Journal Article

# Analytical solutions for some unsteady flows of fluids with linear dependence of viscosity on the pressure

04 Mar 2021-Inverse Problems in Science and Engineering (Taylor & Francis)-Vol. 29, Iss: 3, pp 378-395
Abstract: New exact solutions for unidirectional unsteady flows of incompressible viscous fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates are establ...

Topics: Viscosity (60%)
##### Citations
More

6 results found

Open accessDissertation
01 Mar 2009-
Abstract: Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

Topics: Singular integral (78%), , Integral transform (63%)

304 Citations

Open accessJournal Article
07 Feb 2021-
Abstract: Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.

Topics: Viscosity (62%), Newtonian fluid (61%), Fluid mechanics (60%) ... read more

5 Citations

Journal Article
Abstract: Exact and simple expressions for the permanent solutions corresponding to two oscillatory motions of incompressible upper-convected Maxwell fluids with exponential dependence of viscosity on the pressure between parallel plates have been established using suitable changes of the spatial variable and the unknown function and the Laplace transform technique. The solutions that have been obtained satisfy the boundary conditions and governing equations but are independent of the initial conditions. They are important for those who want to eliminate the transients from their experiments. The similar solutions for the simple Couette flow of the same fluids as well as known results for the Newtonian fluids performing the same motions were obtained as limiting cases. The convergence of starting solutions to the corresponding permanent components that has been graphically proved could constitute an asset on the correctness of obtained results. The influence of pertinent parameters on the fluid motion and the spatial profiles of starting solutions have been graphically depicted and discussed. The oscillations’ amplitude is an increasing function with respect to the dimensionless pressure–viscosity coefficient and the Weissenberg number. It is lower for the shear stress as compared to the fluid velocity. The three-dimensional distribution of the starting velocity fields has been numerically visualized by means of the two-dimensional contour graphs.

Topics: Fluid dynamics (58%), Couette flow (57%), Newtonian fluid (57%) ... read more

1 Citations

Open accessJournal Article
Abstract: Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and "elastic" characteristics in virtue of mechanisms to store energy and produce entropy. Usually the energy storage properties of such fluids are modelled using the same concepts as in the classical theory of nonlinear solids. Recently new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy-Green tensor is expressed as a function of stress is of interest. We show how to use this new perspective it the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of the Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalise to more complex setting wherein the classical approach might be impractical of even inapplicable.

Topics:

##### References
More

27 results found

Journal Article
K L Johnson1, J. L. Tevaarwerk1Institutions (1)
Abstract: A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring 'sinh law' fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus $G$, the zero-rate viscosity $\eta$ and a reference stress $\tau _{0}$. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.

Topics: Viscoelasticity (61%), , Viscosity (57%) ... read more

459 Citations

Book Chapter
01 Jan 2007-
Abstract: T he equations of Fluid Motion commonly employed depend upon the fundamental hypothesis that the mutual action of two adjacent elements of the fluid is normal to the surface which separates them. From this assumption the equality of pressure in all directions is easily deduced, and then the equations of motion are formed according to D'Alembert's principle. This appears to me the most natural light in which to view the subject; for the two principles of the absence of tangential action, and of the equality of pressure in all directions ought not to be assumed as independent hypotheses, as is sometimes done, inasmuch as the latter is a necessary consequence of the former The equations of motion so formed are very complicated, but yet they admit of solution in some instances, especially in the case of small oscillations. The results of the theory agree on the whole with observation, so far as the time of oscillation is concerned. But there is a whole class of motions of which the common theory takes no cognizance whatever, namely, those which depend on the tangential action called into play by the sliding of one portion of a fluid along another, or of a fluid along the surface of a solid, or of a different fluid, that action in fact which performs the same part with fluids that friction does with solids. Thus, when a ball pendulum oscillates in an indefinitely extended fluid, the common theory gives the arc of oscillation constant.

434 Citations

Open accessJournal Article
Kumbakonam R. Rajagopal1Institutions (1)
Abstract: In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).

Topics: , ,  ... read more

337 Citations

Open accessDissertation
01 Mar 2009-
Abstract: Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

Topics: Singular integral (78%), , Integral transform (63%)

304 Citations

Journal Article
01 Mar 1995-Acta Mechanica
Abstract: We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.