Oldroyd Fluid Research Articles

An estimate for the Hausdorff dimension of attractor for two-dimensional equations of Oldroyd fluids

The minimal global B-attractor for a system of equations describing two-dimensional flows of Oldroyd fluids is considered. An asymptotic estimate for the Hausdorff dimension of the attractor is obtained for small values of the viscosity. Bibliography: 11 titles.

Smooth and convergent ε-approximations of the first boundary-value problem for equations of the Kelvin-Voight and oldroyd fluids

Smooth convergent ε-approximations (11)–(13) for the equations of Oldroyd (8) and Kelvin-Voight (9), (10) fluids are constructed. It is shown that the first initial boundary-value problem for two-dimensional system (11) and three-dimensional systems (12) and (13) for every ε>0 has a unique classical solution, and as ε → 0 these solutions converge to classical solutions of the first initial boundary-value problem for Eqs. (8), (9), and (10) respectively. Bibliography: 10 titles.

On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier-Stokes equations, equation of motion of the Oldroyd fluids, and equations of motion of the Kelvin-Voight fluids

Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ (9) $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ (10) .We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.

A variational analytical approach to time dependent flow of oldroyd B fluid in circular tube

In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions.

Asymptotic stability and time periodicity of "small" solutions of the equations of the motion of oldroyd and Kelvin-Voight fluids

One proves the asymptotic stability and the time periodicity of the "small" classical solutions of the systems of equations (1) and (2), describing the motion of the Oldroyd and Kelvin—Voight fluids, respectively.

Towards a theory of global solvability on [0, ?) of initial-boundary value problems for the equations of motion of oldroyd and Kelvin?Voight fluids

Classical global solvability on [0, ∞) is proved for initial-boundary value problems (30), (32), (33), and (31), (32), (33) which describe two-dimensional motion of Oldroyd fluids and three-dimensional motion of Kelvin—Voight fluids of orders L = 2, 3, ⋯ .

The flow of an Oldroyd fluid around a sharp corner

A similarity solution is constructed for the flow of an Oldroyd-B fluid around a 270° re-entrant comer. The velocity is found to vanish like r 5 9 and the stress to be singular like r −2 3 . A simple expression is found for the streamfunction.

Nonlocal problems for filtration equations for nonNewtonian fluids in a porous medium

The following nonlocal problems are studied for the filtration equations of 1) an Oldroyd fluid and 2) of a Kelvin-Voigt fluid: the existence of solutions for the initial boundary problems for the semiaxis t>0 with free terms and the existence of solution periodic in t with period ω with free terms which are also periodic in t with period ω.

Certain nonlocal problems for two-dimensional equations of motion of Oldroyd fluids

The solvability of a boundary-value problem on the semi-axis t≥0 is studied for two-dimensional equations of motion of Oldroyd fluids (1), and with “trivial problem data” a proof is given of the existence of a solution which is periodic with respect to t and has the period ω. This solution has an absolute term which is also periodic with respect to t and has the period ω. Substantiation is given for the principle of linearization (first Liapunov method) in the theory of the exponential stability of solutions at t→∞.

A priori estimates on the semiaxis t≥0 for the solutions of the equations of motion of linear viscoelastic fluids with an infinite Dirichlet integral, and their applications

The solvability on the semiaxis t≥0 of initial-boundary value problems are investigated for the equations of the motion of linear viscoelastic fluids — Oldroyd fluids and Kelvin—Voight fluids—for which the right-hand sides satisfy the conditions f, ft ∈ L∞(R+; L2(Ω)). The existence of “small” stable solutions, periodic with respect to t, is proved for the equations of the motion of Oldroyd fluids and Kelvin—Voight fluids, with a “small” right-hand side f, periodic with respect to t.

Existence and nonexistence of solutions for some rheological models of viscoelastic liquids

We consider the existence of solutions for some models of non‐Newtonian fluid. We have found that Poiseuille and Taylor–Couette problems may have no usual solutions in a large class of Oldroyd models of viscoelastic fluids. That is, axisymmetric Couette‐like solutions and unidirectional Poiseuille‐like solutions fail to exist at sufficiently large Deborah numbers. We have observed that some of the flow features become universal and independent of model parameters in the critical case, i.e., on the boundary between existence and nonexistence regions. We have described a duality between unidirectional flows of the 4‐constant Oldroyd fluid and plane potential flows of an ideal fictitious gas. Based on this duality we have interpreted the nonexistence of solutions as a ‘‘block‐up’’ phenomenon, in the language of gas dynamics.

Thermal instability in Maxwellian and Oldroydian viscoelastic fluids with suspended particles in porous medium

The effect of suspended particles on thermal instability is considered separately in Maxwellian and Oldroydian viscoelastic fluids in a porous medium. The principle of exchange of stabilities is found to hold well under a condition which is the same for Maxwellian as well as Oldroydian fluid. For stationary convection, both the Maxwellian and Oldroydian fluids behave like Newtonian fluid and the medium permeability and the suspended particles have destabilizing effects on the system. The sufficient conditions for the non-existence of overstability for both Maxwellian and Oldroydian viscoelastic fluids are also obtained.

Extending filaments of an oldroyd fluid

Previously reported falling drop experiments such as those on M1 reported by Jones, Hudson and Ferguson (J. Non-Newtonian Fluid Mech., 35 (1990) 263-276) and the extension of a filament of Boger fluid placed on a rotating drum observed by Jones, Williams and Virdi (J. Non-Newtonian Fluid Mech., 21 (1986) 51-64) are analysed. It is found that both viscous and elastic stresses, as described by the Oldroyd equation, are required to model these experiments.

Asymptotic behavior as t→∞ of the solutions of initial-boundary value problems for the equations of motions of linear viscoelastic fluids

It is shown that for the nonstationary equations of motion of the linear viscoelastic fluids, whose defining equation has the form the stationary system is the Navier-Stokes stationary system with viscosity coefficient v: It is proved that for “small” Reynolds numbers the solutions of the initial-boundary value problems for the equations of motion of the Oldroyd fluids (M=L=1, 2, ...) and Kelvin-Voight fluids (M=L + 1, L=0, 1, 2, ...) converge for t→∞ to the solution of the first boundary value problem for the stationary Navier-Stokes system (*).

A finite element procedure coupled with the method of characteristics for simulation of viscoelastic fluid flow

This paper presents numerical results for an upper-convected Oldroyd fluid obtained by using the decoupled approach previously developed by M. Fortin and D. Esselaoui (Int. J. Numer. Methods Fluids, 7 (1987) 1035) for the simulation of viscoelastic fluid flow. Finite element computations performed on the planar four-to-one abrupt contraction show te existence of multiple-solution families and tend to support the existence of an oscillatory flow in the contraction region.

Viscoelastic flow in a curved channel: A similarity solution for the Oldroyd-B fluid

A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.

Solvability in the large on (0, ?) of the fundamental initial-boundary value problem for the two-dimensional equations of the motion of Oldroyd fluids

The unqiue global solvability for t ∈ (0, ∞) is proved for the system describing the two-dimensional motion of an Oldroyd fluid.

Sliding and squeezing flow of a viscoelastic fluid in a wedge

The paper reports an exact three-dimensional similarity solution for the Oldroyd fluid B. The flow involved is generated by closing as well as sliding the boundaries of a two-dimensional wedge. It is found that the squeezing motion is independent of the sliding motion, but not vice-versa. The squeezing load is shown to be a decreasing function of the Weissenberg number, while the frictional coefficient is only weakly-dependent on the Weissenberg number.

Magneto-thermosolutal convection in a viscoelastic fluid in porous medium

The thermosolutal convection in a layer of electrically conducting Oldroyd fluid heated and soluted from below in porous medium is considered to include the effect of a uniform vertical magnetic field. The magnetic field and stable solute gradient are found to have stabilizing effects whereas the medium permeability is shown to have a destabilizing effect. For stationary convection, the Oldroyd fluid behaves like a Newtonian fluid. The sufficient conditions for the nonexistence of overstability are obtained.

Convergent difference schemes for equations of motion of Oldroyd fluids

Stable convergent finite-difference schemes are constructed for systems of differential and integrodifferential equations describing the motion of Oldroyd's linear viscoelastic fluids.