Unsteady Axial Viscoelastic Pipe Flows of an Oldroyd B Fluid

The unsteady flow of a fluid in cylindrical pipes of uniform circular cross-section has appli‐ cations in medicine, chemical and petroleum industries [3,4,5]. For viscoelastic fluids, the unsteady axial decay problem for UCM fluid is considered by Rahman et al. [6]; and for Newtonian fluids as a special case. Rajagopal [7] has studied exact solutions for a class of unsteady unidirectional flows of a second-order fluid under four different flow situations. Atalik et al. [8] furnished a strong numerical evidence that non-linear Poiseuille flow is un‐ stable for UCM, Oldroyd-B and Giesekus models. This fact is supported experimentally by Yesilata, [9]. The unsteady flow of a blood, considered as Oldroyd-B fluid, in tubes of rigid walls under specific APGs is concerned by Pontrelli, [10, 11].


Formulation of the problem
The momentum and continuity equations for an incompressible and homogenous fluid are given by , dq P S dt = -Ñ + Ñ × r (1) and 0, q where ρ is the material density, q is the velocity field, p is the isotropic pressure and S

̿
is the Cauchy or extra-stress tensor.The constitutive equation of Oldroyd-B fluid is written as where T ̿ is the total stress, I ̿ is the unit tensor, μ is a constant viscosity, λ 1 and λ 2 , (0 ≤ λ 2 ≤ λ 1 ) are the material time constants, termed as relaxation and retardation times; respectively.The deformation tensor A .
and "∇ " denotes the upper convected derivative ; i.e. for a symmetric tensor G ̿ we get, .
The symmetry of the problem implies that S ̿ and q depend only on the radial coordinate r in the cylindrical polar coordinates (r,θ,z) where the z-axis is chosen to coincide with the axis of the cylinder.Moreover, the velocity field is assumed to have only a z-component, i.e. (0, 0, ), which satisfies the continuity equation ( 2) identically.The substitution of Eq. ( 6), into Eqs.
(1) and (3) yields the set of equations Equations ( 8) and ( 9) imply that the pressure function takes the form; p = z f (t) + c, so that The elimination of S rz from ( 7) and (8) shows that velocity field w(r, t) is governed by: ) .
The non-slip condition on the wall and the finiteness of w on the axis give 0 and ( , ) 0 0.

| |
r R r w w r t r Introducing the dimensionless quantities where R is the radius of the pipe, ΔP a characteristic pressure difference, L is a characteristic length, We and Re are the Weissenberg and Reynolds numbers; respectively, into Eqs.(10), (11) and ( 12) we get with the BCs.

Pressure gradient varying exponentially with time
We consider the two cases of exponentially increasing and decreasing with time APGs separately.where K and α are constants.The substitution of Eqs. ( 17) and (18) into Eq. ( 14 leads to

Pressure gradient increasing exponentially with time
where I 0 (x) is the modified Bessel-functions of zero-order, and (1 ) . 1 Therefore, the velocity field is given by . ( ) The solution given by Eq. ( 23) processes the following properties:

i.
The time dependence is exponentially increasing such that for η ≠ 1 lim It may be recommendable to choose another APG which increases up to a certain finite limit in order to keep ϕ(η, τ) finite. ii.
The present solution depends on the parameter β in the same form as the solution for the UCM [6].For any value of β the Oldroyd-B fluid exhibits the same form as the UCM-fluid.However, in the present case β depends on λ in addition to H and α 2 .A close inspection show that lim λ→0 β 2 = β 2 for the UCM-fluid while the lim which coincides with the case of the Newtonian fluid, [8]. iii.
The parameter β is inversely proportional to λ where the decay rate increases by increasing the value of H.However, as mentioned above, as λ approaches the value λ = 1 all the curves matches together approaching the value β 2 = α 2 asymptotically.The behavior of β as a function of λ, where H is taken as a parameter is shown in Fig. (1).
For small values of | β | and by using the asymptotic expansion of I 0 (x), it can be shown that the velocity profiles approaches the parabolic distribution; For the case of large | β | the velocity distribution is given as;

Pressure gradient decreasing exponentially with time
The solution at present is obtained from the previous case by changing α 2 by − α −2 .There- fore, where (1 ) 1 The discussion of this solution is similar to the case of increasing APG except that the velocity decays exponentially with time and the value α 2 = 1 / λH is not permissible as it leads to infinite β 1 2 ; i.e.
The two cases of small and large | β 1 | produce similar results as the previous solution.Thus

Constant pressure gradient
Here we consider the flow to be initially at rest and then set in motion by a constant ABG "-K".Hence, Ψ(τ) ; Eq.( 14), subject to BCs. (15) reduces to .

L P K
Therefore, we need to solve the equation subject to the boundary and initial conditions Equation ( 46) can be transformed to a homogenous equation by the assumption ( , ), 4 where Ψ(η,τ) represents the deviation from the steady state solution.Hence, subject to the boundary and initial conditions (1 ) 0.
Τhe initial condition (50) and BCs.(51) will not be sufficient to evaluate the constants A m and B m .Hence, it is required to employ another condition.We assume that G(τ) is smooth about the value τ = 0 and can be expanded in a power series about τ = 0. Assuming G(τ) to be linear function of τ in the domain about τ = 0, then G ″ = 0 in Eq. (52).Hence To determine the constants A m and B m we firstly satisfy the remaining condition (51).Owing to Eq. ( 58) and the initial condition, Eq. ( 51), we notice that, ). 4 Via the Fourier-Bessel series, Eq. (62) leads to, Performing this integration we get From Eqs. ( 61) and ( 64) we obtain : Finally, the velocity field has the series representation The constant-APG velocity field φ(η,τ) as a function of η shown in Fig. (5).

Results and discussion
The behavior of |β| as a function of λ where H is taken as a parameter is shown in Fig. (1).The behavior of β is inversely proportional to λ while it is fast-decreasing for higher H-values.For any β-value, the Oldroyd-B fluid exhibits the same form as the UCM-fluid.A close inspection of β 2 = α 2 (1+α 2 H)/(1+λα 2 H) shows that UCM-fluid is obtained by lim λ→0 β 2 = β 2 while lim λ→0 β 2 = α 2 leads to the case of Newtonian fluid.For small values of |β| as well as | βη| and by using the asymptotic expansion of I 0 (x), it can be shown that the velocity profiles approaches the parabolic distribution.
For decay-APGs, Figs.(2a) and (2b) show that the velocity profiles of Oldroyd-B and UCM fluids are parabolic for small values of |βη| while for large |βη| they are completely different from this situation.The solutions depend on η only in the neighboring of the wall.Therefore, such fluids exhibit boundary layer effects [17]].
For pulsating-APG, the velocity distribution is represented in Figs. ( 3a ) and ( 3b).The smallest value of β in both curves is almost parabolic as shown by Eq. (36) while the largest value exhibits boundary effect as reviled by Eq.( 43 ).To emphasize the oscillating nature of the solution a three-dimensional diagrams (4a) and (4b) for the smallest and largest values of |β| are respectively sketched.
Grigioni, et al [1], studided the behavior of blood as a viscoelastic fluid using the Oldroyd-B model.The results obtained for the velocity distribution stands in agreement with the obtained results in the present work.
solution is completely different from the parabolic distribution and it depends on η on- ly in the neighborhood of the wall.Therefore, such a fluid exhibits boundary effects.The rising-APG velocity field ϕ(η, τ) is plotted in Figs.(2a) and (2b) as a function of η at different values of β for α = 2 and α = 5.