723 research outputs found
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
In this paper, we consider the initial boundary value problem of the two
dimensional multi-term time fractional mixed diffusion and diffusion-wave
equations. An alternating direction implicit (ADI) spectral method is developed
based on Legendre spectral approximation in space and finite difference
discretization in time. Numerical stability and convergence of the schemes are
proved, the optimal error is , where are the
polynomial degree, time step size and the regularity of the exact solution,
respectively. We also consider the non-smooth solution case by adding some
correction terms. Numerical experiments are presented to confirm our
theoretical analysis. These techniques can be used to model diffusion and
transport of viscoelastic non-Newtonian fluids
Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid
In recent years, non-Newtonian fluids have received much attention due to
their numerous applications, such as plastic manufacture and extrusion of
polymer fluids. They are more complex than Newtonian fluids because the
relationship between shear stress and shear rate is nonlinear. One particular
subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is
modelled using terms involving multi-term time fractional diffusion and
reaction. In this paper, we consider the application of the finite difference
method for this class of novel multi-term time fractional viscoelastic
non-Newtonian fluid models. An important contribution of the work is that the
new model not only has a multi-term time derivative, of which the fractional
order indices range from 0 to 2, but also possesses a special time fractional
operator on the spatial derivative that is challenging to approximate. There
appears to be no literature reported on the numerical solution of this type of
equation. We derive two new different finite difference schemes to approximate
the model. Then we establish the stability and convergence analysis of these
schemes based on the discrete norm and prove that their accuracy is of
and ,
respectively. Finally, we verify our methods using two numerical examples and
apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette
flow of a generalized Oldroyd-B fluid model. Our methods are effective and can
be extended to solve other non-Newtonian fluid models such as the generalized
Maxwell fluid model, the generalized second grade fluid model and the
generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table
Analytical solution of a fractional differential equation in the theory of viscoelastic fluids
The aim of this paper is to present analytical solutions of fractional delay differential equations (FDDEs) of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. This results presented in two condition, in first step when 0 †α,ÎČ â€ 1/2 and in the second step we considered 1/2 †α,ÎČ â€ 1, for each step 1,2 for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples
Analytical solution of a fractional differential equation in the theory of viscoelastic fluids
The aim of this paper is to present analytical solutions of fractional delay differential equations (FDDEs) of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. This results presented in two condition, in first step when 0 †α, ÎČ â€ 1/2 and in the second step we considered 1/2 †α, ÎČ â€ 1, for each step 1,2 for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples
Smoothed Particle Hydrodynamics simulations of integral multi-mode and fractional viscoelastic models
To capture specific characteristics of non-Newtonian fluids, during the past
years fractional constitutive models have become increasingly popular. These
models are able to capture in a simple and compact way the complex behaviour of
viscoelastic materials, such as the change in power-law relaxation pattern
during the relaxation process of some materials. Using the Lagrangian
Smoothed-Particle Hydrodynamics (SPH) method we can easily track particle
history; this allows us to solve integral constitutive models in a novel way,
without relying on complex tasks. Hence, we develop here a SPH integral
viscoelastic method which is first validated for simple Maxwell or Oldroyd-B
models under Small Amplitude Oscillatory Shear flows (SAOS). By exploiting the
structure of the integral method, a multi-mode Maxwell model is then
implemented. Finally, the method is extended to include fractional constitutive
models, validating the approach by comparing results with theory under SAOS
Recent Trends in Coatings and Thin FilmâModeling and Application
Over the past four decades, there has been increased attention given to the research of fluid mechanics due to its wide application in industry and phycology. Major advances in the modeling of key topics such Newtonian and non-Newtonian fluids and thin film flows have been made and finally published in the Special Issue of coatings. This is an attempt to edit the Special Issue into a book. Although this book is not a formal textbook, it will definitely be useful for university teachers, research students, industrial researchers and in overcoming the difficulties occurring in the said topic, while dealing with the nonlinear governing equations. For such types of equations, it is often more difficult to find an analytical solution or even a numerical one. This book has successfully handled this challenging job with the latest techniques. In addition, the findings of the simulation are logically realistic and meet the standard of sufficient scientific value
Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent
studies suggest that fractional differential and integral operators are well suited to model
physical phenomena with intrinsic memory retention and anomalous behaviour. The global
property of fractional operators presents difficulties in fnding either closed-form solutions
or accurate numerical solutions to fractional differential equations. In rare cases, when
analytical solutions are available, they often exist only in terms of complex integrals and
special functions, or as infinite series. Similarly, obtaining an accurate numerical solution
to arbitrary order differential equation is often computationally demanding. Fractional
operators are non-local, and so it is practicable that when approximating fractional
operators, non-local methods should be preferred. One such non-local method is the
spectral method. In this thesis, we solve problems that arise in the
ow of non-Newtonian
fluids modelled with fractional differential operators. The recurrent theme in this thesis
is the development, testing and presentation of tractable, accurate and computationally
efficient numerical schemes for various classes of fractional differential equations. The
numerical schemes are built around the pseudo{spectral collocation method and shifted
Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral
methods converge geometrically, are accurate and computationally efficient. The objective
of this thesis is to show, among other results, that these features are true when the method
is applied to a variety of fractional differential equations. A survey of the literature
shows that many studies in which pseudo-spectral methods are used to numerically
approximate the solutions of fractional differential equations often to do this by expanding
the solution in terms of certain orthogonal polynomials and then simultaneously solving
for the coefficients of expansion. In this study, however, the orthogonality condition of
the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature
are used to numerically find the coefficients of the series expansions. This approach is
then applied to solve various fractional differential equations, which include, but are not
limited to time{space fractional differential equations, two{sided fractional differential
equations and distributed order differential equations. A theoretical framework is provided
for the convergence of the numerical schemes of each of the aforementioned classes of
fractional differential equations. The overall results, which include theoretical analysis
and numerical simulations, demonstrate that the numerical method performs well in
comparison to existing studies and is appropriate for any class of arbitrary order differential
equations. The schemes are easy to implement and computationally efficient
Effects of coagulation on the two-phase peristaltic pumping of magnetized Prandtl biofluid through an endoscopic annular geometry containing a porous medium
In this article, motivated by more accurate simulation of electromagnetic blood flow in annular vessel geometries in intravascular thrombosis, a mathematical model is developed for elucidating the effects of coagulation (i.e. a blood clot) on peristaltically induced motion of an electrically-conducting (magnetized) Prandtl fluid physiological suspension through a non-uniform annulus containing a homogenous porous medium. Magnetohydrodynamics is included owing to the presence of iron in the hemoglobin molecule and also the presence of ions in real blood. Hall current which generates a secondary (cross) flow at stronger magnetic field is also considered in the present study. A small annular tube (endoscopic) with sinusoidal peristaltic waves traveling along the inner and outer walls at constant velocity with a clot present is analyzed. The governing conservation equations which comprise the continuity and momentum equations for the fluid phase and particle phase are simplified under lubrication approximations (long wavelength and creeping flow conditions). The moving boundary value problem is normalized and solved analytically (with appropriate wall conditions) for the fluid phase and particle phase using the homotopy perturbation method (HPM) with MATHEMATICA software. Validation is conducted with MAPLE numerical quadrature. A parametric study of the influence of clot height (ÎŽ), particle volume fraction (C), Prandtl fluid material parameters (α, ÎČ), Hartmann number (M),
Hall parameter (m), permeability parameter (k), peristaltic wave amplitude (Ï) and wave number (ÎŽÌ
) on pressure difference and wall shear (friction forces) is included. Pressure rise is elevated
with clot height, medium permeability and Prandtl rheological material parameters whereas it is
reduced with increasing particle volume fraction and magnetic Hartmann number. Friction forces
on the outer and inner tubes of the endoscope annulus are enhanced with clot height and particle volume fraction whereas they are decreased with Prandtl rheological material parameters, Hall parameter and permeability parameter. The simulations provide a good benchmark for more general computational fluid dynamics studies of magnetic endoscopic multi-phase peristaltic pumping
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