Abstract: The unsteady flow of an incompressible viscous fluid, characterized by the motion of a long, circular, cylindrical rod, oscillating both longitudinally and torsionally at different frequencies and amplitudes isexamined, with slip occurring at the surface of the cylindrical rod. Analytical expressions for the velocity field, the tangential drag and the work done by the drag force have been obtained and are displayed graphically using particular values of the flow parameters. These are plotted for different values of slip from perfect slip to no-slip, so as to get some insight into the effects of slip.

INTRODUCTION

Casarella and Laura (1969) investigated the motion of a viscous fluid due to the longitudinal and torsional oscillation of an infinite circular cylindrical rod immersed in a fluid. They were primarily interested in the drag force acting on the rod as this is of practical significance in many ocean engineering problems. They obtained exact solutions for the velocity field, stresses and drag on the rod. Rajagopal (1983) examined a similar problem for a non-Newtonian fluid in particular a fluid of second grade and like Casarella and Laura, obtained an exact solution to the field equation. Ramkissoon and Majumdar (1990) examined the corresponding internal flow problem for viscous fluids. They obtained an exact solution for the flow field and explicit expressions were given for the shear stresses, the drag experienced by the cylinder and the drag coefficient. Ramkissoon et al. (1991) examined the same problem as Casarella and Laura (1969) but for a Polar fluid. Here, an exact solution was obtained for the velocity field. Rahaman (2004, 2005) examined the same problem as Ramkissoon and Majumdar (1990) and Casarella and Laura (1969) but for an upper-convected Maxwell fluid, explicit expressions were obtained for the velocity field, shear stresses and drag in each case. In all of these problems, the frequencies of oscillations in the longitudinal and torsional directions were the same. Furthermore, the magnitude of the oscillations contained a common parameter, implying dependency for these orthogonal motions.

Owen and Rahaman (2006) appear to be the first to relax the common oscillatory frequencies for the two directions of motion in their internal flow problem examined and thus took different frequencies for these oscillations.

The main objective of this research is to investigate a similar problem to that of Casarella and Laura (1969) for a viscous fluid but a more general case. In particular, the motion of a viscous fluid due to the longitudinal and torsional oscillation of an infinite, circular cylindrical rod immersed in the fluid is examined but the two different oscillations of the rod are moving with different frequencies, like that done by Owen and Rahaman (2006). Further, the amplitude dependency of the oscillatory motions, due to the presence of a common factor is removed. Finally, slip at the surface of the cylindrical rod is taken into consideration, as it has long been established that the conventional no-slip boundary condition begins to break down even before the linear stress-strain relationship becomes invalid (Gad-el-Hak, 1999).

Schaaf and Chambre (1961) have shown that the slip parameter, β, otherwise known as the Coefficient of Sliding Friction is given by Eq. 1:

(1)

Where:

It can be shown that Kn≥0 and when Kn≥0.1, the continuum model and therefore the Navier-Stokes equations become invalid in which case, the Boltzman equation has then to be applied to describe the molecular motion properly. When Kn≥0.01, the Navier-Stokes equation is a good approximation. However, when 0.01≤Kn≥0.1 (commonly referred to as the slip-flow regime), the Navier-Stokes equations can still be used, provided tangential slip-velocity boundary conditions are implemented along the solid walls of the flow domain (Schaaf and Chambre, 1961; Aleksandrov et al., 1988; Morinishi, 2006). It is the latter situation, i.e., a continuum flow field with slip boundary conditions that is to be investigated in this study.

MATERIALS AND METHODS

A long, straight, solid, circular cylindrical rod with uniform cross-section, of radius a, undergoing oscillations both longitudinally and torsionally with different frequencies and amplitudes and fully immersed in an incompressible viscous fluid is considered. It will be assumed that the rod is infinite in length, so that rod end effects are neglected, no external forces are acting and the flow is fully established, which is at rest at infinity.

The equation of motion for a viscous fluid (Ramkissoon and Majumdar, 1990) is given by the Navier-Stokes equation:

(2)

Where:

Working in cylindrical polar co-ordinates (R, θ, z) with the z-axis coinciding with the axis of the cylinder, assuming symmetry, then it is fair to assume that the velocity field of the induced flow is of the form

(3)

It is also assumed that the pressure field is independent of the and co-ordinates and is of the form:

(4)

The continuity equation for an incompressible fluid is (Ramkissoon and Majumdar, 1990):

(5)

Substituting Eq. 3 into 2, gives for each component:

(6)

(7)

(8)

The components of stress are given by Batchelor (1967):

(9)

(10)

where, e_ij are the components of the rate of deformation tensor, which in this case are given by Batchelor (1967):

(11)

(12)

(13)

(14)

(15)

(16)

Utilizing Eq. 11-16 in Eq. 9 and 10 gives the components of stress as:

(17)

(18)

(19)

(20)

By solving Eq. 7 and 8, subject to certain conditions, the velocity field can be completely determined. Note that when the velocity field has been determined, the pressure field and the components of stress can also be determined from Eq. 4, 6, 19 and 20.

The velocity of the rod at its surface is taken to be:

(21)

where, Ω₁, Ω₂ represent the different frequencies in the torsional and longitudinal directions respectively. α₁, α₂ are the different amplitudes of the oscillations in the torsional and longitudinal directions respectively.

For the slip condition at the surface of the rod, the well known condition proposed by Basset (1961) is used that is the tangential velocity of a fluid relative to the solid at a point on its surface is proportional to the tangential stress prevailing at that point. For this it will be assumed that the material making up the surface of the rod is uniform, so that the mean degree of roughness per unit area is the same. This implies that there will be equal amounts of slip in any direction and in particular, the longitudinal and torsional directions.

Using this last assumption, 3, 21 and Basset’s slip condition gives:

(22)

and

(23)

where, β is the slip parameter which is assumed to depend only on the nature of the fluid and the solid surface.

Furthermore, β≥0 and in particular, β = 0 represents perfect slip, whereas β→∞ represents no slip. Using Eq. 19 and 20 in 22 and 23, respectively gives:

(24)

and

(25)

Since the flow is at rest at infinity then:

(26)

and

(27)

To determine the velocity field, q, given by (3), Eq. 7 and 8 need to be solved subject to the conditions given by Eq. 24-27. It is noted that Eq. 3 satisfies the continuity Eq. 5, automatically.

Assume that v(R, t) and w(R, t) are of the form:

(28)

and

(29)

where ‘U’ represents the real part of the expression.

On substituting these into Eq. 7 and 8 and solving subject to the stated conditions gives:

(30)

and

(31)

where:

is the Modified Bessel Function of the second kind of order n. The fluid’s velocity field has now been completely determined.

Utilizing Eq. 30 and 31 in Eq. 20 and 19, respectively, the shear stresses on the rod are found to be:

(32)

and

(33)

The tangential drag force, D, acting on the rod per unit length is (Casarella and Laura, 1969):

(34)

Substituting Eq. 32 and 33 into 34 gives:

(35)

The work done W_j by the drag force D on the fluid per half-cycle of torsional and longitudinal motion is given by Casarella and Laura (1969):

(36)

where, j = 1, 2 refer to the torsional and longitudinal motions respectively.

Substituting Eq. 21 and 35 into 36 gives:

(37)

where:

(38)

RESULTS AND DISCUSSION

In order to get some insight on the effects of slip, various graphs involving the components of velocity, drag force and work done are displayed for different values of the slip parameter β, ranging from β = 0 (perfect slip) to β→∞ (no-slip). For some practical oceanographic problems, the parameters v and Ω_j are bounded by Casarella and Laura (1969):

(39)

(40)

Following Casarella and Laura by taking:

(41)

gives on substituting into Eq. 40:

(42)

For seawater, v = 1.17x10^-6 m² sec^-1 and μ = 0.00121 pas (LMNO Engineering, Research and Software, Ltd., 1999), substituting this value of v into Eq. 42 gives approximately:

(43)

Using Eq. 43, the value of a is taken to be:

(44)

Using Eq. 41 with j = 1, Eq. 44 and v for seawater, gives approximately:

(45)

Taking into consideration the minimum value of:

for a 0.00635 m diameter rod, as stated by Casarella and Laura, then selecting:

(46)

along with Eq. 44 and v for seawater, gives approximately:

(47)

The behaviour of the velocity profiles of the -component are displayed in Fig. 1-4 for four different values of the slip parameter β. As observed, the magnitude of the velocity decreases as one moves away from the cylindrical rod, however, the larger the value of β then the greater the magnitude of v (R, t)/α₁. The same is observed in Fig. 5-8 for w (R, t)/α₂ from the -component.

Figure 9-12 show the behaviour of the velocity profiles of the -component at four different times. As observed, the magnitude of the velocity decreases as one moves away from the cylindrical rod.

Fig. 1:	v(R, t)/α1 for β = 1

Fig. 2:	v(R, t)/α1 for β = 5

Fig. 3:	v(R, t)/α1 for β = 20

Fig. 4:	v(R, t)/α1 for β →∞

Fig. 5:	w(R, t)/α2 for β = 1

Fig. 6:	w(R, t)/α2 for β = 5

Fig. 7:	w(R, t)/α2 for β = 20

Fig. 8:	v(R, t)/α2 for β →∞

Fig. 9:	v(R, t)/α1 for t = 0

Fig. 10:	v(R, t)/α1 for t = π/2Ω1

Fig. 11:	v(R, t)/α1 for t = π/Ω1

Fig. 12:	v(R, t)/α1 for t = 3π/2Ω1

Also, the larger the value of β then the greater the magnitude of v (R, t)/α₁ this is consistent with that observed in Fig. 1-4. The same is observed in Fig. 13-16 for w (R, t)/α₂ from the -component.

Figure 17 shows the behaviour of the -component of the velocity field against β for four different values of t on the surface of the cylindrical rod. It can be observed that as β increases, v (R, t)/α₁ tends to the no-slip case. In Fig. 18 a similar observation can be made for the behaviour of the -component.

With respect to the drag, Fig. 19 shows the behaviour of the -component against β at four different times. It is observed that as the value of β increases, the magnitude of -component/α₁ also increases. In Fig. 20, a similar observation is made for the -component/α₂. Figure 21 shows the behaviour of the -component of the drag against t for four different values of the slip parameter β. For each value of β, it can be observed that the graph appears to be periodic and as β increases, the magnitude of the -component/α₁ also increases, which is consistent with the observations from Fig. 19. Corresponding to Fig. 21, the same observation can be made for the -component/α₂ from Fig. 22. This is also consistent with observations from Fig. 20.

Fig. 13:	w (R, t)/α2 for t = 0

Fig. 14:	w (R, t)/α2 for t = π/2Ω2

Fig. 15:	w (R, t)/α2 for t = π/Ω2

Fig. 16:	w (R, t)/α2 for t = 3π/2Ω2

Fig. 17:	v (a, t)/α1 against β

Fig. 18:	w (a, t)/α2 against β

Fig. 19:	-component/α1 of the drag against β

Fig. 20:	-component/α2 of the drag against β

Fig. 21:	-component/α1 of the drag against t

Fig. 22:	-component/α2 of the drag against t

It is of interest to compare the drag in each direction of oscillatory motion to determine which direction experiences greater drag. Thus, it can be shown that the difference in the magnitude of the components for the drag, given by Eq. 35, may be written as:

(48)

and further

(49)

Based on Eq. 48 and 49, it suffices to only examine D₁/α₁ or D₁/α₂.

Figure 23-25 shows D₁/α₂ against β for t = nπ/2Ω₁Ω₂, n = 0,1,2,3 for different values of α₁/α₂. It is observed that when α₁/α₂ = 1/2, the drag in the longitudinal direction of motion is greater than the drag in the torsional direction of motion for each value of t. Conversely, when α₁/α₂ = 2, the drag in the torsional direction of motion is greater than the drag in the longitudinal direction for the same values of t. It is also observed that when α₁/α₂ = 1, the drag in the torsional direction of motion is greater than the drag in the longitudinal direction of motion for t = nπ/2Ω₁Ω₂, n = 0,1,2 but the opposite is observed when t = 3π/2Ω₁Ω₂.

It is noted that the drag in each direction is the same when:

Figure 26-28 show D₁/α₂ against t for four different values of the slip parameter β for different values of α₁/α₂. It is observed that D₁/α₂ varies between positive and negative values, implying that the more efficient direction of motion keeps changing with the progression of time. Since,

Fig. 23:	D1/α2 against β for α1/α2 = 1/2

Fig. 24:	D1/α2 against β for α1/α2 = 1

Fig. 25:	D1/α2 against β for α1/α2 = 2

Fig. 26:	D1/α2 against t for α1/α2 = 1/2

Fig. 27:	D1/α2 against t for α1/α2 = 1

Fig. 28:	D1/α2 against t for α1/α2 = 2

(50)

then one needs to only examine

to explore the behaviour of the work done by the drag force.

As β varies, the behaviour of the work done in each direction is shown in Fig. 29 and 30 for different values of α₁/α₂. It is observed in all cases that there is an increase in magnitude as β increases and using Eq. 37, it can be easily shown, as the graphs suggest that all tend to the limit:

Fig. 29:	W1/α22 against β

Fig. 30:	W2/α22 against β

CONCLUSION

Comparison of the work done in each direction of motion is now examined. Noting (50), it suffices to only examine

From Fig. 31-33, it is observed in all cases that the magnitude of the work done in longitudinal direction is greater than the work done in the torsional direction.

Fig. 31:	Wj/α22 against β for α1/α2 = 1/2

Fig. 32:	Wj/α22 against β for α1/α2 = 1

Fig. 33:	Wj/α22 against β for α1/α2 = 2

Fig. 34:	|W1|-|W2|/α22 against β for α1/α2 = 1/2, 1, 2

Again noting (50), the behaviour of the difference in the work done in each direction, as β varies is displayed in Fig. 34. It is observed that the work done in the longitudinal direction is greater than the work done in the torsional direction, which is consistent with Fig. 31-33.