Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate

https://doi.org/10.1016/j.amc.2007.12.045Get rights and content

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate have been established using Fourier sine and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method has been used. The solutions that have been obtained, presented under integral and series forms in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for Oldroyd-B, Maxwell and Newtonian fluids are obtained as limiting cases of our general solutions.

Introduction

The flow characteristics of non-Newtonian fluids are quite different of those from the Newtonian fluids. Therefore, several constitutive equations have been proposed for these fluids. Among them, the models of rate type have received much attention. As early as Oldroyd [1], developed a systematic procedure for developing rate type viscoelastic fluid models. He was careful to build into his framework the invariance requirements that the model ought to meet, but no thermodynamical issue has been taken into consideration. Recently, Rajagopal and Srinivasa [2] have built a systematic thermodynamic framework within which models for a variety of rate type viscoelastic fluids can be obtained. Amid them, the Oldroyd-B model has had some success in describing the response of some polymeric liquids. This model is amenable to analysis and more importantly experimental. For this reason many papers, regarding these fluids, have already appeared in the literature.

Recently, the fractional calculus has encountered much success in the description of viscoelasticity. Especially, the rheological constitutive equations with fractional derivatives play an important role in the description of the behavior of the polymer solutions and melts. The constitutive equations corresponding to the generalized non-Newtonian fluids are obtained from those for non-Newtonian fluids by replacing the time derivatives of an integer order by the so called Rieman–Liouville fractional operators. More exactly, the ordinary derivatives of first, second or higher order are replaced by fractional derivatives of noninteger order [3]. During the last years, many researchers have studied different problems related to such fluids. Here, we shall refer only to those regarding rate type fluids [4], [5], [6], [7], [8], [9], [10], [11]. Some of the obtained results have been also extended to magnetohydrodynamic flows in porous media [12], [13], [14], [15], [16].

The aim of this paper is to establish exact solutions for the unsteady flow of an incompressible generalized Oldroyd-B fluid (GOF) due to an infinite constantly accelerating plate. These solutions, obtained by means of the Fourier sine and Laplace transforms, are presented under integral and series forms in terms of the generalized G and R functions [17]. The similar solutions for Maxwell fluids with fractional derivatives as well as those for the ordinary models are established as limiting cases of our solutions. We would like to emphasize that the exact solutions for different problems with technical relevance are very important. In addition to serving as approximations to some specific initial-boundary value problems, they can be used as tests to verify numerical schemes that are developed to study more complex unsteady flow problems.

Section snippets

Governing equations

The constitutive equations for an incompressible GOF, are given by [9]T=-pI+S,S+λDSDt=μ1+λrDDtA,where T is the Cauchy stress tensor, -pI denotes the indeterminate spherical stress, S the extra-stress tensor, A=L+LT the first Rivlin–Ericksen tensor, where L is the velocity gradient, μ is the dynamic viscosity of the fluid, λ and λr are the relaxation and retardation times andDSDt=DtαS+V·S-LS-SLT,DADt=DtβA+V·A-LA-ALT.In the above relations V is the velocity, is the gradient operator and Dtα

Exact solution for the velocity field

Multiplying both sides of Eq. (8) by 2/πsin(yξ), integrating then with respect to y from 0 to and taking into account the initial and boundary conditions (9), (10), (11), we find that:(1+λDtα)us(ξ,t)t+νξ2(1+λrDtβ)us(ξ,t)=νAξ2πt+λrt1-βΓ(2-β),ξ,t>0,where the Fourier sine transform us(ξ,t) of u(y,t), has to satisfy the conditions:us(ξ,0)=us(ξ,0)t=0forξ>0.Denoting byu¯s(ξ,q)=L{us(ξ,t)}=0e-qtus(ξ,t)dt,the image function of us(ξ,t), applying the Laplace transform to Eq. (12), using the

Calculation of the shear stress

Applying the Laplace transform to Eq. (6) and using the initial condition (5), we find that:τ¯(y,q)=μ1+λrqβ1+λqαu¯(y,q)y.The image function u¯(y,q) of u(y,t) can be easy obtained from Eqs. (18), (19), (20) and (A.4). Introducing the result into (21), it results that:τ¯(y,q)=-2ρAπ1+λrqβ1+λqα01q-1q+νξ2cos(yξ)ξ2dξ+2νAπμ1+λrqβ1+λqα0ξ2cos(yξ)q+νξ2k=0-νξ2λkj=0-1λj(k+1)jΓ(j+1)λr/λqα(j+b)-a-1qα(j+k)+k+2dξ+2νAπμ1+λrqβ1+λqα0ξ2cos(yξ)q+νξ2k=0m1,l0m+l=k-νξ2λkk!λrmm!l!j=0-1λj(k+1)jΓ(j+1)λr

Limiting cases

1. Making λr0 into Eqs. (18), (32), with τ1(y,t), τ2(y,t) and τ3(y,t) given by (24), (28), (30), we attain to the velocity field:u(y,t)=uN(y,t)-2νAπ00tξsin(yξ)e-ν(t-s)ξ2j,k=0-1λj-νξ2λk×(k+1)jsα(j+k)+k+1Γ(j+1)Γ[α(j+k)+k+2]dsdξ,and the associated shear stress:τ(y,t)=τN(y,t)+2μAπ00tcos(yξ)e-ν(t-s)ξ2Rα,α-1-1λ,0,sdsdξ-2νAπμλ0ξ2cos(yξ)j,k=0-1λj-νξ2λk(k+1)jΓ(j+1)×0t0σe-ν(σ-s)ξ2sα(j+k)+k+1Γ[α(j+k)+k+2]Rα,0-1λ,0,t-σdsdσdξ,corresponding to a generalized Maxwell fluid are recovered. In view

Conclusions

In this paper, closed-form expressions for the velocity field u(y,t) and the shear stress τ(y,t) corresponding to the unsteady flow induced by an infinite constantly accelerating plate in an incompressible generalized Oldroyd-B fluid, have been determined using Fourier sine and Laplace transforms. In order to avoid the lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is applied. The solutions that have been obtained satisfy all imposed

Acknowledgements

The authors acknowledge the support from the Ministry of Education and Research, CNCSIS, through PN II-Ideas, Grant No. 26/28-09-2007, CNCSIS code ID-593. They would also like to express their gratitude to the referees for their careful assessment and for letting know about the similar results in the field.

References (25)

  • W.C. Tan et al.

    A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

    Int. J. Non-Linear Mech.

    (2003)
  • H.T. Qi et al.

    Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders

    Acta Mech. Sin.

    (2006)
  • Cited by (78)

    • Analysis of the time-space fractional bioheat transfer equation for biological tissues during laser irradiation

      2021, International Journal of Heat and Mass Transfer
      Citation Excerpt :

      But all of them have been presented based on the Fourier heat conduction law. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and processes and has found much successful applications in solving real-world problems [30–32], such as viscoelasticity [33,34], fluid flow [35,36] and heat transfer [37]. Many published researches proved that mathematical models with fractional derivatives can be applied to describe the heat transfer process in non-homogeneous and porous mediums [38–41].

    • Theoretical treatment of radiative Oldroyd-B nanofluid with microorganism pass an exponentially stretching sheet

      2020, Surfaces and Interfaces
      Citation Excerpt :

      Maxwell fluid model designated only the relaxation time, but Oldroyd-B fluid model defined both relaxation and retardation time and by the general flow conditions it can capture the viscoelastic feature of dilute polymeric solution. The Oldroyd-B fluid by constantly accelerating plate is carried out by Vieru et al. [1]. Numerical investigation of Oldroyd-B fluid flow with transverse magnetic field through an exponentially stretching sheet is evaluated by Nadeem et al. [2].

    • Effect of slip boundary condition on flow and heat transfer of a double fractional Maxwell fluid

      2020, Chinese Journal of Physics
      Citation Excerpt :

      More analyses on the slip boundary conditions of integer order derivatives are given in the literature [9–13]. Recently, fractional derivatives were found to be quite flexible in describing the viscoelastic constitution and heat conduction law [14–20] due to the long-term memory and hereditary characteristics. Friedrich [21] first formulated the fractional Maxwell model with different order derivatives of stress and strain, which revealed that viscoelastic fluid exhibits a fluid-like behavior in the case of strain derivative.

    View all citing articles on Scopus
    View full text