Research Article The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform Pongsakorn Sunthrayuth , 1 Rasool Shah , 2 A. M. Zidan , 3,4 Shahbaz Khan, 2 and Jeevan Kafle 5 1 Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani, Thailand 2 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan 3 Department of Mathematics, College of Science, King Khalid University, P.O. Box: 9004, Abha 61413, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Al-Azhar University, Assuit, 71511, Egypt 5 Central Department of Mathematics, Tribhuvan University, Kritipur, Kathmandu, Nepal Correspondence should be addressed to Jeevan Kafle; jeevan.kafle@cdmath.tu.edu.np Received 30 April 2021; Accepted 22 July 2021; Published 10 August 2021 Academic Editor: Nehad Ali Shah Copyright © 2021 Pongsakorn Sunthrayuth et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three- dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models. 1. Introduction Fractional calculus is an engineering and sciences area that deals with the integral and derivative of arbitrary orders. Fractional differential equations (FDEs) have gained popu- larity and significance, primarily due to their proven imple- mentations in applied mathematics. For instance, these problems are more and more utilized to an equation in fluid flow, diffusion, rheology, oscillation, relaxation, reaction-dif- fusion, anomalous diffusion, diffusive transport akin to diffu- sion, turbulence, polymer physics, electric networks, electrochemistry of corrosion, chemical physics, propagation of seismic waves, relaxation processes in complex systems, porous structures, and dynamical processes in self-similar and various other physical procedure. The most significant benefit of FDEs in these and other uses is their nonlocal property. It is well-known that the differential operator of integer-order is a local operator, but the fractional-order dif- ferential operator is nonlocal [1–5]. For example, the nonlin- ear oscillation of earthquakes can be modeled with fractional derivatives [6], and the fluid dynamic traffic model with frac- tional derivatives [7] can calculate the insufficiency arising from the assumption of continuum traffic flow. Experimental data fractional partial differential equations for seepage flow in porous media are shown in [8]. Fractional differential equations have created attention among the researcher due to the exact description of nonlinear phenomena, especially in nanohydrodynamics where continuum assumption does not well, and fractional model can be considered the best can- didate. These findings invoked the growing interest in studies of fractal calculus in many branches of engineering and sci- ence. This is more realistic and it is one reason why fractional calculus has become more and more popular [9–17]. A famous governing equation of motion of viscous fluid flow called the Navier-Stokes (NS) equation has been derived in 1822 [18]. The equation can be regarded as the Momen- tum equation and is a combination of Newton’s second law of motion for fluid substances, the energy equation, and con- tinuity equation. This equation describes many physical things such as ocean currents, liquid flow in pipes, blood Hindawi Journal of Function Spaces Volume 2021, Article ID 1029196, 15 pages https://doi.org/10.1155/2021/1029196