Unsteady Unidirectional Flow of Second-Grade Fluid through a Microtube with Wall Slip and Different Given Volume Flow Rate

The second-grade flows through a microtube with wall slip are solved by Laplace transform technique. The effects of rarefaction and elastic coefficient are considered with an unsteady flow through a microtube for a given but arbitrary inlet volume flow rate with time. Five cases of inlet volume flow rate are as follows: 1 trapezoidal piston motion, 2 constant acceleration, 3 impulsively started flow, 4 impulsively blocked fully developed flow, and 5 oscillatory flow. The results obtained are compared to those solutions under no-slip and slip condition.


Introduction
During the past decades, a great deal of literatures used the Navier-Stokes equation to describe the Newtonian fluid. However, the Newtonian fluid is the simplest to be solved and its application is very limited. In practice, many complex fluids such as blood, soaps, clay coatings, certain oils and greases, elastomers, suspensions, and many emulsions have been treated as non-Newtonian fluids. From literatures, the non-Newtonian fluids are mainly classified into three categories on the basis of their behavior in shear. a The shear stress of the fluids depends only on the rate of shear. b A fluid with a relationship between the shear stress and shear rate. c The fluids possess both elastic and viscous properties. One of the most popular models for non-Newtonian fluids is called the second-grade fluid. Erdogan andİmrak 1 used the second-order Rivlin-Ericksen fluid, or the so-called second-grade fluid to model the non-Newtonian fluid. They solved some unsteady flows, such as unsteady flow over a plane wall, unsteady Couette flow, and unsteady Poiseuille flow. Hayat et al. 2 analysed the influence of variable viscosity and viscous dissipation on the non-Newtonian flow. Bandelli and Rajagopal 3 solved various startup flows of second-grade fluids in domains with one finite dimension by integral transform method. Some unsteady flows of the fluids of second grade have been investigated by many authors 4-7 . Microfluidics is the significant technologies developed in the engineering field. As the microflow is considered, the no-slip condition is not sufficient for a fluid of second grade. Rarefaction phenomena should be considered when fluid flows in a microtube. The typically flow field can be divided into the four regimes by Knudsen number 8 : Kn < 10 −3 , continuum flow; 10 −3 ≤ Kn < 10 −1 , slipflow; 10 −1 ≤ Kn < 10, transition flow; and 10 ≤ Kn free molecular flow. Much research in the literature does not consider the effect of rarefaction in the secondgrade fluid. The study examined the effects of rarefaction of an unsteady flow through a microtube by Chen et al. 9 .
In practice application, generally the inlet volume flow rate is a given condition instead of pressure gradient. Das and Arakeri 10 solved the unsteady laminar duct flow with a given volume flow rate variation. They discussed the problem with various types of given inlet piston motion in the channel and duct. Also Das and Arakeri 11 verified their earlier experimental work. Chen et al. 12-14 extended Das and Arakeri's work by considering various non-Newtonian fluids. Several studies 15, 16 have suggested the noslip condition that is deduced as the limiting cases when the slip parameter is equal to zero. Hayat et al. 17 considered the unsteady flow of an incompressible second-grade fluid in a circular duct with a given volume flow rate variation. The effects of Hall current are taken into account. For the above reason, this study considers the wall slip condition and the second-grade fluid with different given volume flow rate. For α 1 → 0, they reduce to the similar solutions for Newtonian fluids. The results show that the analytical solutions of velocity profile and pressure gradient are affected by the slip conditions and the viscoelastic parameter.

Constitutive Equations
The constitutive equation of second-grade Rivlin-Ericksen fluid is in the following form: where T is the stress tensor, I is identity tensor, p is the static fluid pressure, μ is the dynamic viscosity coefficient, α 1 is the elastic coefficient and α 2 is the transverse viscosity coefficient, and A 1 and A 2 represent the Rivlin-Ericksen tensors. Here, μ, α 1 and α 2 are material modules which are assumed constant. The kinematic tensors A 1 and A 2 are defined as where ρ is the density of the fluid and b is the body force. In the sense of the Clausius-Duhem inequality and the condition that the Helmholtz free energy is a minimum when the fluid is at rest, then the material modules must be satisfied 7 as follows: In our study, we use the cylindrical polar coordinates r, θ, x , where r is radial distance from the center of the pipe, θ is the angular direction, and x is the axial direction. Velocity in the x, r, and θ-direction are u, u r , and u θ , respectively. We also investigate the fluid rarefaction effect in a microtube, the Knudsen number is an important nondimensional parameter where λ is the molecular mean free path, which is defined as the mean secondary collision distance of a gas molecule, and L is the characteristic length. In order to find the fluid of second grade for unsteady unidirectional flows, we seek a velocity field of the form u u r, t , u r 0, u θ 0. where ν is the kinematic viscosity. This implies that the pressure gradient is a function of time only.

Methodology of Solution
The problem can be solved if the governing equation, boundary condition, and initial condition are known. This third-order nonhomogeneous partial differential equation is not convenient to use the method of separation of variable to solve. In this paper, we give the Laplace transform method reducing the two variables into single variable. In other words, we transform PDE into ODE that will effectively reduce the original difficult equation.
The governing equation of motion in x-direction is where ν μ/ρ and θ α 1 /ρ. With R is the radius of duct, the boundary conditions are where β ν λ is the velocity slip coefficient and is defined as and F ν is the tangential momentum accommodation coefficient that describes the interaction between fluid and wall and is related to constituents of fluid, temperature, velocity, wall temperature, roughness, and chemical status. F ν is defined as where u i , u re , and U w are tangential momentum of incoming molecules, reflected molecules, and re-emitted molecules, respectively. We need an initial condition for the velocity, u r, 0 , and the problem can be solved if the pressure gradient function is known. In our case, we determined the pressure gradient indirectly by the volume flow rate, which is given. The velocity is related to the inlet volume flow rate by R 0 2πru r, t dr u p t πR 2 Q t , 3.5 where u p is the average inlet velocity.

Mathematical Problems in Engineering 5
Using the Laplace transform technique of 3.1 , 3.2 , and 3.5 yields the following equations: Equation 3.6 is a second-order inhomogeneous ordinary differential equation. The homogeneous part is the modified Bessel's equation of zeroth order and assuming the particular integral as Ψ p , the general solution is where m s/ ν sθ . Using the boundary conditions 3.7 and 3.8 into 3.6 to solve these two unknown coefficients C 1 and C 2 , substituting C 1 and C 2 into 3.6 give u r, s where I 1 is the modified Bessel's equation of the first order.
To solve for the unknown Ψ p , we substitute 3.11 into 3.9 and Ψ p is obtained as Furthermore, the pressure gradient is found by substituting 3.11 into 3.6 to obtain We obtain the expressions for the variation of nondimensional pressure gradient with time by taking the inverse transform formula.

Illustration of Examples
We consider some examples proposed by Das and Arakeri 10 with the second-grade fluid and the effect of wall-slip conditions on the unsteady flow patterns in a microtube. For the following case, the velocity moves with a constant acceleration of the piston starting from rest, and the other one, the piston suddenly starts from rest and then keeping this velocity. These two solutions we apply to the trapezoidal motion, that is, the piston has three stages: constant acceleration of the piston starting from rest, a period of constant velocity, and a constant deceleration of the piston to a stop.

Trapezoidal Piston Motion
We get the solution for the three stages piston velocities which vary with time as follows: Mathematical Problems in Engineering 7 where U p is the constant velocity after acceleration, and t 0 , t 1 , and t 2 are the time periods for changing piston velocity. We use the Heaviside unit step function to describe the piston motion as follows:

4.2
For the constant acceleration period 0 ≤ t ≤ t 0 , taking the Laplace transform of u p t U p tH t /t 0 , we get From 3.16 expression, the integration is determined using complex variable theory, as discussed by Arparci 18 . We obtain the velocity distribution where R j is the residual of poles of U p e st G r, s /t 0 s 2 . It can be easily observed that s 0 is a pole of order 2. Therefore, the residue at s 0 is

4.5
The other singular points are the zeroes of Setting mR iλ, we find that If λ n , n 1, 2, 3, . . . , ∞ are zeros of 4.7 , then s n −λ 2 n ν/ R 2 θλ 2 n , n 1, 2, 3, . . . , ∞ are the simple poles. Since all λ n are symmetrically placed about zero on the real axis, all the poles 8 Mathematical Problems in Engineering s n lie on the negative real axis. These are simple poles, and residues at all these poles can be obtained as Res s n U p R 2 t 0 ν 2 J 0 λ n − J 0 r/R λ n − αλ n J 1 λ n 1 2α λ 3 n J 1 λ n αλ 4 n J 0 λ n e −νλ 2 n t/ R 2 θλ 2 n .

4.8
Adding Res 0 and Res s n , the dimensionless velocity distribution is obtained as where u * u p /U p , c r/R, α β ν Kn, t * tν/R 2 , t * 0 t 0 ν/R 2 , β θ/R 2 . By the same method, the dimensionless velocity profile during the constant piston velocity t 0 ≤ t ≤ t 1 is obtained as Mathematical Problems in Engineering 9 And after the piston has stopped t 2 ≤ t ≤ ∞ ,

4.12
We also obtain the expressions of the dimensionless pressure gradient during these four different stages. During the constant acceleration period 0 ≤ t ≤ t 0 , n t * / 1 βλ 2 n J 0 λ n − αλ n J 1 λ n 1 2α λ n J 1 λ n αλ 2 n J 0 λ n ,

4.16
Above these infinite series, equations are convergent and we set the n 50 as enough for the cases. For trapezoidal piston motion with different nondimensional times t * tν/R 2 are t 0 ν/R 2 , t 1 ν/R 2 and t 2 ν/R 2 0.0012, 0.0305, and 0.0366, respectively. The velocity profiles at Kn 0.1 and β 0.05 are illustrated in Figure 1.
These values are chosen for the purpose of comparing the results obtained by Das and Arakeri 10 and Chen et al. 9 . When α ≈ Kn 0 no-slip condition and β 0 no-elastic effect , the velocity profiles in 4.9 to 4.12 are exactly like Das and Chen's results. Figure 1 shows the second-grade flow with slippage on the microtube wall during four different time periods. The development of velocity is similar to that in Das et al. and Chen et al.'s works. However, the elastic coefficient retarded the change of velocity in the microtube. Because the effect of slippage, the shift of velocity from the wall to centerline is smoother than that in Das et al. and Chen et al.'s works. During the time period when the piston decelerates and stops at time t * 2 , it is observed that the flow reverses its direction near the wall see Figure 1 c . After the piston motion ceases, the velocity profile see Figure 1 d continues to have reverse flow near the wall to satisfy the zero mass flow condition. Figure 2 shows the variation of nondimensional pressure gradients with time at Kn 0.1 and β 0.05. During the acceleration and deceleration stages, the pressure gradients are large mainly because of fluid inertia. Finally, when the piston stops, the pressure gradient slowly decays to zero. The degree of smoothness is proportional to the β value. In the special case, it is worth mentioning that when β → 0 means that α 1 → 0 , corresponding to Newtonian fluids, all solutions that have been obtained are going to be those for Newtonian fluids performing the same motions. Figure 4 shows the effect of Kn various values Kn 0, 0.05, 0.1 on the velocity profiles at β 0.05. The analytical result demonstrates that a larger Kn value will flatten the velocity profile. It is observed that the slip condition occurs near the wall.

Constant Acceleration Case
For a piston with constant acceleration can be described by the following equation: where a p is the constant acceleration, U p is the final velocity after acceleration, and t 0 is the time period of acceleration. The velocity profile can be obtained by putting t t 0 of 4.9 as follows: and when the pressure gradient, as time approaches infinity, is dp * dx * −

Suddenly Started Flow
The solution to the suddenly started flow in a circular duct is as follows: where U p is the constant velocity

Suddenly Blocked Fully Developed Flow
The exact solution of this problem with no-slip wall condition was considered by Weinbaum and Parker 19 . The initial condition for this problem is u r, 0 1 − c 2 , and the mass flow condition is

Oscillatory Flow
Here, the oscillating piston motion starting from rest is considered. The piston motion is defined as for t ≤ 0, U p sin ωt , for t > 0.