1,683 research outputs found
Stokes' first problem for some non-Newtonian fluids: Results and mistakes
The well-known problem of unidirectional plane flow of a fluid in a
half-space due to the impulsive motion of the plate it rests upon is discussed
in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The
governing equations are derived from the conservation laws of mass and momentum
and three correct known representations of their exact solutions given. Common
mistakes made in the literature are identified. Simple numerical schemes that
corroborate the analytical solutions are constructed.Comment: 10 pages, 2 figures; accepted for publication in Mechanics Research
Communications; v2 corrects a few typo
Symmetric factorization of the conformation tensor in viscoelastic fluid models
The positive definite symmetric polymer conformation tensor possesses a
unique symmetric square root that satisfies a closed evolution equation in the
Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms
of the velocity field and the symmetric square root of the conformation tensor,
these models' equations of motion formally constitute an evolution in a Hilbert
space with a total energy functional that defines a norm. Moreover, this
formulation is easily implemented in direct numerical simulations resulting in
significant practical advantages in terms of both accuracy and stability.Comment: 7 pages, 5 figure
Comments on: "Starting solutions for some unsteady unidirectional flows of a second grade fluid," [Int. J. Eng. Sci. 43 (2005) 781]
A significant mathematical error is identified and corrected in a recent
highly-cited paper on oscillatory flows of second-grade fluids [Fetecau &
Fetecau (2005). Int. J. Eng. Sci., 43, 781--789]. The corrected solutions are
shown to agree identically with numerical ones generated by a finite-difference
scheme, while the original ones of Fetecau & Fetecau do not. A list of other
recent papers in the literature that commit the error corrected in this Comment
is compiled. Finally, a summary of related erroneous papers in this journal is
presented as an Appendix.Comment: 8 pages, 2 figures (4 images), elsarticle class; accepted for
publication in International Journal of Engineering Scienc
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
Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity
In this note we analyze a model for a unidirectional unsteady flow of a
viscous incompressible fluid with time dependent viscosity. A possible way to
take into account such behaviour is to introduce a memory formalism, including
thus the time dependent viscosity by using an integro-differential term and
therefore generalizing the classical equation of a Newtonian viscous fluid. A
possible useful choice, in this framework, is to use a rheology based on
stress/strain relation generalized by fractional calculus modelling. This is a
model that can be used in applied problems, taking into account a power law
time variability of the viscosity coefficient. We find analytic solutions of
initial value problems in an unbounded and bounded domain. Furthermore, we
discuss the explicit solution in a meaningful particular case
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