New exact analytical solutions for Stokes’ first problem of Maxwell fluid with fractional derivative approach

Jamil, Muhammad Kamran; Rauf, Abdul; Zafar, Azhar Ali; Khan, Nasir

doi:10.1016/j.camwa.2011.03.022

Cited by 51 publications

(30 citation statements)

References 29 publications

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“…In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method will be used. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] …”

Section: Development Of the Governing Equationsmentioning

confidence: 99%

“…In general, fractional model of viscoelastic fluids is derived from well known ordinary model by replacing the ordinary time derivatives to fractional order time derivatives and this plays an important role to study the valuable tool of viscoelastic properties. Some authors [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] have investigated unsteady flows of fractionalized viscoelastic second grade, Maxwell, Oldroyed-B, Burgers' and generalized Burgers' model through walls/channel/ annulus and solutions for velocity field and the associated shear stress are obtained by using discrete Laplace transform, Fourier transform, Weber transform and finite Hankel transforms.…”

Section: Introductionmentioning

confidence: 99%

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Helical flows of fractionalized Burgers' fluids

Jamil

Khan

2012

AIP Advances

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The unsteady flows of Burgers’ fluid with fractional derivatives model, through a circular cylinder, is studied by means of the Laplace and finite Hankel transforms. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Ut. The solutions that have been obtained, presented in series form in terms of the generalized Ga,b,c(•, t) functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for fractionalized Oldroyd-B, Maxwell and second grade fluids appear as special cases of the present results. Furthermore, the solutions for ordinary Burgers’, Oldroyd-B, Maxwell, second grade and Newtonian performing the same motion, are also obtained as special cases of general solutions by substituting fractional parameters α = β = 1. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison among models, is shown by graphical illustrations

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Section: Development Of the Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Helical flows of fractionalized Burgers' fluids

Jamil

Khan

2012

AIP Advances

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“…Let us note that the paper [18] also presents plots of the solution of Stokes' first problem for fractional Maxwell fluids with β = 1. However, the plots in Figures 5a, 7a and 9a of [18] do not seem to be monotonically decreasing with respect to the spatial variable, which is a contradiction with the expected monotonic behavior of the solution.…”

Section: Explicit Representation Of Solution and Numerical Resultsmentioning

confidence: 99%

“…For studies on Stokes' first problem for classical Maxwell fluids we refer to [2,3] and the references cited therein. Stokes' first problem for generalized fractional Maxwell fluids is studied in [17,18], where explicit solutions in the form of series expansions are obtained (in [18] only the particular case β = 1 is considered). Explicit solutions of other types of problems for unidirectional flows of fractional Maxwell fluids with constitutive Equation (2) can be found in [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

Bazhlekova

Bazhlekov

2017

Fractal Fract

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Stokes' first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress-strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders in the interval (0, 1]. Explicit integral representation of the solution is derived and some of its characteristics are discussed: non-negativity and monotonicity, asymptotic behavior, analyticity, finite/infinite propagation speed, and absence of wave front. To illustrate analytical findings, numerical results for different values of the parameters are presented.

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“…UCM fluid has been the topic of several research Renardy, 3 Rao and Rajagopal, 4 Qi and Xu, 5 Hayat and Sajid, 6 Hayat et al, 7 Jamil et al, 8 Salah et al 9 and Hayat et al 10 Wang and Tan 11 discussed the flow of Maxwell fluid in a porous medium. Zierep and Fetecau 12 studied Rayleigh-Stokes problem using Maxwell fluid and find exact solution.…”

Section: Sarpakayamentioning

confidence: 99%

Application of homotopy perturbation method for MHD boundary layer flow of an upper-convected Maxwell fluid in a porous medium

Jhankal

2016

Chem. Eng. Res. Bull.

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Abstract:The magneto-hydrodynamics (MHD) boundary layer flow of an electrically conducting upper-convected Maxwell (UCM) fluid in porous medium is studied. The governing similarity equation is solved by He's Homotopy perturbation method (HPM). The main advantage of HPM is that it does not require the small parameters in the equations and hence the limitations of traditional perturbation can be eliminated. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems. The effects of various physical parameters on the flow presented and discussed through graphs.

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New exact analytical solutions for Stokes’ first problem of Maxwell fluid with fractional derivative approach

Cited by 51 publications

References 29 publications

Helical flows of fractionalized Burgers' fluids

Helical flows of fractionalized Burgers' fluids

Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

Application of homotopy perturbation method for MHD boundary layer flow of an upper-convected Maxwell fluid in a porous medium

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