广义Oldroyd-B粘弹性流体Stokes第一问题
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Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B
model1
Haitao Qi Department of Applied Mathematics and Statistics, Shandong University at Weihai
=
μ
⎡⎢⎣1 + θ
D Dt
⎤ ⎥⎦
A,
(2.2)
where λ and θ are relaxation and retardation times, μ is the dynamic viscosity of fluid, T is the Cauchy stress tensor, − pI denotes the indeterminate spherical stress, A = L + LT with L the velocity gradient and D / Dt (upper convected time derivative) operating on any tensor B is defined
Weihai, P. R. China 264209 htqi@sdu.edu.cn
Mingyu Xu School of Mathematics and Systematical Science, Shandong University
Jinan, P.R. China, 250100
Abstract
1
http://www.paper.edu.cn
பைடு நூலகம்
classical Newtonian fluid and the Maxwell fluid. For some special flows the model of second grade fluid is also included. Here, we mention some of the studies such as [2-5].
Recently, fractional calculus has encountered much success in the description of the complex dynamics. In particular it has proved to be a valuable tool to handle viscoelastic properties. The starting point of the fractional derivative model of viscoelastic fluid is usually a classical differential equation which is modified by replacing the classical, time derivatives of an integer order by the fractional calculus operators. This generalization allows one to define precisely non-integer order integrals or derivatives [6]. Fractional calculus has been found to be quite flexible in describing viscoelastic behavior [7-10]. More recently, Huang et al. [11-16] discussed some unsteady flows of the generalized second grade fluid. The unidirectional flow of a viscoelastic fluid with the fractional Maxwell model is considered by Tan et al. [17,18]. The flows with a generalized Jeffreys model in an annular pipe is also studied by Tong et al. [19,20].
1 Supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai.
The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach has been taken into account in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox-H function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.
Keywords: Generalized Oldroyd-B fluid, Stokes' first problem, Fractional calculus, Exact solution, Fox- H function.
1 Introduction
Navier-Stokes equations are the most fundamental motion equations in fluid dynamics. However, there are few cases in which their exact analytical solutions can be obtained. Exact solutions are very important not only because they are solutions of some fundamental flows, but also because they serve as accuracy checks for experimental, numerical, and asymptotic methods. The inadequacy of the classical Navier-Stokes theory to describe rheologically complex fluids such as polymer solutions, blood and heavy oils, has led to the development of several theories of non-Newtonian fluids. In order to describe the non-linear relationship between the stress and the rate of strain, numerous models or constitutive equations have been proposed. The model of differential type and those of rate type have received much attention [1]. In recent years, the Oldroyd-B fluid has acquired a special status amongst the many fluids of the rate type, as it includes as special cases the
On the other hand, almost every one made a study of fluid mechanics is familiar with Stokes’ celebrated paper on pendulums (in 1851) in which he describes a kind of classical problems of the impulsive and oscillatory motion of an infinite plate in its own plane. Nowadays, the flows over a plane wall which is initially at rest and is suddenly moved in its own plane with a constant velocity and with a harmonic vibration are termed Stokes’ first (or Rayleigh-type) and second problems, respectively [21,22]. At present, one paid more and more attention to the Stokes problems due to important theoretical and practice meanings [23-25]. The aim of this paper is to investigate the Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model. The fractional calculus approach has been taken into account in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox-H function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.
2 Basic equations
First, we know that the constitutive equation of an incompressible, ordinary Oldroyd-B fluid is of the form [1]
T = − pI + S,
(2.1)
S
+
λ
DS Dt
Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B
model1
Haitao Qi Department of Applied Mathematics and Statistics, Shandong University at Weihai
=
μ
⎡⎢⎣1 + θ
D Dt
⎤ ⎥⎦
A,
(2.2)
where λ and θ are relaxation and retardation times, μ is the dynamic viscosity of fluid, T is the Cauchy stress tensor, − pI denotes the indeterminate spherical stress, A = L + LT with L the velocity gradient and D / Dt (upper convected time derivative) operating on any tensor B is defined
Weihai, P. R. China 264209 htqi@sdu.edu.cn
Mingyu Xu School of Mathematics and Systematical Science, Shandong University
Jinan, P.R. China, 250100
Abstract
1
http://www.paper.edu.cn
பைடு நூலகம்
classical Newtonian fluid and the Maxwell fluid. For some special flows the model of second grade fluid is also included. Here, we mention some of the studies such as [2-5].
Recently, fractional calculus has encountered much success in the description of the complex dynamics. In particular it has proved to be a valuable tool to handle viscoelastic properties. The starting point of the fractional derivative model of viscoelastic fluid is usually a classical differential equation which is modified by replacing the classical, time derivatives of an integer order by the fractional calculus operators. This generalization allows one to define precisely non-integer order integrals or derivatives [6]. Fractional calculus has been found to be quite flexible in describing viscoelastic behavior [7-10]. More recently, Huang et al. [11-16] discussed some unsteady flows of the generalized second grade fluid. The unidirectional flow of a viscoelastic fluid with the fractional Maxwell model is considered by Tan et al. [17,18]. The flows with a generalized Jeffreys model in an annular pipe is also studied by Tong et al. [19,20].
1 Supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai.
The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach has been taken into account in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox-H function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.
Keywords: Generalized Oldroyd-B fluid, Stokes' first problem, Fractional calculus, Exact solution, Fox- H function.
1 Introduction
Navier-Stokes equations are the most fundamental motion equations in fluid dynamics. However, there are few cases in which their exact analytical solutions can be obtained. Exact solutions are very important not only because they are solutions of some fundamental flows, but also because they serve as accuracy checks for experimental, numerical, and asymptotic methods. The inadequacy of the classical Navier-Stokes theory to describe rheologically complex fluids such as polymer solutions, blood and heavy oils, has led to the development of several theories of non-Newtonian fluids. In order to describe the non-linear relationship between the stress and the rate of strain, numerous models or constitutive equations have been proposed. The model of differential type and those of rate type have received much attention [1]. In recent years, the Oldroyd-B fluid has acquired a special status amongst the many fluids of the rate type, as it includes as special cases the
On the other hand, almost every one made a study of fluid mechanics is familiar with Stokes’ celebrated paper on pendulums (in 1851) in which he describes a kind of classical problems of the impulsive and oscillatory motion of an infinite plate in its own plane. Nowadays, the flows over a plane wall which is initially at rest and is suddenly moved in its own plane with a constant velocity and with a harmonic vibration are termed Stokes’ first (or Rayleigh-type) and second problems, respectively [21,22]. At present, one paid more and more attention to the Stokes problems due to important theoretical and practice meanings [23-25]. The aim of this paper is to investigate the Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model. The fractional calculus approach has been taken into account in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox-H function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.
2 Basic equations
First, we know that the constitutive equation of an incompressible, ordinary Oldroyd-B fluid is of the form [1]
T = − pI + S,
(2.1)
S
+
λ
DS Dt