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Journal ArticleDOI

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
TL;DR: In this article, 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 established.
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...
Citations
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Dissertation
01 Mar 2009
TL;DR: In this paper, the relationship between these transforms and their properties was discussed and some important applications in physics and engineering were given, as well as their properties and applications in various domains.
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.

383 citations

Journal ArticleDOI
07 Feb 2021
TL;DR: In this paper, the authors derived exact solutions in terms of standard Bessel functions 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.
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.

8 citations

Journal ArticleDOI
TL;DR: In this paper, 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.
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.

4 citations

Journal ArticleDOI
22 Mar 2021-Fluids
TL;DR: In this paper, the authors propose a new model for non-Newtonian viscoelastic fluids based on implicit constitutive relations, where the left Cauchy-Green tensor is expressed as a function of stress.
Abstract: Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled 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 the 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 in 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 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 generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.

3 citations

Journal ArticleDOI
24 Jan 2022-Symmetry
TL;DR: In this paper , the modified Stokes second problem for upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated.
Abstract: The modified Stokes second problem for incompressible upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated. The fluid motion, between infinite horizontal parallel plates, is generated by the lower wall, which oscillates in its plane. The movement region of the fluid is symmetric with respect to the median plane, but its motion is asymmetric due to the boundary conditions. Closed-form expressions are found for the steady-state components of start-up solutions for non-dimensional velocity and the corresponding non-trivial shear and normal stresses. Similar solutions for the simple Couette flow are obtained as limiting cases of the solutions corresponding to the motion due to cosine oscillations of the wall. For validation, it is graphically proved that the start-up solutions (numerical solutions) converge to their steady-state components. Solutions for motions of ordinary incompressible UCM fluids performing the same motions are obtained as special cases of present results using asymptotic approximations of standard Bessel functions. The time needed to reach the permanent or steady state is also determined. This time is higher for motions of ordinary fluids, compared with motions of liquids with pressure-dependent viscosity. The impact of physical parameters on the fluid motion and the spatial–temporal distribution of start-up solutions are graphically investigated and discussed. Ordinary fluids move slower than fluids with pressure-dependent viscosity.

1 citations

References
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Book ChapterDOI
01 Jan 2007
TL;DR: In this paper, it was shown that the theory of D'Alembert's principle of equality of pressure in all directions is a necessary consequence of the absence of tangential action.
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.

494 citations

Journal ArticleDOI
TL;DR: In this article, a simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. But the model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain.
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.

476 citations


"Analytical solutions for some unste..." refers background in this paper

  • ...There have been recent experiments by Cutler et al. [8], Griest et al. [9], Johnson and Cameron [10], Johnson and Greenwood [11], Johnson and Tevaarwerk [12], Bain and Winer [13], which confirm the strong dependence of viscosity on the pressure....

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  • ...[9], Johnson and Cameron [10], Johnson and Greenwood [11], Johnson and Tevaarwerk [12], Bain and Winer [13], which confirm the strong dependence of viscosity on the pressure....

    [...]

Dissertation
01 Mar 2009
TL;DR: In this paper, the relationship between these transforms and their properties was discussed and some important applications in physics and engineering were given, as well as their properties and applications in various domains.
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.

383 citations


"Analytical solutions for some unste..." refers background or methods in this paper

  • ...and J0(·) and Y0(·) are standard Bessel functions, integrating the result with respect to r from a to b and bearing in mind the conditions (27) and (28) and the known result [27] ∫ b...

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  • ...Solving the ordinary differential equation (31) with the initial condition (32) and applying the inverse Hankel transform to the result, we obtain (see [27], the page 482)...

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Journal ArticleDOI
TL;DR: In this paper, the authors discuss implicit constitutive theories for the Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process.
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).

376 citations


"Analytical solutions for some unste..." refers background in this paper

  • ...During the flows of fluids with pressure dependent viscosity, it is possible for concentrations of vorticity to occur adjacent to solid boundaries, and at times even in the interior of the flow, at low Reynolds number (see Rajagopal [18,19])....

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  • ...The authors also would like to thank Prof. K.R. Rajagopal for bringing the present problem to their attention and for some fruitful and valuable suggestions....

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  • ...Some numerical solutions, but in the non-steady case, have been developed by Massoudi and Phuoc [23] and Srinivasan and Rajagopal [24]....

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  • ...It is worth pointing out the fact that the structure of present solutions is completely different of that of Prusa [25] or Rajagopal et al. [26] solutions....

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  • ...General solutions for the same problems in terms of a suitable system of eigenfunctions and eigenvalues, as well as qualitative and uniqueness results, have been established by Rajagopal et al. [26]....

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Journal ArticleDOI
TL;DR: In this article, the authors derive trois solutions des equations de l'elasticite non lineaire and determine les deformations qui sont possibles dans un solide isotrope and incompressible quelconque.
Abstract: Nous avons derive trois solutions des equations de l'elasticite non lineaire. Nous avons determine les deformations qui sont possibles dans un solide isotrope et incompressible quelconque.

230 citations


"Analytical solutions for some unste..." refers background in this paper

  • ...CONTACT Constantin Fetecau c_fetecau@yahoo.com Section of Mathematics, Academy of Romanian Scientists, 050094 Bucharest, Romania © 2020 Informa UK Limited, trading as Taylor & Francis Group An important class of semi-inverse solutions contains the universal solutions established within the context of non-linear elasticity (see Ericksen [1,2]) and those of simplematerials (see Wineman [3], Carroll [4], Fosdick [5])....

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  • ...An important class of semi-inverse solutions contains the universal solutions established within the context of non-linear elasticity (see Ericksen [1,2]) and those of simplematerials (see Wineman [3], Carroll [4], Fosdick [5])....

    [...]