Time-fractional Navier Stokes Equations Research Articles

A new approach to solve time-fractional Navier–Stokes equation

This article presents a new technique for the solution of Time Fractional Navier–Stokes equation. The approach is a combination of Adomian decomposition method (ADM) with Kamal integral transform (KIT). The proposed method is implemented on time-fractional Navier–Stokes equation (TFNSEs) to yield the analytical solution. Illustrative examples in TFNSEs are discussed to validate the applicability of the technique. The graphical visualization of the solutions is also presented. Further, the obtained results are compared with the existing solution methods.

Well-Posedness and Dependence on the Initial Value of the Time-Fractional Navier–Stokes Equations on the Heisenberg Group

Well-Posedness and Dependence on the Initial Value of the Time-Fractional Navier–Stokes Equations on the Heisenberg Group

ON SOLUTIONS TO THE TIME-FRACTIONAL NAVIER–STOKES EQUATIONS WITH DAMPING

ON SOLUTIONS TO THE TIME-FRACTIONAL NAVIER–STOKES EQUATIONS WITH DAMPING

Variational iteration method for n-dimensional time-fractional Navier–Stokes equation

In this paper, a modified method is used to approximate the solution to the time-fractional n-dimensional Navier–Stokes equation. The modified method is the Variational Iteration Transform Method, which is implemented in the equation whose fractional order derivative is described in the Caputo sense. The proposed method's findings are presented and examined using figures. It is demonstrated that the proposed method is efficient, dependable, and simple to apply to various science and engineering applications.

A Note on the Time-Fractional Navier–Stokes Equation and the Double Sumudu-Generalized Laplace Transform Decomposition Method

In this work, the time-fractional Navier–Stokes equation is discussed using a calculational method, which is called the Sumudu-generalized Laplace transform decomposition method (DGLTDM). The fractional derivatives are defined in the Caputo sense. The (DGLTDM) is a hybrid of the Sumudu-generalized Laplace transform and the decomposition method. Three examples of the time-fractional Navier–Stokes equation are studied to check the validity and demonstrate the effectiveness of the current method. The results show that the suggested method succeeds remarkably well in terms of proficiency and can be utilized to study more problems in the field of nonlinear fractional differential equations (FDEs).

Efficient simulation of two-dimensional time-fractional Navier–Stokes equations using RBF-FD approach

This study is focused on the Navier–Stokes equations. The numerical method employed in this study for solving the Navier–Stokes equations is a meshless approach that combines the radial basis function (RBF) and finite difference (FD) schemes on two-dimensional arbitrary domains. This method, known as radial basis function-generated finite differences (RBF-FD), utilizes the finite difference technique to discretize the time domain and the RBF-FD approximation to discretize the space domain. The RBF-FD method offers several advantages, including high-order convergence rates and the ability to effectively handle sparse node layouts within arbitrary domains. The efficacy and accuracy of the suggested approach are evaluated through the analysis of four unique instances: comprising Taylor–Green vortices, lid-driven cavity flow, backward-facing step flow, and flow around a square cylinder. In the last example, we examined a simulation of air pollution around buildings.

Classical and non-classical symmetries of time-fractional Navier–Stokes equation

Classical and non-classical symmetries of time-fractional Navier–Stokes equation

An Approach for Approximating Analytical Solutions of the Navier-Stokes Time-Fractional Equation Using the Homotopy Perturbation Sumudu Transform’s Strategy

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the SuT with the homotopy perturbation method using He’s polynomials. This approach proves to be powerful and practical for solving various linear and nonlinear fractional partial differential equations (FPDEs) in scientific and engineering fields. We demonstrate the efficiency and simplicity of this method through examples, showcasing its ability to approximate solutions for FPDEs. Additionally, we compare the numerical results obtained using this technique for different values of alpha, showing that as the value moves from a fractional order to an integer order, the solution becomes increasingly similar to the exact solution. Furthermore, we provide the tabular representations of the solution for each example.

On the uniqueness of mild solutions to the time-fractional Navier–Stokes equations in $$L^{N} \left( \mathbb {R} ^{N}\right) ^{N}$$

In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier–Stokes equations in class \(C\left( [0,\infty );L^{N}\left( \mathbb {R}^{N}\right) ^{N}\right) \) by means of the estimates \(L^{p}-L^{q}\) of Giga-Shor inequality and the Gronwall inequality.

MODIFICATION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD FOR MULTI-DIMENSIONAL TIME-FRACTIONAL MODEL OF NAVIER–STOKES EQUATION

In this paper, we introduce a new method to create a series solution to the time-fractional Navier–Stokes equations using a combination of the Laplace transform with the optimal homotopy asymptotic method. Caputo’s derivative was applied in the present technique. Three test problems are carried out to validate and demonstrate the method’s efficacy. The system has been found to be a very reliable, effective and powerful technique for solving the proposed model as well as a variety of engineering and science problems.

A study of the time fractional Navier-Stokes equations for vertical flow

<abstract><p>Navier-Stokes (NS) equations dealing with gravitational force with time-fractional derivatives are discussed in this paper. These equations can be used to predict fluid velocity and pressure for a given geometry. This paper investigates the local and global existence and uniqueness of mild solutions to NS equations for the time fractional differential operator. We also work on the regularity effects of such types of equations were caused by orthogonal flow.</p></abstract>

Mild Solution for the Time-Fractional Navier–Stokes Equation Incorporating MHD Effects

The Navier–Stokes (NS) equations involving MHD effects with time-fractional derivatives are discussed in this paper. This paper investigates the local and global existence and uniqueness of the mild solution to the NS equations for the time fractional differential operator. In addition, we work on the regularity effects of such types of equations which are caused by MHD flow.

Weak solutions to the time-fractional g-Bénard equations

The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is constant. The main goal of this paper is to, first of all, introduce the g-Bénard problem with time-fractional derivative of order alpha in (0,1). This formulation is new even in the classical Bénard problem, that is with constant g. The second goal of this paper is to prove the existence and uniqueness of a weak solution by means of the Faedo–Galerkin approximation method. Some recent works on time-fractional Navier–Stokes equations have opened new perspectives in studying variational aspects in problems involving time-fractional derivatives.

Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces

Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces

Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

Approximate controllability for mild solution of time-fractional Navier–Stokes equations with delay

This paper addresses some results about mild solution to time-fractional Navier–Stokes equations with bounded delay existed in the convective term and the external force. Based on the Schauder’s fixed point theorem, we establish the sufficient conditions for the existence and uniqueness of the global mild solution in Banach spaces. Moreover, the polynomial decay estimate of the mild solution is given. Furthermore, the approximate controllability for the mild solution is obtained by constructing a Cauchy sequence.

Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation

In this paper, we present the unsteady translational motion of a porous spherical particle in an incompressible viscous fluid. In this case, the modified Navier–Stokes equation with fractional order time derivative is used for conservation of momentum external to the particle whereas modified Brinkman equation with fractional order time derivative is used internal to the particle to govern the fluid flow. Stress jump condition for the tangential stress along with continuity of normal stress and continuity of velocity vectors is used at the porous–liquid interface. The integral Laplace transform technique is employed to solve the governing equations in fluid and porous regions. Numerical inversion code in MATLAB is used to obtain the solution of the problem in the physical domain. Drag force experienced by the particle is obtained. The numerical results have been discussed with the aid of graphs for some specific flows, namely damping oscillation, sine oscillation and sudden motion. Our result shows a significant contribution of the jump coefficient and the fractional order parameter to the drag force.

Numerical solution of the time-fractional Navier–Stokes equations for incompressible flow in a lid-driven cavity

This paper addresses a computational technique for solving 2D unsteady Navier–Stokes equations (NSEs) with time-fractional order in the Caputo sense in the formulation of stream function-vorticity. The finite difference-based method of lines is used to discretize the time-fractional NSEs on a collocated grid that construct a fractional differential algebraic equations system. After solving the discretized complementary Poisson’s equation, this system is reduced to a system of fractional differential equations (FDEs). The resulting FDEs are solved by fractional backward differentiation formulas. The flow in a square lid-driven cavity is considered as the model problem.

Lattice BGK model for time-fractional incompressible Navier–Stokes equations

In this paper, a novel D2Q9 lattice Boltzmann model with BGK operator (LBGK) is proposed for the incompressible time-fractional Navier–Stokes equations with Caputo-type fractional derivative. First the fractional derivative is divided into the history part and the local part, in which the former is approximated using the efficient algorithm for the evaluation of the Caputo fractional derivative, while the latter is simply approximated by rectangle formula to keep the time-dependent characteristics of Navier–Stokes equations as the evolution equations. Then, a LBGK model is constructed for Navier–Stokes equations after approximation of the Caputo derivative. Through Chapman–Enskog analysis the macroscopic equations can be recovered from this model in the small Mach number limit. At the end of this paper, a numerical example with analytic solutions is carried out to show that the LBGK model is efficient.

Analytical solutions for Navier–Stokes equations with Caputo fractional derivative

This work aims to use the homotopy analysis method to obtain analytical solutions of the linear time-fractional Navier–Stokes equation with cylindrical coordinates and also of a system of nonlinear time-fractional Navier–Stokes equations with Cartesian coordinates. These equations are described by means of $$\psi $$ -Caputo fractional derivative in time and contain results of particular cases given in the literature. The solutions obtained for time-fractional Navier–Stokes equations are graphically presented.