Here, we generalize the governing equations of unsteady multi-dimensional incompressible and compressible NSE to fractional time and multi-fractional space. As such, fractional Navier–Stokes equations have been generalized by researchers in the last decades by time fractional NSE (tfNSE) and/or space fractional NSE (sfNSE). Fractional flow and transport models showed superiority in describing anomalous diffusion 9, 29, 30, intermittent turbulence 31, chaos-induced turbulence diffusion 32, and multifractal behavior of velocity fields of turbulent fluids at low viscosity 33. Recently, a number of models in applied mathematics was also reported based on fractional derivatives, for example, to model propagation of long waves by fractional Korteweg-de Vries equation 24– 26, and to model diffusion by fractional Burgers’ equation 27, 28.Įxpressing the conservation of mass and momentum, the NSE govern the motion of fluids in above mentioned subfields of science including flows and turbulence in the atmosphere, rivers, lakes and soil. Evidence of fractional flow and transport behavior in various fields of science was already reported, for example in hydrology and hydraulics 11– 15, anomalous transport of solutes in porous media 16– 19, and climate science 20– 23. Navier Stokes equations (NSE) are the origin of the governing equations of flow and transport. Such applications are found in rheology, electrochemistry, chemical physics, finance, bioengineering, continuum mechanics, image and signal processing, plasma physics, diffusion and advection phenomena 6– 10. However, starting with the last quarter of twentieth century, the fractional differentiation found numerous applications in science, mainly due to its nonlocal nature (see the next section). Until recently, fractional calculus has been considered as a mathematical theory without real-life applications. Later, Euler 1, Lagrange 2, Liouville 3, Grünwald 4, and Riemann 5 made significant contributions to fractional calculus 6. The origin of fractional differentiation goes back to a letter between Leibniz and l’Hôpital in late seventeenth century. It is shown that the developed equations are capable of simulating anomalous sub- and super-diffusion due to their nonlocal behavior in time and space. The numerical simulations are also performed to investigate the merits of the proposed fractional governing equations. When their derivative fractional powers are specified to unit integers, these equations are shown to reduce to the classical Euler equations. For the frictionless flow conditions, the corresponding fractional governing equations were also developed as a special case of the fractional governing equations of incompressible flow. Therefore, they can quantify the effects of initial and boundary conditions better than the classical Navier–Stokes equations. The derived governing equations of fluid flow in fractional differentiation framework herein are nonlocal in time and space. As such, these fractional governing equations for fluid flow may be interpreted as generalizations of the classical Navier–Stokes equations. When their fractional powers in time and in multi-fractional space are specified to unit integer values, the developed fractional equations of continuity and momentum for incompressible and compressible fluid flow reduce to the classical Navier–Stokes equations. Then check the initial value.This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. \ Hintįirst verify that \(y\) solves the differential equation. +4\sin t\) is a solution to the initial-value problem
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