Numerical solution for Stokes' first problem for a heated generalized second grade fluid with fractional derivative

2009 ◽  
Vol 59 (10) ◽  
pp. 2571-2583 ◽  
Author(s):  
Chunhong Wu
Keyword(s):  
Fractional Derivative ◽  
Numerical Solution ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Advanced Implicit Meshless Approaches for the Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

2018 ◽  
Vol 15 (05) ◽  
pp. 1850032 ◽  
Author(s):  
Xiaolei Bi ◽  
Shanjun Mu ◽  
Qingxia Liu ◽  
Quanzhen Liu ◽  
Baoquan Liu ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Bounded Domain ◽  
Stokes Problem ◽  
Second Order ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Discrete Scheme ◽  
First Order ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

To solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain is important in the research for diffusion processes. In this paper, novel implicit meshless approaches based on the moving least squares (MLS) approximation for spatial discretization and two different time discrete schemes, which are the first-order semi-discrete scheme and the second-order semi-discrete scheme for time, are developed for the numerical simulation of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain. Based on these two time discretization schemes, the newly developed meshless approaches will have the first-order and the second-order accuracy in time, respectively. The stability and convergence of the implicit MLS meshless approaches are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approaches. It has found that the newly developed meshless approaches are accurate and convergent for fractional partial differential equations (FPDEs). Most importantly, the meshless approaches are robust for arbitrarily distributed nodes and complex domains.

scholarly journals On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝ N

2021 ◽  
Vol 11 (1) ◽  
pp. 580-597
Author(s):  
Jia Wei He ◽  
Yong Zhou ◽  
Li Peng ◽  
Bashir Ahmad
Keyword(s):  
Fractional Derivative ◽  
Blow Up ◽  
Stokes Problem ◽  
Second Grade ◽  
Laplacian Operator ◽  
Well Posedness ◽  
Second Grade Fluid ◽  
Liouville Fractional Derivative ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Abstract We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on the Gagliardo-Nirenberg inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.

An Unconditionally Stable and High-Order Convergent Difference Scheme for Stokes’ First Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

2017 ◽  
Vol 10 (3) ◽  
pp. 597-613 ◽  
Author(s):  
Cuicui Ji ◽  
Zhizhong Sun
Keyword(s):  
Fractional Derivative ◽  
Difference Scheme ◽  
Second Order ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Unconditionally Stable ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid ◽  
Second Order Convergence ◽  
Convergent Difference Scheme

AbstractThis article is intended to fill in the blank of the numerical schemes with second-order convergence accuracy in time for nonlinear Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. A linearized difference scheme is proposed. The time fractional-order derivative is discretized by second-order shifted and weighted Gr¨unwald-Letnikov difference operator. The convergence accuracy in space is improved by performing the average operator. The presented numerical method is unconditionally stable with the global convergence order of in maximum norm, where τ and h are the step sizes in time and space, respectively. Finally, numerical examples are carried out to verify the theoretical results, showing that our scheme is efficient indeed.

Exact Solutions of Electro-Osmotic Flow of Generalized Second-Grade Fluid with Fractional Derivative in a Straight Pipe of Circular Cross Section

2014 ◽  
Vol 69 (12) ◽  
pp. 697-704 ◽  
Author(s):  
Shaowei Wang ◽  
Moli Zhao ◽  
Xicheng Li ◽  
Xi Chen ◽  
Yanhui Ge
Keyword(s):  
Fractional Derivative ◽  
Integral Transform ◽  
Capillary Tube ◽  
Circular Cross Section ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Osmotic Flow ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid ◽  
Electro Osmotic Flow

AbstractThe transient electro-osmotic flow of generalized second-grade fluid with fractional derivative in a narrow capillary tube is examined. With the help of the integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of retardation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically.

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