229 
 

American Academic Scientific Research Journal for Engineering, Technology,  and Sciences   

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

 

http://asrjetsjournal.org/  

 

Internal Newtonian Flow Due to a Cylinder Undergoing 

Longitudinal and Torsional Oscillations of Different 

Frequencies 

Robin Ramlal
a*

, Karim Rahaman
b
 

a
Senior Lecturer in Mathematics, College of Science, Technology and Applied Arts of Trinidad and Tobago - 

COSTAATT 

b
Senior Lecturer in Mathematics, Department of Mathematics and Statistics, The University of the West Indies, 

Trinidad and Tobago 

a
Email: robinramlal@outlook.com 

b
Email: karim.rahaman@sta.uwi.edu 

Abstract 

The unsteady flow of an incompressible viscous fluid contained in a cylinder of infinite length, subject to 

longitudinal and torsional oscillations of different frequencies is examined. Analytical expressions for the 

velocity field, shear stresses, drag on the cylinder, work done and the drag coefficients are obtained.  

Keywords: Viscous; unsteady flow; Longitudinal; Torsional; Oscillation; Different Frequencies. 

1. Introduction 

As early as 1886, an exact solution for the velocity field due to an infinite rod rotating in a Newtonian fluid was 

determined by Stokes [1].  Later, Casarella and Laura [2] determined analytical expressions for the velocity 

components of a Newtonian fluid and the viscous drag forces acting on a cylindrical rod-like cable which is 

undergoing both longitudinal and torsional oscillations.  Ramkissoon and Majumdar [3] looked at the 

corresponding internal problem to that done in [2], and determined the corresponding results.  A modification to 

[3], which considered independent amplitudes of the oscillations, was looked at by Phillips and Rahaman [4], 

here, expressions for the velocity field, shear stresses, drag forces, drag coefficient and work done by the drag 

forces were obtained.  Due to much interests in the field of non-Newtonian fluids and the important applications 

of such, Calmelet-Eluhu and Majumdar [5] considered the problem for a micropolar fluid. Analytical 

expressions of the fluid velocity and micro-rotation were obtained, along with explicit expressions for the shear 

stresses and drag force acting at the wall of the cylinder.  Rahaman [6] considered a similar situation using an 

upper-convected Maxwell fluid; the velocity field, shear stresses and drag were obtained and comparisons made 

with its Newtonian counterpart.  

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

230 
 

Reference [7] considered the case for different frequencies of oscillations with an Oldroyd-B fluid, analytic 

solutions were obtained for the velocity components, shear stresses and drag on the cylinder.  Different 

frequencies was also considered by [8] for a micropolar fluid, analytical expressions for the velocity and 

microrotation components were obtained in terms of modified Bessel's functions, along with the drag force 

acting on the wall of the cylinder.  The main objective of this research is to investigate an extension to that done 

by [3], in particular, the longitudinal and torsional oscillations of the cylinder is considered to have different 

frequencies.  Analytical expressions for the velocity field, shear stresses, drag on the cylinder, work done and 

the drag coefficient are obtained. The behaviour of the velocity components, the drag and the work done are 

illustrated graphically and conclusions made. 

2. Statement of the Problem. 

The unsteady flow of an incompressible viscous fluid contained in a cylinder which is infinite in length and 

radius ‘a’, is undergoing longitudinal and torsional oscillations with different frequencies.  Due to the nature of 

the flow, cylindrical polar coordinates, (𝑅, 𝜃, 𝑧), will be used, with the axis of the cylinder coinciding with the z 

axis.   Similar to that done in [2] and [3], the velocity of the cylinder, 𝑞𝑏, at 𝑅 = 𝑎 takes the form, 

𝑞𝑏 = 𝑞0 cos 𝛽 cos(Ω1𝑡) �̂� +  𝑞0 sin 𝛽 cos(Ω2𝑡)�̂�     (2.1) 

where 𝑞0, 𝛽, Ω1 and Ω2 are real constants. 

It is noted that when 𝛽 = 0 or 𝜋 the oscillations are purely torsional and when 𝛽 =
𝜋

2
 or

3𝜋

2
  they are purely 

longitudinal.  Due to the motion of the cylinder, it is fair to assume that the radial component of the fluid’s 

velocity is zero.  Further, it will be assumed that the flow is axisymmetric about the z-axis. Hence, the velocity 

of the fluid takes the form, 

𝑞 = v(𝑅, 𝑡)�̂� + 𝑤(𝑅, 𝑡) �̂�       (2.2) 

Since the fluid is incompressible, the continuity equation [9] is, 

∇ ⋅ 𝑞 = 0          (2.3) 

which is satisfied by (2.2). 

In the absence of external forces, the Navier-Stokes equation [9] to be solved is, 

−
1

𝜌
∇𝑝 + 𝜈∇2𝑞 =

𝜕𝑞

𝜕𝑡
+ (𝑞 ⋅ ∇) 𝑞      (2.4) 

where 𝜌 is the density, 𝑝 is the pressure and 𝜈 is the kinematic viscosity. 

 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

231 
 

3. Velocity Components, Stresses and Drag 

On substituting (2.2) into (2.4) gives the following linear system of equations, 

−
1

𝜌

𝜕𝑝

𝜕𝑅
= −

v2

𝑅
        (3.1) 

𝜈 (
1

𝑅

𝜕v

𝜕𝑅
+

𝜕2v

𝜕𝑅2 −
v

𝑅2) =
𝜕v

𝜕𝑡
       (3.2) 

 

𝜈 (
1

𝑅

𝜕𝑤

𝜕𝑅
+

𝜕2𝑤

𝜕𝑅2) =
𝜕𝑤

𝜕𝑡
        (3.3) 

Assuming that the �̂� component of the velocity field takes the form, 

v(𝑅, 𝑡) = Re[𝑓(𝑅)𝑒𝑖Ω1𝑡]        (3.4) 

where Re refers to the real part, along with the fact that the velocity must remain finite as 𝑅 → 0, one gets, on 

solving (3.2) and using the no-slip condition with (2.1), 

v(𝑅, 𝑡) = Re [𝑞0 cos 𝛽
𝐼1(√𝑖𝜎1𝑅)

𝐼1(√𝑖𝜎1𝑎)
𝑒𝑖Ω1𝑡]      (3.5) 

where 𝜎1 = √
Ω1

𝜈
  

In a similar manner, the �̂� component of the velocity is,  

𝑤(𝑅, 𝑡) = Re [𝑞0 sin 𝛽
𝐼0(√𝑖𝜎2𝑅)

𝐼0(√𝑖𝜎2𝑎)
𝑒𝑖Ω2𝑡]      (3.6) 

where 𝜎2 = √
Ω2

𝜈
 

 The tangential stresses on the wall of the cylinder are given by [10], 

𝜏𝑅𝜃|𝑅=𝑎 = 𝜇 [
𝜕v

𝜕𝑅
−

v

𝑅
]
𝑅=𝑎

        (3.7) 

𝜏𝑅𝑧|𝑅=𝑎 = 𝜇 [
𝜕𝑤

𝜕𝑅
]
𝑅=𝑎

        (3.8) 

Substituting (3.5) into (3.7) gives for the torsional stress on the cylinder’s wall, 

𝜏𝑅𝜃 = Re [𝜇 𝑞0 cos 𝛽
√𝑖𝜎1 𝐼0(√𝑖𝛼1)−

2

𝑎
 𝐼1(√𝑖𝛼1)

 𝐼1(√𝑖𝛼1)
𝑒𝑖Ω1𝑡]    (3.9) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

232 
 

where 𝛼1 = 𝜎1𝑎. 

Also, substituting (3.6) into (3.8) gives for the longitudinal stress on the cylinder’s wall,  

𝜏𝑅𝑧 =  Re [𝜇 𝑞0 sin 𝛽
√𝑖𝜎2 𝐼1(√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
 𝑒𝑖Ω2𝑡]     (3.10) 

where 𝛼2 = 𝜎2𝑎. 

The tangential drag per unit length acting on the cylinder is given by [3], 

𝐷 = −2𝜋𝑎(𝜏𝑅𝜃  �̂� +  𝜏𝑅𝑧  �̂�)|𝑅=𝑎
      (3.11) 

which on using (3.9) and (3.10) gives, 

𝐷 = −2𝜋𝑎𝜇𝑞0Re [cos 𝛽
√𝑖𝜎1 𝐼0(√𝑖𝛼1)−

2

𝑎
 𝐼1(√𝑖𝛼1)

𝐼1(√𝑖𝛼1)
𝑒𝑖Ω1𝑡  �̂� + sin 𝛽

√𝑖𝜎2 𝐼1(√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
𝑒𝑖Ω2𝑡  �̂�] (3.12) 

It is noted that in the case of the same frequencies of oscillations, (3.5), (3.6), (3.9), (3.10) and (3.12) reduce to 

that obtained in [3]. 

4. Alternative Expressions for the Velocity and Stress Components 

Using that given in [11], one gets in terms of the real-valued Kelvin functions 𝑏𝑒𝑟𝜔𝑥 and 𝑏𝑒𝑖𝜔𝑥, 

𝑒𝑖
3𝜔𝜋

2 𝐼𝜔(√𝑖𝜎𝑅) = 𝑏𝑒𝑟𝜔(−𝜎𝑅) + 𝑖 𝑏𝑒𝑖𝜔(−𝜎𝑅)     (4.1) 

From [11], 

𝑏𝑒𝑟𝑛(−𝑥) = (−)𝑛𝑏𝑒𝑟𝑛𝑥,   𝑏𝑒𝑖𝑛(−𝑥) = (−)𝑛𝑏𝑒𝑖𝑛𝑥     (4.2) 

which on using in (4.1), gives, 

𝐼0(√𝑖𝜎𝑅) = 𝑏𝑒𝑟0(𝜎𝑅) + 𝑖 𝑏𝑒𝑖0 (𝜎𝑅)      (4.3) 

and 

𝑖 𝐼1(√𝑖𝜎𝑅) = 𝑏𝑒𝑟1(𝜎𝑅) +  𝑖 𝑏𝑒𝑖1(𝜎𝑅)      (4.4) 

If 𝑀𝜔(𝑥) and 𝜃𝜔(𝑥) are the modulus and argument respectively of 𝐼𝜔(𝑥) then [11], 

𝑀𝜔(𝑥) = √𝑏𝑒𝑟𝜔
2(𝑥) + 𝑏𝑒𝑖𝜔

2 (𝑥)       (4.5) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

233 
 

𝜃𝜔(𝑥) = tan−1 (
𝑏𝑒𝑖𝜔𝑥

𝑏𝑒𝑟𝜔𝑥
)        (4.6) 

which gives, 

𝐼0(√𝑖𝜎𝑅) = 𝑀0(𝜎𝑅)𝑒𝑖𝜃0(𝜎𝑅)        (4.7) 

and 

𝐼1(√𝑖𝜎𝑅) = −𝑖 𝑀1(𝜎𝑅)𝑒𝑖𝜃1(𝜎𝑅)       (4.8) 

resulting in, 

v(𝑅, 𝑡) = Re [
𝑀1(𝜎1𝑅)

𝑀1(𝛼1)
𝑒𝑖[𝜃1(𝜎1𝑅)−𝜃1(𝛼1)]𝑞0 cos 𝛽 𝑒𝑖Ω1𝑡]     (4.9) 

and 

𝑤(𝑅, 𝑡) = Re [
𝑀0(𝜎2𝑅)

𝑀0(𝛼2)
𝑒𝑖[𝜃0(𝜎2𝑅)−𝜃0(𝛼2)]𝑞0 sin 𝛽 𝑒𝑖Ω2𝑡]     (4.10) 

It is observed again that for the same frequencies of oscillations, (4.9) and (4.10) reduce to that obtained in [3]. 

Similarly, from (3.9), 

𝜏𝑅𝜃|𝑅=𝑎 =
𝜇𝛼1

𝑎
𝑞0 cos 𝛽

𝑀0(𝛼1)

𝑀1(𝛼1)
𝐿 cos(Ω1𝑡 + 𝛿)     (4.11) 

where 

𝐿2 = (cos 𝜂 −
2

𝛼1

𝑀1(𝛼1)

𝑀0(𝛼1)
)

2

+ sin2 𝜂       (4.12) 

tan 𝛿 =
sin 𝜂

cos𝜂− 
2

𝛼1

𝑀1(𝛼1)

𝑀0(𝛼1)

        (4.13) 

𝜂 = 𝜃0(𝛼1) − 𝜃1(𝛼1) +
3𝜋

4
        (4.14) 

Also, from (3.10),  

𝜏𝑅𝑧|𝑅=𝑎 =
𝜇𝛼2

𝑎
 𝑞0 sin 𝛽

𝑀1(𝛼2)

𝑀0(𝛼2)
cos(Ω2𝑡 + 𝜉)      (4.15) 

where 

𝜉 = 𝜃1(𝛼2) − 𝜃0(𝛼2) −
𝜋

4
         (4.16) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

234 
 

5. Work Done and the Drag Coefficient 

(3.11) can be written as,  

𝐷 =  −2𝜋𝑎(𝑇 cos 𝜙 �̂� + 𝑇 sin𝜙 �̂�)       (5.1) 

where  

𝜏𝑅𝜃|𝑅=𝑎 = 𝑇 cos𝜙        (5.2) 

and 

𝜏𝑅𝑧|𝑅=𝑎 = 𝑇 sin 𝜙        (5.3) 

Using (3.9) and (3.10), it follows that, 

𝑇2 = ( Re [
√𝑖𝜎1 𝐼0(√𝑖𝛼1)− 

2

𝑎
 𝐼1(√𝑖𝛼1)

𝐼1(√𝑖𝛼1)
𝜇𝑞0 cos 𝛽 𝑒𝑖Ω1𝑡])

2

+ ( Re [
√𝑖𝜎2𝐼1 (√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
𝜇𝑞0 sin 𝛽  𝑒𝑖Ω2𝑡])

2

 (5.4) 

and, with the use of (5.2) and (5.3), 

tan𝜙 =
Re [

√𝑖𝜎2𝐼1(√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
𝜇𝑞0 sin 𝛽  𝑒𝑖Ω2𝑡]

Re [
√𝑖𝜎1 𝐼0(√𝑖𝛼1)− 

2

𝑎
 𝐼1(√𝑖𝛼1)

𝐼1(√𝑖𝛼1)
𝜇𝑞0 cos 𝛽 𝑒𝑖Ω1𝑡]

⁄   (5.5) 

The work done on the fluid per half-cycle of motion is [3], 

𝑊𝑗 = − ∫ 𝐷 ⋅ 𝑞(𝑎, 𝑡)d𝑡

𝜋

Ω𝑗

0
        (5.6) 

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

With the use of (2.1) and (3.12), (5.6) gives the work done, Wj , by the drag force, 𝐷, per half cycle of torsional 

and longitudinal motion as, 

𝑊𝑗 =

−𝜋𝑎𝜇𝑞0
2 [𝑅𝑒 {

√𝑖𝜎1 𝐼0(√𝑖𝛼1)− 
2

𝑎
 𝐼1(√𝑖𝛼1)

𝐼1(√𝑖𝛼1)
cos(Ω1𝑡) cos2 𝛽 𝐼(Ω𝑗 , Ω1)} +

              𝑅𝑒 {
√𝑖𝜎2𝐼1(√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
cos(Ω2𝑡) sin2 𝛽 𝐼(Ω𝑗 , Ω2)}]     (5.7) 

where 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

235 
 

𝐼(Ω𝑗 , Ω) = 2∫ 𝑒𝑖Ω𝑡 cos(Ω𝑡)

𝜋
Ω𝑗

0

𝑑𝑡 =

Ω𝑗 cos (
πΩ
Ω𝑗

) sin (
πΩ
Ω𝑗

) + πΩ − iΩ𝑗 cos2 (
πΩ
Ω𝑗

) + 𝑖Ω𝑗

ΩΩ𝑗

 

(5.8) 

Substituting (2.1) and (5.1) into (5.6) gives an alternative expression for the work done, 

𝑊𝑗 = 2𝜋𝑎𝑞0 ∫ 𝑇[cos𝜙 cos 𝛽 cos(Ω1𝑡) + sin𝜙 sin 𝛽 cos(Ω2𝑡)]

𝜋
Ω𝑗

0

𝑑𝑡 

(5.9) 

It is noted that for the same frequencies of oscillations, this reduces to that obtained in [3].  

The drag coefficient, C, can be obtained by equating the work done on the fluid by a hypothesized drag force, 

which is given by [3], 

 𝐷 𝐻 = −𝐶𝑞𝑏
𝑛(cos 𝛽 �̂� + sin 𝛽 �̂�)  

Using this in (5.6) and equating it to (5.7) gives on solving, 

𝐶 = 

−𝜋𝑎𝜇𝑞0
2 [𝑅𝑒 {

√𝑖𝜎1 𝐼0(√𝑖𝛼1) − 
2
𝑎
 𝐼1(√𝑖𝛼1)

𝐼1(√𝑖𝛼1)
cos(Ω1𝑡) cos2 𝛽 𝐼(Ω𝑗 , Ω1)} + 𝑅𝑒 {

√𝑖𝜎2𝐼1(√𝑖𝛼2)

𝐼0(√𝑖𝛼2)
cos(Ω2𝑡) sin2 𝛽 𝐼(Ω𝑗 , Ω2)}]  ⁄  

[
 
 
 
 

∫ [𝑞0
𝑛+1 cos𝑛+2 𝛽 cos𝑛+1(Ω1𝑡) + 𝑞0 cos2 𝛽 cos𝑛(Ω2𝑡) sin𝑛 𝛽 cos(Ω1𝑡)

𝜋
Ω𝑗

0

+ 𝑞0 sin
𝑛+2 𝛽 cos𝑛+1(Ω2𝑡)+𝑞0

𝑛+1 cos𝑛 𝛽 cos𝑛(Ω1𝑡) sin2 𝛽 cos(Ω2𝑡)] 𝑑𝑡

]
 
 
 
 

 

(5.10) 

An alternate expression for the drag coefficient is obtained by substituting (5.9) into (5.6) which gives, 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

236 
 

𝐶 = 2𝜋𝑎𝑞0 ∫ 𝑇[cos𝜙 cos 𝛽 cos(Ω1𝑡) + sin 𝜙 sin 𝛽 cos(Ω2𝑡)]

𝜋
Ω𝑗

0

𝑑𝑡  ⁄  

[
 
 
 
 

∫ [𝑞0
𝑛+1 cos𝑛+2 𝛽 cos𝑛+1(Ω1𝑡) + 𝑞0 cos2 𝛽 cos𝑛(Ω2𝑡) sin𝑛 𝛽 cos(Ω1𝑡)

𝜋
Ω𝑗

0

+ 𝑞0 sin
𝑛+2 𝛽 cos𝑛+1(Ω2𝑡)+𝑞0

𝑛+1 cos𝑛 𝛽 cos𝑛(Ω1𝑡) sin2 𝛽 cos(Ω2𝑡)] 𝑑𝑡

]
 
 
 
 

 

(5.11) 

 

In the case when that frequencies of oscillations are the same, one gets that obtained in [3].  

6. Graphical Results 

For the following graphs, the effects of having independent oscillating frequencies is examined.  Due to some 

practical oceanographic problems, [2] gives, 1 ≤
Ω𝑗

2𝜋
≤ 10 Hz  and 9.30 × 10−7 ≤ 𝜈 ≤ 1.86 × 10−7 m2s−1  

Hence, based on this, the values Ω𝑗  and 𝜈  have been chosen. It should be noted that various values of 𝛼𝑖 

correspond to different frequencies. 

Figures 1 and 2 illustrate the effects of different frequencies of oscillations in the torsional and longitudinal 

directions of the velocity components respectively.  The torsional frequency is taken to be the same as that in 

[3], which is different to the higher frequency longitudinal oscillation in figure 2.  It is observed, when also 

comparing with figure 4 which has the same longitudinal frequency as in [3], that the magnitude of the higher 

frequency longitudinal component of the velocity field is smaller closer to the centre of the cylinder.  As a 

result, the overall velocity field in this case would have a greater contribution from the torsional component. The 

general characteristic shapes of the curves are however similar.  Figures 3 and 4 also show the effects of 

different frequencies of oscillations in the torsional and longitudinal directions of the velocity components 

respectively. 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

237 
 

The longitudinal frequency is now taken to be the same as that in [3], which is different to the torsionafrequency 

Figure 1: 
𝑅

𝑎
 versus 

𝑣(𝑅,𝑡)

𝑞0 cos𝛽
, 𝛼1 = 6 

 

Figure 2: 
𝑅

𝑎
 versus 

𝑤(𝑅,𝑡)

𝑞0 sin 𝛽
, 𝛼2 = 10 

 

                    Figure 3: 
𝑅

𝑎
 versus 

𝑣(𝑅,𝑡)

𝑞0 cos 𝛽
, 𝛼1 = 10 

 

Figure 4: 
𝑅

𝑎
 versus 

𝑤(𝑅,𝑡)

𝑞0 sin 𝛽
, 𝛼2 = 6 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

238 
 

in figure 3.  It is observed, when also comparing with figure 1, which has the same torsional frequency as in [3], 

that the magnitude of the higher frequency torsional component of the velocity field is smaller closer to the 

centre of the cylinder.  As a result, the overall velocity field in this case would have a greater contribution from 

longitudinal component. The general characteristic shapes of the curves are again similar. As a result, it seems 

that in each case of different frequencies, that with the higher one seems to suppress the magnitude of that 

component of the velocity field closer to the centre of the oscillating cylinder. 

In figures 5 and 6, the �̂� component of the drag is depicted graphically for different values of 𝛼1.  Similarly, 

figures 7 and 8 display the �̂�  component of the drag for different values of 𝛼2.  The magnitude of the drag in the 

�̂�   direction oscillates between negative values, while the magnitude in the �̂�  direction oscillates between 

positive and negative values.  It is noted that for each component, when the oscillating frequencies increase, 

such results in an increase in the respective magnitudes.   

 

 

Figure 5: 
𝐷𝜃

𝑞0 μ cos 𝛽
 versus Ω1𝑡, 𝛼1 = 6 

     

Figure 6: 
𝐷𝜃

𝑞0 μ cos 𝛽
 versus Ω1𝑡, 𝛼1 = 10 

Figure 7: 
𝐷𝑧

𝑞0 μ sin 𝛽
 versus Ω2𝑡, 𝛼2 = 6 Figure 8: 

𝐷𝑧

𝑞0 μ sin 𝛽
 versus Ω2𝑡, 𝛼2 = 10 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

239 
 

 

In Figure 9, the work done in the �̂� direction is depicted graphically, where two scenarios are considered; a plot 

when Ω1 = 30 and Ω2 = 60 , and another when Ω1 = 60 and Ω2 = 30.  Both graphs are periodic and it is 

observed that the magnitudes are negative, with the amplitude for Ω1 = 30 and Ω2 = 60  being greater than that 

for Ω1 = 60 and Ω2 = 30, which indicates more work is being done.  In figure 10, the work done in the �̂� 

direction is examined for two cases, in particular, when  Ω1 = 30  and Ω2 = 60 , and when Ω1 = 60  and 

Ω2 = 30.  These plots appear to be periodic, each with a negative magnitude. The amplitude of the work done in 

this �̂� direction when Ω1 = 60 and Ω2 = 30 is observed to be larger than when Ω1 = 30 and Ω2 = 60, which 

shows that more work is being done. 

6. Conclusion 

The velocity of the fluid is affected if the frequencies of the cylinder’s oscillations are different, as was observed 

when compared to when the frequencies were the same.  In particular, it was noted that the higher oscillating 

frequency tended to suppress the magnitude of the corresponding component of the velocity field closer to the 

centre of the cylinder.  The magnitude of the drag increased with an increase in the oscillating frequency.  The 

work done in the �̂� direction is more when the torsional frequency is less than that of the longitudinal one.  In 

the �̂�  direction, the work done is less when the longitudinal frequency is more than that of the torsional 

oscillation. 

References 

[1]. G. G. Stokes. “On the effect of rotation of cylinders and spheres about their own axes in increasing the 

logarithmic decrement of the arc of vibration”.  Mathematical and Philosophical Papers 5 Cambridge: 

Cambridge University Press, England, pp. 207-214, 1886. 

[2]. M.J. Casarella, P.A. Laura.  “Drag on an Oscillating Rod with Longitudinal and Torsional Motion”.  

Journal of Hydronautics, v. 3, n. 4, pp. 180-183, 1969.  

[3]. H. Ramkissoon, S.R. Majumdar.  “Flow due to the Longitudinal and Torsional Oscillation of a 

Figure 9: 
𝑊1

𝑞0
2μ

 versus 𝑡 Figure 10: 
𝑊2

𝑞0
2μ

 versus 𝑡 

 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 

 

240 
 

Cylinder”.   Journal of Applied Mathematics and Physics, v. 41, pp. 598-603, 1990. 

[4]. W. Phillips, K. Rahaman.  “Motion of a Viscous Fluid contained in a Cylinder of Infinite Length 

subjected to Longitudinal and Torsional Oscillations of Independent Amplitudes”.  American 

International Journal of Contemporary Research, v. 2, n. 12, pp. 58-71, 2012. 

[5]. C. Calmelet-Eluhu, D. Majumdar.  “Flow of a micropolar fluid through a circular cylinder subject to 

longitudinal and torsional oscillations”.  Mathematical and Computer Modelling, v. 27, n. 8, pp. 69-78, 

1998. 

[6]. K. Rahaman.  “Internal Flow due to the Longitudinal and Torsional Oscillation of a Cylinder”.  Asia 

Journal of Information Technology, v. 3, n. 10, pp. 985-991, 2004. 

[7]. D. Owen, K. Rahaman.  “On the flow of an Oldroyd-B liquid through a straight tube performing 

longitudinal and torsional oscillations of different frequencies”.  Math: Ensenanza Univ., v. 14, pp. 34-

43, 2006. 

[8]. J. V. R. Murthy, N. K. Bahali.  “Micropolar Fluid flow in a Straight Circular Cylinder performing 

Longitudinal and Torsional Oscillations”.  Computational Thermal Sciences: An International Journal, 

v. 3, n. 2, pp. 123-131, 2011. 

[9]. G.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, pp. 75, 1967. 

[10]. W. Hughes.  An Introduction to Viscous Flows. New York. Hemisphere Publishing Corporation, 1979. 

[11]. M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and 

Mathematical Tables. U.S. Government Printing Office, pp. 379-382, 1972.