Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms

Year 2022, Volume: 2 Issue: 1, 13 - 25, 31.03.2022

Ndolane Sene

https://doi.org/10.53391/mmnsa.2022.01.002

Cited By: 23

Abstract

In this paper, we consider the constructive equations of the fractional second-grade fluid. The considered fluid model is described by the Caputo derivative. The problem consists to determine the exact analytical solution using the Laplace transform method. The influence of the order of the used fractional operator has been presented in this paper. We also analyze the influence of the Prandtl number in the dynamics of the temperature distribution according to the variation of the order of the Caputo derivative. The impact of the second-grade parameter and the Grashof number in the dynamics of the velocity has been presented and discussed. The influences of the parameters used in the modeling have been interpreted in terms of a fractional context. In general, it is shown that the order of the fractional operator influences the diffusivity of the considered fluid. This influence can cause an increase or decrease in the temperature and velocity distributions. The main findings of the paper have been illustrated using the graphical representations of the considered distributions according to the order of the fractional operator.

Keywords

Second-grade fluid, Grashof number, Prandtl number, Laplace transform

References

• Atangana, A., & Araz, S.İ. Extension of Atangana-Seda numerical method to partial differential equations with integer and non-integer order. Alexandria Engineering Journal, 59(4), 2355-2370, (2020).
• Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. Theory and applications of fractional differential equations (Vol. 204). Elsevier, (2006).
• Podlubny, I. Fractional Differential Equations, Academic Press: New York, NY, USA (1999).
• Atangana, A. & Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Sciences, 20(2), 763-769, (2016).
• Caputo, M., & Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 1-13, (2015).
• Saad, K.M., Baleanu, D., & Atangana, A. New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Computational and Applied Mathematics, 37(4), 5203-5216, (2018).
• Sene, N. Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(1), 20-38, (2022).
• Wang, X., Wang, Z., Huang, X., & Li, Y. Dynamic analysis of a delayed fractional-order SIR model with saturated incidence and treatment functions. International Journal of Bifurcation and Chaos, 28(14), 1850180, (2018).
• Qureshi, S., Yusuf, A., Shaikh, A.A., & Inc, M. Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data. Physica A: Statistical Mechanics and its Applications, 534, 122149, (2019).
• Sene, N. Fractional SIRI Model with Delay in Context of the Generalized Liouville–Caputo Fractional Derivative. In Mathematical Modeling and Soft Computing in Epidemiology (pp. 107-125). CRC Press, (2020).
• Shen, Z.H., Chu, Y.M., Khan, M.A., Muhammad, S., Al-Hartomy, O.A., & Higazy, M. Mathematical modeling and optimal control of the COVID-19 dynamics. Results in Physics, 31, 105028, (2021).
• Li, X.P., Wang, Y., Khan, M.A., Alshahrani, M.Y., & Muhammad, T. A dynamical study of SARS-COV-2: A study of third wave. Results in Physics, 29, 104705, (2021).
• Yavuz, M., & Sene, N. Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate. Fractal and Fractional, 4(3), 35, (2020).
• Kumar, P., & Erturk, V.S. Dynamics of cholera disease by using two recent fractional numerical methods. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 102-111, (2021).
• Ali, F., Gohar, M., & Khan, I. MHD flow of water-based Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable surface velocity, temperature and concentration. Journal of Molecular Liquids, 223, 412-419, (2016).
• Abro, K.A. A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology. The European Physical Journal Plus, 135(1), 1-14, (2020).
• Sene, N. Stability and Convergence Analysis of Numerical Scheme for the Generalized Fractional Diffusion-Reaction Equation. In Advanced Numerical Methods for Differential Equations (pp. 1-16). CRC Press, (2021).
• Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
• Rashid, M., Kalsoom, A., Ghaffar, A., Inc, M., & Sene, N. A Multiple Fixed Point Result for-Type Contractions in the Partially OrderedDistance Spaces with an Application. Journal of Function Spaces, 2022, 6202981, (2022).
• Shah, N.A., Khan, I., Aleem, M., & Imran, M.A. Influence of magnetic field on double convection problem of fractional viscous fluid over an exponentially moving vertical plate: New trends of Caputo time-fractional derivative model. Advances in Mechanical Engineering, 11(7), 1-11, (2019).
• Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A., Alshomrani, A.S., & Alghamdi, M.S. Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results in physics, 7, 789-800, (2017).
• Ahmad, H., Khan, T.A., Ahmad, I., Stanimirović, P.S., & Chu, Y.M. A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Results in Physics, 19, 103462, (2020).
• Yavuz, M., & Sene, N. Approximate solutions of the model describing fluid flow using generalized ρ-Laplace transform method and heat balance integral method. Axioms, 9(4), 123, (2020).
• Li, J.F., Ahmad, I., Ahmad, H., Shah, D., Chu, Y.M., Thounthong, P., & Ayaz, M. Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method. Open Physics, 18(1), 1063-1072, (2020).
• Naik, P.A., Eskandari, Z., & Shahraki, H.E. Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 95-101, (2021).
• Yavuz, M., & Sene, N. Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model. Journal of Ocean Engineering and Science, 6(2), 196-205, (2021).
• Panda, S.K., Ravichandran, C., & Hazarika, B. Results on system of Atangana–Baleanu fractional order Willis aneurysm and nonlinear singularly perturbed boundary value problems. Chaos, Solitons & Fractals, 142, 110390, (2021).
• Tahir, M., Imran, M.A., Raza, N., Abdullah, M., & Aleem, M. Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives. Results in physics, 7, 1887-1898, (2017).
• Ahmad, I., Ahmad, H., Thounthong, P., Chu, Y.M., & Cesarano, C. Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry, 12(7), 1195, (2020).
• Li, X.P., Gul, N., Khan, M.A., Bilal, R., Ali, A., Alshahrani, M.Y., ... & Islam, S. A new Hepatitis B model in light of asymptomatic carriers and vaccination study through Atangana–Baleanu derivative. Results in Physics, 29, 104603, (2021).
• Nisar, K.S., Jothimani, K., Kaliraj, K., & Ravichandran, C. An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain. Chaos, Solitons & Fractals, 146, 110915, (2021).
• Bonyah, E., Yavuz, M., Baleanu, D., & Kumar, S. A robust study on the listeriosis disease by adopting fractal-fractional operators. Alexandria Engineering Journal, 61(3), 2016-2028, (2022).
• Vieru, D., Fetecau, C., & Fetecau, C. Time-fractional free convection flow near a vertical plate with Newtonian heating and mass diffusion. Thermal Science, 19(1), 85-98, (2015).
• Sene, N. Fractional diffusion equation with reaction term described by Caputo-Liouville generalized fractional derivative, Journal of Fractional Calculus and Applications 13(1), 42-57, (2022).
• Ali, F., Saqib, M., Khan, I., & Sheikh, N.A. Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model. The European Physical Journal Plus, 131(10), 1-10, (2016).
• Saqib, M., Ali, F., Khan, I., Sheikh, N.A., & Jan, S.A.A. Exact solutions for free convection flow of generalized Jeffrey fluid: a CaputoFabrizio fractional model. Alexandria engineering journal, 57(3), 1849-1858, (2018).
• Imran, M.A., Khan, I., Ahmad, M., Shah, N.A., & Nazar, M. Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. Journal of Molecular Liquids, 229, 67-75, (2017).
• Khalid, A., Khan, I., Khan, A., & Shafie, S. Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium. Engineering Science and Technology, an International Journal, 18(3), 309-317, (2015).
• Nadeem, S., Haq, R.U., & Lee, C. MHD flow of a Casson fluid over an exponentially shrinking sheet. Scientia Iranica, 19(6), 1550-1553, (2012).
• Narahari, M., & Dutta, B.K. Effects of thermal radiation and mass diffusion on free convection flow near a vertical plate with Newtonian heating. Chemical Engineering Communications, 199(5), 628-643, (2012).
• Shah, N.A., & Khan, I. Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. The European Physical Journal C, 76(7), 1-11, (2016).