificial Life”, Ms.C. Thesis, A Thesis Submitted in Partial Fulfillment of the Requirements , 1997.
19. Meade Jr, A. J., and Fernandez, A. A., “The Numerical Solution of Linear Ordinary Differential Equations by Feed-forward Neural Networks”, Mathematical and Computer Modelling, Vol. 19, pp. 1-25, 1994.
20. Lagaris, I. E., Likas, A., and Fotiadis, D. I., “Artificial Neural Networks for Solving Ordinary and Partial Differential Equations”, Neural Networks, IEEE Transactions on, Vol. 9, pp. 987-1000, 1998.
21. Malek, A., and Shekari Beidokhti, R. “Numerical Solution for High Order Differential Equations using a Hybrid Neural Network-optimization Method”, Applied Mathematics and Computation, Vol. 183, pp. 260-271, 2006.
22. Noghrehabadi, A., Ghalambaz, M., Ghalambaz, M., and Vosough, A., “A Hybrid Power Series-Cuckoo Search Optimization Algorithm to Electrostatic Deflection of Micro Fixed-fixed Actuators”, International Journal of Multidsciplinary Sciences and Engineering, Vol. 2, pp. 22-26, 2011.
23. Noghrehabadi, A., Ghalambaz, M., and Ghanbarzadeh, A., “A Hybrid Power Series Artificial Bee Colony Algorithm to Obtain a Solution for Buckling of Multiwall Carbon Nanotube Cantilevers Near Small Layers of Graphite Sheets”, Applied Computational Intelligence and Soft Computing, Vol. 2012, pp. 1-6, 2012.
24. Behrang, M. A., Ghalambaz, M., Assareh, E., and Noghrehabadi, A., “A New Solution for Natural Convection of Darcian Fluid About a Vertical Full Cone Embedded in Porous Media Prescribed Wall”, World Academy of Science, Engineering and Technology, Vol. 49, pp. 871-876, 2011.
25. Yekrangi, A., Ghalambaz, M., Noghrehabadi, A., Beni, Y. T., Abadyan, M., and Abadi, M. N., “An Approximate Solution for a Simple Pendulum Beyond the Small Angles Regimes using Hybrid Artificial Neural Network and Particle Swarm Optimization Algorithm”, Procedia Engineering, Vol. 10, pp. 3734-3740, 2011.
26. Ghalambaz, M., Noghrehabadi, A., Behrang, M. A., Assareh, E., Ghanbarzadeh, A., and Hedayat, N., “A Hybrid Neural Network and Gravitational Search Algorithm (HNNGSA) Method to Solve Well Known Wessinger's Equation”, World Academy of Science, Engineering and Technology, Vol. 5, pp. 609-614, 2011.
27. Biglari, M., Assareh, E., Poultangari, I., and Nedaei, M., “Solving Blasius Differential Equation by using Hybrid Neural Network and Gravitational Search Algorithm (HNNGSA)”, Global Journal of Science, Engineering and Technology, Vol. 11, pp. 29-36, 2013.
28. Ahmad, I., and Bilal, M., “Numerical Solution of Blasius Equation Through Neural Networks Algorithm”, American Journal of Computational Mathematics, Vol. 4, No. 03, p. 223, 2014.
29. Assareh, E., Behrang, Ghalambaz, M., Noghrehabadi, A., and Ghanbarzadeh, M. A., “A New Approach to Solve Blasius Equation using Parameter Identification of Nonlinear Functions Based on the Bees Algorithm (BA)”, International Journal of Mechanical and Mechatronics Engineering, Vol. 5, pp. 255-257, 2011.
30. Minsky, M., and Papert, A., Perceptrons: an Introduction to Computational Geometry, Expanded edition, M.I.T. Press, Cambridge, 1997.
31. El-Bouri, A., Balakrishnan, S., and Popplewell, N., Sequencing Jobs on a Single Machine: A Neural Network Approach, European Journal of Operational Research, Vol. 126, pp. 474-490, 2000.
32. Hornik, K., Stinchcombe, M., and White, H., “Multilayer Feedforward Networks Are Universal Approximators”, Neural Networks, Vol. 2, pp. 359-366, 1984.
33. Kennedy, J., and Eberhart, R., “Particle Swarm Optimization”, IEEE International Conference, Vol. 4, pp. 1942-1948, 1995.
34. Kuznetsov, A. V., and Nield, D. A., “Natural Convective Boundary-layer Flow of a Nanofluid Past a Vertical Plate”, International Journal of Thermal Sciences, Vol. 49, pp. 243-247, 2010.
35. Goyal, M., and Bhargava, R., “Boundary Layer Flow and Heat Transfer of Viscoelastic Nanofluids Past a Stretching Sheet with Partial Slip Conditions”, Applied Nanoscience, Vol. 4, pp. 761-767, 2014.
36. Papanastasiou, T., Georgiou, G., and Alexandrou, A. N., Viscous Fluid Flow, CRC Press, Boca Raton. 2000.
2. Sheu, L. J., Chiou, H. S., Weng, W. T., and Lee, S. R., “The Onset of Convection in a Viscoelastic Nanofluid Layer”, International Conference on Electronic & Mechanical Engineering and Information Technology, Vol. 2011, pp. 2044-2047, 2011.
3. Chhabra, R. P., and Richardson, J. F., Non-Newtonian Flow in the Process Industries, Oxford: Butterworth-Heinemann, 1999.
4. Kevorkian, J., and Cole, J. D., Perturbation Methods in Applied Mathematics, New York: Springer-Verlag, 1981.
5. Liao, S., “Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems”, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
6. Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, 1994.
7. Noghrehabadi, A., Ghalambaz, M., and Ghanbarzadeh, A., “Heat Transfer of Magnetohydrodynamic Viscous Nanofluids Over an Isothermal Stretching Sheet”, Journal of Thermophysics and Heat Transfer, Vol. 26, pp. 686-689, 2012.
8. Noghrehabadi, A., Pourrajab, R., and Ghalambaz, M., “Effect of Partial Slip Boundary Condition on the Flow and Heat Transfer of Nanofluids Past Stretching Sheet Prescribed Constant Wall Temperature”, International Journal of Thermal Sciences, Vol. 54, pp. 253-261, 2012.
9. Noghrehabadi, A., Pourrajab, R., and Ghalambaz, M., “Flow and Heat Transfer of Nanofluids Over Stretching Sheet Taking Into Account Partial Slip and Thermal Convective Boundary Conditions”, Heat and Mass Transfer, Vol. 49, pp. 1357-1366, 2013.
10. Noghrehabadi, A., Izadpanahi, E., and Ghalambaz, M., “Analyze of Fluid Flow and Heat Transfer of Nanofluids Over a Stretching Sheet Near the Extrusion Slit, Computers & Fluids, Vol. 100, pp. 227-236, 2014.
11. Sadeghy, K., and Sharifi, M., “Local Similarity Solution for the Flow of a Second-grade Viscoelastic Fluid Above a Moving Plate”, International Journal of Non-Linear Mechanics, Vol. 39, pp. 1265-1273, 2004.
12. Cortell, R., “Analysing Flow and Heat Transfer of a Viscoelastic Fluid over a Semi-infinite Horizontal Moving Flat Plate”, International Journal of Non-Linear Mechanics, Vol. 43, pp. 772-778, 2008.
13. Munawar, S., Mehmood, A., and Ali, A., “Comment on “Analysing Flow and Heat Transfer of a Viscoelastic Fluid over a Semi-infinite Horizontal Moving Flat Plate”, IJNLM, 43 (2008) 772”, International Journal of Non-Linear Mechanics,Vol. 46, pp. 1280-1282, 2011.
14. Madani Tonekaboni, S. A., Abkar, R., and Khoeilar, R., “On the Study of Viscoelastic Walters' B Fluid in Boundary Layer Flows”, Mathematical Problems in Engineering, Vol. 2012, pp. 1-18, 2012.
15. Choi, S. U. S., and Eastman, J. A., “Enhancing Thermal Conductivity of Fluids with Nanoparticles”, American Society of Mechanical Engineers, pp. 99-105, 1995.
16. Das, S. K., Choi, S. U. S., Yu, W., and Pradeep, T., Nanofluids: Science and Technology, John Wiley & Sons Inc, 2007.
17. Buongiorno, J., 2005. “Convective Transport in Nanofluids”, Journal of Heat Transfer, Vol. 128, pp. 240-250, 2005.
18. Cavuto, D. J., “An Exploration and Development of Current Artificial Neural Network Theory and Applications with Emphasis on Artificial Life”, Ms.C. Thesis, A Thesis Submitted in Partial Fulfillment of the Requirements , 1997.
19. Meade Jr, A. J., and Fernandez, A. A., “The Numerical Solution of Linear Ordinary Differential Equations by Feed-forward Neural Networks”, Mathematical and Computer Modelling, Vol. 19, pp. 1-25, 1994.
20. Lagaris, I. E., Likas, A., and Fotiadis, D. I., “Artificial Neural Networks for Solving Ordinary and Partial Differential Equations”, Neural Networks, IEEE Transactions on, Vol. 9, pp. 987-1000, 1998.
21. Malek, A., and Shekari Beidokhti, R. “Numerical Solution for High Order Differential Equations using a Hybrid Neural Network-optimization Method”, Applied Mathematics and Computation, Vol. 183, pp. 260-271, 2006.
22. Noghrehabadi, A., Ghalambaz, M., Ghalambaz, M., and Vosough, A., “A Hybrid Power Series-Cuckoo Search Optimization Algorithm to Electrostatic Deflection of Micro Fixed-fixed Actuators”, International Journal of Multidsciplinary Sciences and Engineering, Vol. 2, pp. 22-26, 2011.
23. Noghrehabadi, A., Ghalambaz, M., and Ghanbarzadeh, A., “A Hybrid Power Series Artificial Bee Colony Algorithm to Obtain a Solution for Buckling of Multiwall Carbon Nanotube Cantilevers Near Small Layers of Graphite Sheets”, Applied Computational Intelligence and Soft Computing, Vol. 2012, pp. 1-6, 2012.
24. Behrang, M. A., Ghalambaz, M., Assareh, E., and Noghrehabadi, A., “A New Solution for Natural Convection of Darcian Fluid About a Vertical Full Cone Embedded in Porous Media Prescribed Wall”, World Academy of Science, Engineering and Technology, Vol. 49, pp. 871-876, 2011.
25. Yekrangi, A., Ghalambaz, M., Noghrehabadi, A., Beni, Y. T., Abadyan, M., and Abadi, M. N., “An Approximate Solution for a Simple Pendulum Beyond the Small Angles Regimes using Hybrid Artificial Neural Network and Particle Swarm Optimization Algorithm”, Procedia Engineering, Vol. 10, pp. 3734-3740, 2011.
26. Ghalambaz, M., Noghrehabadi, A., Behrang, M. A., Assareh, E., Ghanbarzadeh, A., and Hedayat, N., “A Hybrid Neural Network and Gravitational Search Algorithm (HNNGSA) Method to Solve Well Known Wessinger's Equation”, World Academy of Science, Engineering and Technology, Vol. 5, pp. 609-614, 2011.
27. Biglari, M., Assareh, E., Poultangari, I., and Nedaei, M., “Solving Blasius Differential Equation by using Hybrid Neural Network and Gravitational Search Algorithm (HNNGSA)”, Global Journal of Science, Engineering and Technology, Vol. 11, pp. 29-36, 2013.
28. Ahmad, I., and Bilal, M., “Numerical Solution of Blasius Equation Through Neural Networks Algorithm”, American Journal of Computational Mathematics, Vol. 4, No. 03, p. 223, 2014.
29. Assareh, E., Behrang, Ghalambaz, M., Noghrehabadi, A., and Ghanbarzadeh, M. A., “A New Approach to Solve Blasius Equation using Parameter Identification of Nonlinear Functions Based on the Bees Algorithm (BA)”, International Journal of Mechanical and Mechatronics Engineering, Vol. 5, pp. 255-257, 2011.
30. Minsky, M., and Papert, A., Perceptrons: an Introduction to Computational Geometry, Expanded edition, M.I.T. Press, Cambridge, 1997.
31. El-Bouri, A., Balakrishnan, S., and Popplewell, N., Sequencing Jobs on a Single Machine: A Neural Network Approach, European Journal of Operational Research, Vol. 126, pp. 474-490, 2000.
32. Hornik, K., Stinchcombe, M., and White, H., “Multilayer Feedforward Networks Are Universal Approximators”, Neural Networks, Vol. 2, pp. 359-366, 1984.
33. Kennedy, J., and Eberhart, R., “Particle Swarm Optimization”, IEEE International Conference, Vol. 4, pp. 1942-1948, 1995.
34. Kuznetsov, A. V., and Nield, D. A., “Natural Convective Boundary-layer Flow of a Nanofluid Past a Vertical Plate”, International Journal of Thermal Sciences, Vol. 49, pp. 243-247, 2010.
35. Goyal, M., and Bhargava, R., “Boundary Layer Flow and Heat Transfer of Viscoelastic Nanofluids Past a Stretching Sheet with Partial Slip Conditions”, Applied Nanoscience, Vol. 4, pp. 761-767, 2014.
36. Papanastasiou, T., Georgiou, G., and Alexandrou, A. N., Viscous Fluid Flow, CRC Press, Boca Raton. 2000.
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