1 Introduction

Non-Newtonian fluids have played an important role in industry and our daily life, and have been widely used in a lot of fields^[1-6], e.g., physiology, chemical, and food industries. Among these applications, pipe flow is the most widely used in the fluid dynamical models, and Poiseuille flow is the most widely used in the basic models. Fundamentally, we consider the steady motion of the viscous incompressible Newtonian fluid in a circular pipe at low Reynolds numbers. For simplification, the nonlinear terms in the equations are easy to remove. Then, the equations become linear, and their the exact solution can be obtained.

There are already many papers for the flow of such non-Newtonian fluids. Rao et al.^[7] considered the flow of a Johnson-Segalman (J-S) fluid through a circular pipe, and found the discontinuous velocity profiles. Qiu et al.^[8-10] studied the effects of the material parameters, including the Weissenberg number, the ratio of viscosities, and the slip parameters, on the shear stress and streamwise velocity of the Poiseuille flow of a J-S fluid, and found that the characteristics of the velocity gradient jumped discontinuously, which allowed for the non-monotonic relationship between the shear stress and the velocity gradient. Oliveira and Pinho^[11] derived the analytic expressions in the fully developed channel and pipe flow of Phan-Thien-Tanner fluids, and showed that the wall shear stress of the Phan-Thien-Tanner fluids was substantially smaller than the corresponding value for a Newtonian or upper-convected Maxwell fluid. Leteliera and Siginer^[12] investigated the fully developed pipe flow of Phan-Thien-Tanner fluids, derived the analytical expressions for the stress components, the friction factor, and the velocity field, and demonstrated that the friction factor, which depended on the Deborah and Reynolds numbers, was substantially smaller than the corresponding value for the Newtonian flow field with the implications concerning the volume flow rate.

The flow of a viscoelastic fluid in a tube with a slowly varying cross section is a basic one with the applications in physiology and engineering. The flow in a blood vessel, the shear stress distribution on the wall, and the velocity distribution of the blood flow are extremely important to investigate the arteriosclerosis of the blood vessel, because the radius of the blood vessel often varies slowly in our body. Manton^[14] obtained an asymptotic series solution for the low Reynolds number flow through an axisymmetric tube, whose radius varied slowly in the axial direction. Sahu et al.^[15] investigated the stability of the flow through a slowly diverging pipe through the examination of the possible role of small local divergences in the transition process.

The J-S fluid model is a kind of representative models for non-Newtonian fluids, which has been widely used in many applications, and the simulation results fit the experimental data in a wide range^[14-19]. In this paper, the flow of a viscoelastic fluid obeying the J-S equation in an axisymmetric pipe is studied, whose radius varies slowly in the axial direction. In Section 2, the physical model and base equations are given. In Section 3, the derivation of the asymptotic solution for the Newtonian fluid is given. In Section 4, according to the governing equations of the J-S model, the perturbation solutions of the stream function ϕ and the vorticity component π to the first-order are derived. In Section 5, according to ϕ and π, we can further get the the velocity profile of the flow.

2 Physical model and basic equations

We consider a circular pipe flow with the radius slowly varying in the domain D (see Fig. 1), where −∞ < z < +∞, 0 ≤ r ≤ R(z), 0 ≤ θ ≤ 2π, (z, r, θ) are the cylindrical polar coordinates, u_z, u_r, and u_θ are the axial, radial, and azimuthal velocities, respectively, and

(1)

For the viscous fluid in a slowly varying pipe, let us recall the work by Manton^[14]. By supposing that the fluid is incompressible and the motion is axisymmetric and steady, we can write the equations for the conservation of mass and momentum as follows:

(2)

(3)

(4)

where u_{z, z} and u_{z, r} are the axial derivative and the radial derivative of u_z, respectively, and u_{r, z} and u_{r, r} are the axial derivative and the radial derivative of u_r, respectively. We introduce a stream function ψ and a vorticity component Ω such that

(5)

The pressure terms can be removed from the momentum equations to get the conservation equation of the vorticity, i.e.,

(6)

Thus, the motion within the domain D obeys Eqs. (5) and (6).

The following conditions must be satisfied on the boundaries of D:

(7)

(8)

(9)

(10)

(11)

Equation (7) means that there is no tangential fluid motion at the wall, while Eqs. (8) and (9) specify the constant flow rate through the tube. Equation (8) implies that there is no fluid motion normal to the tube wall. Equations (10) and (11) ensure that the solution is regular at the axis of the tube. Substituting Eq. (5) into Eqs. (7)-(11) yields the boundary conditions as follows:

(12)

(13)

(14)

(15)

(16)

The required boundary-based problem is therefore completely specified by Eqs. (5)-(16).

Since the solution domain D is infinite in the z-direction, an arbitrary "initial" profile cannot be specified on some cross section of the tube. Hence, the solution can be compared only with the experimental results, which are insensitive to the entry conditions of the tube.

Suppose that we are now seeking a solution for the case when the radius of the pipe R varies slowly in the axial direction. Particularly, the function R is assumed to depend on a small parameter ε, i.e.,

(17)

where R₀ is the characteristic radius of the pipe, and is the particular formula of the varying radius depending on the axial position, the small parameter, and the initial radius. When ε→ 0, the radius of the tube is constant, and the streamfunction is given by the familiar Poiseuille relation as follows:

(18)

i.e., the solution is independent of z. When ε increases from zero, the stream function is expected to depend upon $\varepsilon z$ rather than z alone. Thus, we look for a solution with the form as follows:

(19)

where r' = r/R₀ is the normalized radial coordinate, and z' = εz/R₀ is a slowly varying normalized axial coordinate. Then, we get the normalized equations of motion (dropping the dashes) as follows:

(20)

(21)

which are subjected to the conditions

(22)

(23)

where Re = ψ₀/(R₀ν) is the characteristic Reynolds number of the flow.

Equations (20)-(23) imply that, because ε = o(1), two distinct ranges exist for the order of magnitude of Re, i.e., (ⅰ) εRe = o(1) and (ⅱ) εRe = O(1) or larger. In Case (ⅰ), low Reynolds number viscous solutions are given, in which the viscous effects dominate the non-linear inertial effects. In Case (ⅱ), the inertial terms are comparable with or greater than the viscous terms, and hence gives high Reynolds number inertial solutions. Here, we consider only Case (ⅰ).

3 Asymptotic solution for the Newtonian fluid

We suppose that there exists an asymptotic solution with power series in ε (dropping the dashes) as follows^[13-14]:

(24)

(25)

which are then substituted into the system (20)-(23). We equate the coefficients of like powers of ε of the obtained result. Then, the coefficients ϕ(n) and ω(n) are determined successively from the following systems of equations:

(26)

(27)

(28)

(29)

etc., which are subjected to the boundary conditions as follows:

(30)

(31)

(32)

(33)

(34)

The zeroth-order equation (26) may be readily solved to yield

(35)

where A, B, C, and D are functions of r, which must be determined from the boundary conditions. From the boundary conditions (30)-(34), we can get

(36)

Comparing Eq. (36) with Eq. (18), we can see that the zeroth-order solution is the Poiseuille flow appropriate to the local radius of the pipe. Similarly, the first-order solution is found to be

(37)

This represents an inertial correction allowing for the advection of the zeroth-order vortex lines by the zeroth-order axial velocity component. Therefore, we can get the second-order solution as follows:

(38)

where

(39)

(40)

The first terms in ϕ⁽²⁾ represent an inertial correction accounting for the advection and stretching of the zeroth-and first-order vortex lines by the first-and zeroth-order velocity components, respectively. The second term in Eq. (38) is a viscous correction, which accounts for the curvature in the axial direction of the zeroth-order vortex lines.

Then, we get the solutions for the vorticity as follows:

(41)

(42)

(43)

4 Flow of a J-S fluid through a slowly varying pipe 4.1 Equations and simplifications

The constitutive equation of the J-S model^[20] is

(44)

where

(45)

(46)

(47)

In the above equation, is the material time derivative, D is the symmetric part of the velocity gradient, and W is the skew symmetric part of the velocity gradient, i.e.,

(48)

(49)

−pI denotes the indeterminate part of the stress due to the constraint of incompressibility, μ and η are viscosities, λ is the relaxation time, and a is the slip parameter. When a = 1, the J-S model reduces to the Oldroyd-B model. When μ = 0 and a = 1, the model reduces to a Maxwell fluid. When λ = 0, the model reduces to the classical Navier-Stokes fluid.

Here, we assume μ = 0 to simplify the calculation.

(50)

Then, the momentum equations become

(51)

(52)

4.2 Solution for a J-S fluid in a varying pipe

For the viscoelastic fluid in the expanding pipe, we introduce a stream function Ψ and a vorticity component Π. Then, we look for a solution of the form as follows:

(53)

(54)

Here, we introduce the non-dimensionalized quantities as follows:

(55)

(56)

Then, the constitutive equation will take the form as follows (dropping the dashes):

(57)

where α = λΨ₀/R₀³ is the viscoelastic parameter. When α = 0, we have the Newtonian fluid. If we expand ψ, u_r, u_z, and S in powers of α, we have S⁽⁰⁾ = 2D⁽⁰⁾ at the zeroth-order.

Suppose that

(58)

(59)

where Re = Ψ₀/(R₀ν) is the characteristic Reynolds number of the flow. Then, we have

(60)

(61)

From the momentum equations, we have

(62)

Substituting

(63)

into Eq. (62) and non-dimensionalizing the equation, we have

(64)

Ignoring the second-order terms of ε, we can get the terms of the zeroth-and first-orders of α one by one. Suppose that we have A₀ + αA₁ + O(α₂), where A₀ and A₁ are functions of r, z, and ε. Then, we have

(65)

Suppose that we have B₀ + αB₁ + O(α₂), where B₀ and B₁ are functions of r, z, and ε. Then, we have

(66)

As we know,

(67)

Based on A₀=B₀ and A₁=B₁, ignoring the second-order terms of ε, we get

(68)

From the asymptotic solution of the Newtonian fluid, we get

(69)

Then, we get

(70)

We can obtain the solution of the expanding flow for the viscoelastic fluid (ignoring the second-order terms of ε) as follows:

(71)

(72)

Moreover, we can get the expressions of the axial velocity and radial velocity. According to Eqs. (1), (5), and (71), we have

(73)

(74)

In this section, we consider the effects of the parameters such as the axial distance z, the slowly varying angle ε, and the Reynolds number Re on the axial velocity u_z and the radial velocity u_r of the J-S fluid through a slowly varying pipe. Let R₀ = 1 at the initiation of the variable coss-section, Ψ₀=1/16, and the density of the fluid ρ = 1 000.

First, we would like to check out the effects of z on the distributions of u_z and u_r. Let α=0.05, ε= 1, and Re=100. We will obtain the relationship between the velocity and the radius when the axial distance reaches a certain value. The distribution curves of u_z and u_r for z=2, 8, 14, and 20 are shown in Figs. 2 and 3 after normalization of the radius. From Fig. 2, we can see that u_z reaches the maximum value in the center of the pipe, while the velocity almost reduced to zero at the tube wall. For the same r/R, there will be a smaller u_z with a larger z. From Fig. 3, we can see that u_r is zero in the center of the pipe and at the pipe wall, and reaches the maximum value at r/R=0.6. Moreover, u_r becomes smaller when the axial distance becomes larger at the same r/R.

Then, we can check the effects of the parameter ε by fixing α=0.05 and Re=100 at the cross-section z=6. Figures 4 and 5 show the distribution curves of u_z and u_r for ε=1, 2, 3, and 5, respectively. Figures 4 and 2 have similar variation tendencies. From the figures, we find that, the larger ε is, the smaller u_z is for the same r/R. On the contrary, at the same r/R, u_r becomes larger when the slowly varying angle becomes larger (see Fig. 5), and the maximum value also appears at r/R=0.6.

Finally, we check the effects of the parameter Re on the velocity by fixing α=0.05 and Ψ₀=1/16 while z=6. The distribution curves of u_z and u_r for Re=100, 500, 1 000, and 1 500 are plotted in Figs. 6 and 7. From Fig. 6, we can see that these distribution curves meet at r/R= 0.56, u_z increases with the increase in Re when 0 < r/R < 0.56, and u_z decreases with the increase in Re when 0.56 < r/R < 1. From Fig. 7, we can see that these distribution curves meet at r/R=0.64, u_r increases with the increase in Re when 0 < r/R < 0.64, and u_r decreases with the increase in Re when 0.64 < r/R < 1. Different values of Re have different positions where the maximum value appears. When Re increases, the distance between the position and the center of the pipe decreases.

5 Conclusions

In this paper, the flow of a J-S fluid through a slowly varying pipe is investigated. The asymptotic solutions of the stream function ϕ and the vorticity component π are derived with double perturbation strategy. First, we derive the second-order solution for the Newtonian fluid in the slowly varying pipe by selecting the slowly varying angle ε for the perturbation paragram. Second, with this solution, we derive the first-order solution for the J-S fluid in the slowly varying pipe by selecting the viscoelastic parameter α for the perturbation paragram. Finally, we get the expression of the velocity distribution. The effects of different paragrams, including z, ε, and Re, on the velocity distribution are considered. The increases in both z and ε cause u_z slow down. However, the radial velocity u_r increases with the increase in ε, and decreases with the increase in z. When the effects of Re is studied, we observe some special positions. There are different trends on both sides of the special positions. These results show a good way to get the exact velocity distributions inside the tube, and will lead to a good direction to study the drag reduction problem in non-Newtonian fluid dynamics.

From the stream function ϕ and vorticity ω, we can further get the the pressure distributions and shear stress at the tube wall. From the pressure distribution, we can study the pressure loss in the tube. For the shear stress, as we know, if the shear stress equals zero, we can determine the Reynolds number, at which separation occurs. This will lead to a good direction to study the drag reduction problem in non-Newtonian fluid dynamics.