Unsteady flow of fractional Oldroyd-B fluids through rotating annulus

1 Introduction

Humans benefit from fluid flow but also need to understand the negative effects in nature, e.g. hurricanes and cyclones. The topographic conditions and the sufficient supply of rain flow is important in the cultivation of crops. Cultivation in deserts can also be achieved with artificial irrigation. Artificial breathing machines, artificial hearts and dialysis systems are designed using fluid dynamics. Hence fluids are vital.

The stress and rate of strain of a fluid is related: fluid is Newtonian if this relationship is linear, if not, the fluid is non-Newtonian. The non-Newtonian fluids, like polymers, plasma, paste and pulps are very important in industry. Therefore, the discussion of non-Newtonian fluids in fluid mechanics is important. All existing fluids cannot be described by the Navier-stocks equations. By categorizing different fluids, we have different mathematical models to represent fluids. Oldroyd-B fluid is one of these rate-type models.

The non-Newtonian fluids possessing shear dependent viscosity are known as viscoelastic fluids. The first rate type fluid model is a Maxwell model that is still in used. Fetecau et. al [1] derived the results of the velocity field and shear stress of an Oldroyd-B fluid on a plate, accelerating constantly. Hayat et. al [2] determined the solutions for MHD flow over an infinite oscillatory plate about an axis that is normal to the oscillatory plate. Some authors determined solutions for a viscoelastic fluid model by reproducing a kernel method (RKM) [3,4,5,6]. The unsteady incompressible Oldroyd-B fluid is analyzed by Jamil et. al [7] and special cases are also obtained for Maxwell, second grade and Newtonian fluids. Similarly, Burdujan [8] used this idea on Taylor-Couette flow and calculated the velocity field and shear stress of a rotating fluid between annuls of double cylinders. Numerical solutions of fractional fluids are also under attention [9, 10].

A lot of interest is given to a rotating circular domain corresponding to helical flow for second grade, Maxwell fluids and Oldroyd-B [11,12,13,14]. The flow of a polymer in a circular cylinder by considering pulsatile APG was analyzed by Barnes et. al [15, 16]. Davies et. al [17] and Phan-Thien [18] presented the same problem for a White-Metzner fluid. To solve the viscoelastic motion of different fluid models, one needs to learn about fractional calculus [19, 20]. Fractional calculus is very useful in fields of mathematics and physics [21,22,23]. Furthermore, the interested reader can consult the references [24,25,26,27,28,29,30].

In this proposal, we consider the fractional Oldroyd-B fluid model and study the flow due to the circulation of two pipes around its axial. Integral transforms are used to obtain the general solutions and the final result is in the form of hypergeometric functions. There are some limiting cases by applying the limits on physical parameters, i.e ς →1, λ_r →0 and λ → 0, solutions for the ordinary Oldroyd-B, fractional and ordinary Maxwell and Newtonian fluids are discussed respectively.

2 Development of the flow

Consider a fractional Oldroyd-B fluid (OBFFD) between two coaxial pipes. The radii of the pipes are R₁ and R₂(>R₁), after time t = 0⁺, both pipes and the fluid between them start rotating along their axis. For the considered problem, the z-axis is along the axis of the pipes, the r-axis is perpendicular to the axis of the pipe and θ is along the tangent to the boundary. Let v_r, v_θ and v_z be the velocity components in the direction of the r, θ and z axes respectively.

The continuity and linear momentum equations for incompressible flow is

where V, t, ρ, b, d/dt is velocity, time, density, body force and the material time derivative respectively.

For an incompressible Oldroyd-B fluid [36]

where p is the hydrostatic pressure, I denotes the identity tensor, S is the extra-stress tensor, λ and λ_r are relaxation and retardation times, superposed dot indicates the material time derivative, L represents the velocity gradient, A = L + L^T is the first Rivlin Ericsen tensor, μ is the dynamic viscosity of the fluid and the superscript T denotes the transpose operation. For the problem in consideration, the velocity field and the extra stress tensor has the following form [31]

For such flows, (1) is true.

The governing equations for a Oldroyd-B fluid, corresponding to the above defined motions are [31]

where ν is the kinematic viscosity and τ(r, t) is the non zero shear stress.

The governing equations for a fractional Oldroyd-B fluid is obtained by using a fractional differential operator Dtς, defined by [32, 33]

in governing equations (5) and (6) we have

3 Initial and boundary conditions

We need three conditions to solve a fractional partial differential equation (9). The relevant initial and boundary conditions are

where ℧₁, ℧₂ and κ are real numbers.

4 Limiting cases

1. Letting ς → 1 into Eqs. ((24)) and ((28)) the velocity field

   ω(r,t)=℧1R12(R22−r2)+℧2R22(r2−R12)r(R22−R12)tκ−πλ×∑n=1∞J12(R1rn)A(r,rn)℧2R2J1(R1rn)−℧1R1J1(R2rn)J12(R1rn)−J12(R2rn)×∑k=0∞∑m=0kk!m!(k−m)!−νrn2λk×λrm[G1,m−k−κ−1,k+1−λ−1,t+λG1,1+m−k−κ−1,k+1−λ−1,t],(29)

   and the shear stress

   τ(r,t)=2μR12R22(℧2−℧1)λ(R22−R12)r2[R1,−κ−1−λ−1,t+λr×R1,−κ−λ−1,t]+μπλ×∑n=1∞2rA(r,rn)−rnA∗(r,rn)J1(R1rn)J12(R1rn)−J12(R2rn)×℧2R2J1(R1rn)−℧1R1J1(R2rn)∑k=0∞∑m=0kk!m!(k−m)!×−νrn2λkλrm[G1,m−k−κ−1,k+1−λ−1,t+λrG1,1+m−k−κ−1,k+1−λ−1,t],(30)

   for the ordinary Oldroyd-B fluid.

2. Making λ_r →0 in Eqs. ((24)) and ((28)), the velocity field

   ω(r,t)=℧1R12(R22−r2)+℧2R22(r2−R12)r(R22−R12)tκ−πλς∑n=1∞J1(R1rn)A(r,rn)J12(R1rn)−J12(R2rn){℧2R2J1(R1rn)−℧1R1J1(R2rn)}∑k=0∞−νrn2λςk×{Gς,−k−κ−1,k+1−λ−ς,t+λGς,ς−k−κ−1,k+1−λ−ς,t},(31)

   and the shear stress

   τ(r,t)=2μR12R22(℧2−℧1)λr2(R22−R12)Rς,−2−λ−1,t+πμλς∑n=1∞J1(R1rn)2rA(r,rn)−rnA∗(r,rn)J12(R1rn)−J12(R2rn)××℧2R2J1(R1rn)−℧1R1J1(R2rn)×∑k=0∞−νrn2λςkGς,−k−2,k+1−λ−ς,t,(32)

   corresponding to a fractional Maxwell fluid.

3. Making ς → 1 in Eqs. ((31)) and ((32)), the velocity field is

   ω(r,t)=℧1R12(R22−r2)+℧2R22(r2−R12)r(R22−R12)tκ−πλς∑n=1∞J1(R1rn)A(r,rn)J12(R1rn)−J12(R2rn){℧2R2J1(R1rn)−℧1R1J1(R2rn)}∑k=0∞−νrn2λk×{G1,−k−κ−1,k+1−λ−1,t+λG1,1−k−κ−1,k+1−λ−1,t},(33)

   and the shear stress is

   τ(r,t)=2μR12R22(℧2−℧1)λr2(R22−R12)Rς,−2−λ−1,t+πμλς∑n=1∞J1(R1rn)2rA(r,rn)−rnA∗(r,rn)J12(R1rn)−J12(R2rn)××℧2R2J1(R1rn)−℧1R1J1(R2rn)×∑k=0∞−νrn2λkG1,−k−2,k+1−λ−1,t,(34)

   for the ordinary Maxwell fluid.

   Using Appendix A₂, the expressions (33) and (34) in the simplified form for κ = 1, in terms of expositional functions, are

   ωM(r,t)=℧1R12(R22−r2)+℧2R22(r2−R12)r(R22−R12)t−πν∑n=1∞J1(R1rn)A(r,rn)rn2[J12(R1rn)−J12(R2rn)]{℧2R2J1(R1rn)−℧1R1J1(R2rn)}1−λq1n2eq2nt−q2n2eq1ntq2n−q1n,(35)

   and

   τM(r,t)=2μR12R22(℧2−℧1)r2(R22−R12)t−λ1−e−t/λ+πρ∑n=1∞J1(R1rn)2rA(r,rn)−rnA∗(r,rn)rn2[J12(R1rn)−J12(R2rn)]×℧2R2J1(R1rn)−℧1R1J1(R2rn)×1+q1neq2nt−q2neq1ntq2n−q1n.(36)

4. By now letting λ→ 0 in Eqs. (35) and (36) or ς → 1 and λ→ 0 in Eqs. (31) and (32) and using limλ→0⁡1λkG1,b,k−1λ,t=t−b−1Γ(−b);b < 0, the velocity field is

   ωN(r,t)=℧1R12(R22−r2)+℧2R22(r2−R12)r(R22−R12)t−πν∑n=1∞J1(R1rn)A(r,rn)rn2[J12(R1rn)−J12(R2rn)]{℧2R2J1(R1rn)−℧1R1J1(R2rn)}1−e−νrn2t,(37)

   and the shear stress is

   τN(r,t)=2μR12R22(℧2−℧1)r2(R22−R12)t+πρ∑n=1∞J1(R1rn)2rA(r,rn)−rnA∗(r,rn)rn2[J12(R1rn)−J12(R2rn)]××℧2R2J1(R1rn)−℧1R1J1(R2rn)1−e−νrn2t,(38)

   corresponding to a Newtonian fluid.

5 Flow through a circular pipe

The finite Hankel transform of the function f(r, t) is

and the inverse Hankel transform of f_H(rn′) is

where J₁(R₂r) = 0 as r_n are the positive roots of the equation.

By making J₁(R₂r_n) = 0, and using the identities J₀(z) + J₂(z) = 2zJ1 (z) in Eq. (15) we get

Writing the Integral

Introducing Eq. (41) and taking R₁ = 0, the above integral takes the form

as a result from (24) and (28) we recovered the expansions for the velocity field

and

through a cylinder already obtained by Kamran et al. [31] in Equation (21) and (27) for p = 1.

6 Conclusions and results

In this article, exact solutions for a fractional Oldroyd-B fluid between two rotating pipes are determined. Fluid motion is produced due to rotation of the pipes around their axis with time dependent angular velocities. The solutions determined by the use of integral transforms and presented in terms of Bessel functions and hypergeometric functions which are free of integrals. The final result satisfies the initial and boundary conditions. In the limiting cases, the corresponding results for Oldroyd-B fluid, Maxwell and Newtonian fluid, are obtained from general results. Moreover, the solution for the fluid through a circular pipe is obtained as a special case.

The velocity profiles and shear stress are illustrated in Figures 2 — 7 for different values of parameters. Fig. 2 for the effect of t, Fig. 3 for the effect of ν, Fig. 4 for the effect of λ, Fig. 5 for the effect of λ_r and Fig. 6 shows the effect of the fractional parameter ς on the fluid motion. Clearly, the velocity as well as shear stress are increasing as a function of t and ν, The effect is qualitatively the same for fractional parameter ς and λ, more exactly the velocity ω(r, t) is a decreasing function with regards to ς.

Figure 1

Problem Geometry.

Figure 2

Profiles of the velocity field ω(r, t) given by Eqs. (24) and (28) for R₁ = 1, R₂ = 7.4, ℧₁ = 1, ℧₂ = −1, λ = 4, λ_r = 1, ν = 0.002, ς = 0.1 and different values of time

Figure 3

Profiles of the velocity field ω(r, t) given by Eqs. (24) and (28) for t = 7, R₁ = 1, R₂ = 7.4, ℧₁ = 1,℧₂ = −1, λ = 20, λ_r = 5, ς = 0.9 and different values of ν

Figure 4

Profiles of the velocity field ω(r, t) given by Eqs. (24) and (28) for t = 7, R₁ = 1, R₂ = 7.4,℧₁ = 1, ℧₂ = −1, ν = 0.2,λ_r = 1, ς = 0.5 and different values of λ

Figure 5

Profiles of the velocity field ω(r, t) given by Eqs. (24) and (28) for t = 8, R₁ = 1, R₂ = 7.4, ℧₁ = 1, ℧₂ = −1,ν = 0.2,λ = 10,ς = 0.5 and different values of λ_r

Figure 6

Profiles of the velocity field ω(r, t) given by Eqs. (24) and (28) for t = 20, R₁ = 1, R₂ = 7.4,℧₁ = 1, ℧₂ = −1, ν = 0.001, λ = 4,λ_r = 1 and different values of ς

Finally, Fig. 7 is the comparison of ω(r, t) corresponding to the motion of the Newtonian, fractional and ordinary Maxwell, and fractional and ordinary Oldroyd-B for the same values of time and for the same material and fractional parameters. From this comparison, the Oldroyd-B fluid is the slowest and the Newtonian fluid is the fastest on the whole flow domain. SI units are used for all figures, and the roots rn=nπ(R2−R1)..

Figure 7

Profiles of the velocity field ω(r, t) corresponding to the Newtonian, Maxwell, Fractional Maxwell, Oldroyd-B and fractional Oldroyd-B fluids, for t = 8, R₁ = 1, R₂ = 7.4, ℧₁ = 1, ℧₂ = −1,ν = 0.001,λ = 3.6,λ_r = 3 and ς = 0.8