Abstract
An exact solution of an incompressible second-grade fluid for flow between two coaxial cylinders with porous walls is given. It is assumed that the inner cylinder is rotating with a constant angular velocity and the outer one is at rest. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number and the elastic number. The solutions for , , and values of the cross-Reynolds number are obtained and a comparison with those of the Newtonian fluid is given. Furthermore, the torque exerted by the fluid on the inner cylinder is calculated. It is shown that the moment coefficient depends on the cross-Reynolds number, the elastic number, and the ratio of the radii of the cylinders. The variation of the moment coefficient with these numbers is discussed.
1. Introduction
The flow considered in this paper is for flow of a second-grade fluid in an annulus with porous walls. For a Newtonian fluid, the flow in an annulus has been investigated by Berman [1, 2] and Yuan [3]. The extensions of this flow to non-Newtonian fluids have been studied by many authors [4–6]. However, the results for non-Newtonian fluids have been given by using a perturbation method. Therefore, it is not possible to compare with the exact solution of a Newtonian fluid for all values of the cross-Reynolds number which is defined by the suction velocity at the surface of the cylinder, the radius of the cylinder, and the kinematic viscosity of the fluid. The exact solution given in this paper provides comparison with the exact solution of a Newtonian fluid for all values of the cross-Reynolds number. The non-Newtonian fluid model in this paper is that of the second-grade fluid [7]. Although there are some criticisms on the applications of this model, many papers have been published and a listing of some of them may be found in the literature. Furthermore, it has been shown by Walters [8] that for many types of problems in which the flow is slow enough in the viscoelastic sense, the results given by Oldroyd’s constitutive equation will be substantially similar to those of fluids of the second or third grade. Therefore, it would seem reasonable to use the second- or third-grade models in carrying out the calculations. This is particularly so in view of the fact that the calculation is generally simpler. The constitutive equation for an incompressible second-grade fluid has three coefficients. There are some restrictions on these coefficients due to the Clausius-Duhem inequality and due to the assumption that the Helmhotlz free energy is minimum in equilibrium. A comprehensive discussion on the restrictions for these coefficients has been given by Dunn and Fosdick [9] and Dunn and Rajagopal [10]. The sign of the material moduli is the subject of much controversy [11, 12].
The no-slip boundary condition is sufficient for a Newtonian fluid, but may not be sufficient for a fluid of second grade. Therefore, one needs an additional condition at boundary. However, if one uses a perturbation expansion in terms of the coefficient appearing in the higher-order derivative of the governing equation, the no-slip boundary condition is sufficient. A critical review on the boundary conditions, the existence, and uniqueness of the solution has been given by Rajagopal [13]. In order to overcome the difficulty on the boundary condition, several workers have studied to give an additional condition. Frater [14] has studied the problem without using a perturbation expansion in terms of the coefficient of the higher-order term of the governing equation. He imposed an additional condition so that the solution tends to the Newtonian value as the coefficient of the higher derivative in the equation approaches zero. This can be used provided the explicit solution or the numerical solution should exist. In this paper, the solution is obtained in terms of the confluent hypergeometric functions and has three coefficients: two of them can be determined by the no-slip boundary condition and the other can be determined by using the properties of the confluent hypergeometric functions.
The problem considered in this paper is an extension of a viscous fluid in an annulus with uniform porous walls to the problem for the flow of a second-grade fluid. The fluid injection rate at one wall is taken equal to the withdrawal rate at the outer wall. Further, because of this restriction on the cross wall, the velocity components do not vary along the axis of the cylinder. The flow in an annulus with porous walls has been investigated by many authors [4–6]. In order to obtain the solution in [4, 6], a perturbation method has been used, therefore, the results obtained can be applicable for small values of the cross-Reynolds number. The axial flow in an annulus with porous walls of a second-grade fluid has been investigated in [15]. It can be shown that the axial and the azimuthal velocities are superposable. The solution for the flow of a second-grade fluid between two parallel porous plates has been given in [16]. The flow of a second-grade fluid generated by a rotating porous cylinder has been investigated in [17]. The flow of a second-grade fluid between eccentric rotating cylinders was examined in [18]. Assuming that the eccentricity is small, the secondary flow for a third-grade fluid between two eccentrically placed cylinders has been investigated in [19]. The flow of a second-grade fluid in a porous cylindrical tube was examined for suction or injection through the surface of the tube, under the assumption that the second-order parameter is sufficiently small [20]. In this paper, the solution is obtained in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number and the elastic number. The solutions for , , and values of the cross-Reynolds number are discussed. The solution for value of the cross-Reynolds number is given in terms of the modified Bessel functions of the first kind of order one. The solution for value of the cross-Reynolds number is given in terms of the modified Bessel function of the second kind of order one. The solution for value of the cross-Reynolds number is given in terms of the exponential integral.
The torque exerted by the fluid on the inner cylinder is calculated. This can be used to obtain the values of the material coefficients. It is shown that the moment coefficient depends on the cross-Reynolds number, the elastic number, and the ratio of the radii of the cylinders. The variation of the moment coefficient with these numbers is discussed. There are appreciable differences between that of the Newtonian fluid and that of the second-grade fluid.
2. Basic Equations
The equation of motion for a fluid in the absence of body forces is where is the density of the fluid, is the velocity, is the stress tensor, and represents the material derivative. The continuity equation for the velocity is Equations (2.1) and (2.2) can be applied to all types of incompressible Newtonian and non-Newtonian fluids. The stress depends on the local properties of the fluid. The relation between the stress and the local properties of the fluid is called constitutive equation. The constitutive equation for an incompressible second-grade fluid is in the following form [7]: where , , and are material constants, and represents the Rivlin-Eriksen tensor defined as [21] where is time, is pressure, and is the identity tensor. The Clausius-Duhem inequality and the condition that Helmholtz free energy is minimum in equilibrium provide the following restrictions [9, 10]: A comprehensive discussion on the restriction for , , and can be found in the work by Dunn and Rajagopal [10]. The sign of the material moduli and is the subject of much controversy [11, 12]. The experiments have not confirmed these restriction on and . Thus, the conclusion is that the fluids which have been tested are not fluids of second grade and characterized by a different constitutive structure.
Fully developed laminar flow of an incompressible fluid of second grade in an annulus with porous walls is considered. The cylindrical polar coordinates are used. The radii of the porous cylinders are and . The rate of fluid withdrawal at one wall of the annulus is assumed to be equal to the rate of injection of fluid at the other wall, and that these rates are independent of axial position in annulus. The condition that the suction rate at wall is equal to the injection rate at the outer is satisfied if where and are the radial velocities at walls of smaller and larger pipes, respectively. Equation (2.6) provides that the velocity is independent of the axial position in annulus. The velocity field is assumed to be in the following form: where are components of the velocity in cylindrical polar coordinates, is positive for injection at the inner cylinder and negative for suction at the inner one. The velocity field given by (2.7) is not a special case of the flow that is given in [22]. The velocity field in [22] is a superposition of the stagnation point flow and the rotating flow. Equation (2.2) is satisfied identically by the velocity given by (2.7). Inserting the velocity given by (2.7) into the expression of the stress, the components of the stress tensor, in cylindrical polar coordinates, can be written in the following forms: where , , ; the primes denote differentiation with respect to . Inserting the stress components and the velocity given by (2.7) into (2.1), one obtains and the boundary conditions are where is the radius of the inner cylinder and is the radius of the outer cylinder. The solution to (2.9) has three coefficients: two of them can be determined by the no-slip condition (2.10) and the other can be determined by the confluent hypergeometric functions. It is expected that when the viscoelastic parameter goes to zero, the solution should give the Newtonian case [23], leading to entirely two different kinds of flow behavior. Similar discussions can be found in [24–26].
3. Solution of the Problem
After some manipulations, (2.9) takes the form where where prime denotes the differentiation with respect to , is the new function, is the new variable, is the cross-Reynolds number, and is the elastic number and the boundary conditions are Equation (3.3) denotes that the inner cylinder is rotating with angular velocity and the outer one is at rest. The solution of (3.1) can be written in the following form: where and are the confluent hypergeometric functions [27, 28]. The solution has three coefficients; therefore, one needs three boundary conditions. Thus, unless an additional condition is prescribed over the conditions (2.10), one has a parametric solution. However, the third coefficient can be determined by the properties of the confluent hypergeometric functions. For becomes negative; then for is not acceptable and therefore must be zero and (3.4) takes the form Using the identity in [28], The integration gives where . Using the identity in [28] which is and by the boundary condition (3.3), one obtains where and . When goes to zero, using the asymptotic expression of in [28], becomes or which is the expression of the velocity of a Newtonian fluid. The variation of with respect to for various values of and is illustrated in Figure 1. The value of is taken as and the values of are taken as and . Equation (3.9) is valid for all values of and . When goes to infinity, using the expression given in [28] which is
equation (3.9) takes the following form: where is the Gamma function and is the modified Bessel function of the first kind of order one. The variation of with respect to for various values of is illustrated in Figure 1. Since the asymptotic form of is when goes to zero, (3.13) becomes Equation (3.14) satisfies the boundary condition at , but it does not satisfy the boundary condition at . It is clearly seen from Figure 1 that when goes to , the flow shows a boundary layer character near the outer cylinder for a Newtonian fluid, but it does not behave the same way for a fluid of second grade. A similar situation occurs for the axial flow between two circular cylinders with porous walls of a second-grade fluid [15].
For , is positive; then for , becomes an increasing function of ; therefore, must be zero, then (3.4) takes the form where . Using the identity in [28] which is and by the boundary condition, the integration gives where . When goes to zero, using the asymptotic expression of in [28], one obtains which is the expression of the velocity for a Newtonian fluid. The variation of , with respect to for various values of and , is illustrated in Figure 1. The value of is taken as and the values of are taken as and . Equation (3.17) is valid for all values of and . When goes to infinity, using the asymptotic expression given in [28] which is Equation (3.17) takes the following form: where is the modified Bessel function of the second kind of order one. The variation of with respect to for various values of is illustrated in Figure 1. Since the asymptotic form of is when goes to zero, (3.20) becomes Equation (3.21) satisfies the boundary condition at , but it does not satisfy the boundary condition at . The reason is that a boundary layer occurs at for a Newtonian fluid, but it does not behave the same way for a fluid of second grade.
The case of , namely , cannot be obtained by (3.17). For , by the definition of , can be found as where is the exponential integral defined in the following form Inserting into the expression of and by integration and by using the boundary conditions, one finds The variation of with respect to for various values of is illustrated in Figure 1. The value of is taken as and the values of are taken as and . When goes to zero, the argument of the exponential integral becomes large, then can be written as and (3.24) becomes
4. The Torque Exerted by the Fluid on the Inner Cylinder
The torque exerted by the flow on the inner cylinder is given in the following form: where h is the length of the inner cylinder. The moment coefficient is Equation (4.2) shows that the moment coefficient depends on the cross-Reynolds number, the elastic number, and the ratio of the radii of the cylinders. The variation of the moment coefficient with these numbers is illustrated in Figure 2. There are appreciable differences between that of the Newtonian fluid and that of the second-grade fluid. The curves in Figure 2 show the effect of the second-grade fluid. The moment coefficient decreases with the cross-Reynolds number for a Newtonian fluid and increases with the cross-Reynolds number for a second-grade fluid, for all values of the ratio of the radii of the cylinders.
5. Conclusions
An exact solution of the governing equation for the flow of a fluid of second grade between two coaxial cylinders with porous walls is given. The flow of a second-grade fluid for which the inner cylinder is rotating and the outer one is at rest is considered. The solution is given in terms of the confluent hypergeometric functions and is valid for all values of the cross-Reynolds number and the elastic parameter. The boundary conditions are the no-slip boundary conditions and an additional boundary condition. The additional boundary condition is that the solution tends to the Newtonian value as the elastic number in the solution approaches zero. The solution is discussed for positive, negative, infinite, and values of the cross-Reynolds number. The torque exerted by the fluid on the inner cylinder is calculated. It is shown that the moment coefficient depends on the cross-Reynolds number, the elastic number, and the ratio of the radii of the cylinders. For a given value of this ratio, it decreases with the cross-Reynolds number for a Newtonian fluid and increases with the cross-Reynolds number for a second-grade fluid.
Acknowledgments
It is pleasure to acknowledge many stimulating correspondences with Professor K. R. Rajagopal and the authors would like to express their thanks to the referees for the valuable suggestions.