Time-dependent fractional second-grade fluid flow through a channel influenced by unsteady motion of a bottom plate

Zehba Raizah; Arshad Khan; Saadat Hussain Awan; Anwar Saeed; Ahmed M. Galal; Wajaree Weera; Zehba Raizah; Arshad Khan; Saadat Hussain Awan; Anwar Saeed; Ahmed M. Galal; Wajaree Weera

doi:10.3934/math.2023020

AIMS Mathematics

2023, Volume 8, Issue 1: 423-446. doi: 10.3934/math.2023020

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Research article

Time-dependent fractional second-grade fluid flow through a channel influenced by unsteady motion of a bottom plate

1.
Department of Mathematics, College of Science, Abha, King Khalid University, Saudi Arabia
2.
College of Aeronautical Engineering, National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan
3.
Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Khyber, Pakhtunkhwa, Pakistan
4.
Department of Mechanical Engineering, College of Engineering in Wadi Alddawasir, Prince Sattam bin Abdulaziz University, Saudi Arabia
5.
Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, P.O 35516, Mansoura, Egypt
6.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Received: 29 June 2022 Revised: 06 September 2022 Accepted: 20 September 2022 Published: 08 October 2022
MSC : 26A33, 42A38, 44A10, 44A35

This investigation theoretically describes the exact solution of an unsteady fractional a second-grade fluid upon a bottom plate constrained by two walls at the sides which are parallel to each other and are normal to the bottom plate. The flow in the fluid is induced by the time dependent motion of the bottom plate. Initially the flow equation along with boundary and initial conditions are considered which are then transformed to dimensionless notations using suitable set of variables. The Laplace as well as Fourier transformations have been employed to recover the exact solution of flow equation. The time fractional differential operator of Caputo-Fabrizio has been employed to have constitutive equations of fractional order for second-grade fluid. After obtaining the general exact solutions for flow characteristics, three different cases at the surface of bottom plate are discussed; namely (i) Stokes first problem (ii) Accelerating flow (iii) Stokes second problem. It has noticed in this study that, for higher values of Reynolds number the flow characteristics have augmented in all the three cases. Moreover, higher values of time variable have supported the flow of fractional fluid for impulsive and constantly accelerated motion and have opposeed the flow for sine and cosine oscillations.
- Caputo-Fabrizio Operator,
- Non-Newtonian fluid,
- Channel flow,
- unsteady flow,
- Laplace transform,
- Fourier transform,
- second grade fluid
Citation: Zehba Raizah, Arshad Khan, Saadat Hussain Awan, Anwar Saeed, Ahmed M. Galal, Wajaree Weera. Time-dependent fractional second-grade fluid flow through a channel influenced by unsteady motion of a bottom plate[J]. AIMS Mathematics, 2023, 8(1): 423-446. doi: 10.3934/math.2023020

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Abstract

This investigation theoretically describes the exact solution of an unsteady fractional a second-grade fluid upon a bottom plate constrained by two walls at the sides which are parallel to each other and are normal to the bottom plate. The flow in the fluid is induced by the time dependent motion of the bottom plate. Initially the flow equation along with boundary and initial conditions are considered which are then transformed to dimensionless notations using suitable set of variables. The Laplace as well as Fourier transformations have been employed to recover the exact solution of flow equation. The time fractional differential operator of Caputo-Fabrizio has been employed to have constitutive equations of fractional order for second-grade fluid. After obtaining the general exact solutions for flow characteristics, three different cases at the surface of bottom plate are discussed; namely (i) Stokes first problem (ii) Accelerating flow (iii) Stokes second problem. It has noticed in this study that, for higher values of Reynolds number the flow characteristics have augmented in all the three cases. Moreover, higher values of time variable have supported the flow of fractional fluid for impulsive and constantly accelerated motion and have opposeed the flow for sine and cosine oscillations.

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