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Hydrodynamic Force Expression with Couple Stress Drag Over Sphere 

at Small Reynolds Numbers 

 

K Madhukar1, Kallur V Vijayakumar2*, Chethan A.S. 3, Muneshwara M.S. 4 
1Department of Mathematics, BMS College of Engineering, Bengaluru – 560019, Karnataka, India. 

madhukar.maths@bmsce.ac.in. 

2Department of Mathematics, BMS Institute of Technology and Management, Bengaluru  – 560064, Karnataka, India. 

kallurvijayakumar@bmsit.in. 

3Department of Mathematics, BMS Institute of Technology and Management, Bengaluru – 560064, India. 

aschethan@bmsit.in. 

4Department of CSE, BMS Institute of Technology and Management Bengaluru – 560019, Karnataka,, India. 

muneshwarams@bmsit.in 

* Corresponding author E-mail: kallurvijayakumar@bmsit.in . 

 

Article History: 

Received: 20-04-2024 

Revised: 10-06-2024 

Accepted: 24-06-2024 

Abstract:  

Introduction: The flow past a sphere has been an important problem since Stokes and the 

Navier Stokes (NS) equation is the basic governing equation for it. The main reason for its 

importance is that the NS equation is not yet solvable analytically and has been solved in the 

absence of either or both unsteady inertia and convective inertia. The hydrodynamic force on 

a spherical particle at low Reynolds numbers has been meticulously derived by Lovalenti and 

Brady [2] and Stokes [1], has derived the same for a couple stress fluid. 

Objectives: In this problem we extend the works of Stokes [1] and Lovalenti and Brady [2] 

to obtain an expression for the hydrodynamic force on a couple stress flow over a rigid sphere 

at low Reynolds numbers. 

Methods: We consider both the expressions for the hydrodynamic forces and combine them 

to get a new expression for the effects of couple stress fluids at low Reynolds numbers on a 

rigid sphere. Then the expression is validated by reproducing the results of both the authors.  

Results: We consider both the extremes and obtain the plots of Stokes [1] and Lovalenti and 

Brady [2] and also obtain the transition from Newtonian fluid to Couple stress fluid at small 

Reynolds numbers. We note that the hydrodynamic force on the rigid sphere is the effect of 

viscous force and pressure and at zero couple stress! (an ideal situation) the fluid behaves as 

a Newtonian fluid with zero or constant flow when fluid is quiescent and as the couple stress 

parameters are introduces the force tend to grow from constant to exponential flow.  

Conclusions: This work fills the gap in the literature for the expression for force on a couple 

stress fluid on a rigid sphere at small finite Reynolds number. 

Keywords: Couple stresses, low Reynolds number, force, rigid spheres. 

 

𝜗 Kinematic viscosity 

𝑅𝑒𝐹 Amplitude of the periodic force 

𝑈𝑐 Characteristic particle slip velocity 

mailto:kallurvijayakumar@bmsit.in
mailto:aschethan@bmsit.in
mailto:muneshwarams@bmsit.in


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𝑭𝒔
𝑯∥(𝑡) = −6𝜋𝑼𝒔. 𝒑𝒑 Tangential part of Stokes’ drag 

FH Hydrodynamic force 

𝑎 Characteristic particle dimension 

𝑆𝑙 =
𝑎

𝑈𝑐
/𝜏𝑐 Strouhal number 

𝜏𝑐 Characteristic time scale 

𝑈𝑝𝑥
̅̅ ̅̅ ̅ Mean of Upx and similarly for other corresponding 

values 

𝑭𝒔
𝑯 Pseudo steady Stokes’ drag 

𝑅𝑒 = 𝑎𝑈𝑐/𝜗 Reynolds number 

𝑭𝒔
𝑯⊥(𝑡) = −6𝜋𝑼𝒔. (𝜹 − 𝒑𝒑) Normal part of Stokes’ Drag 

𝒑 =
𝒀𝒔(𝑡) − 𝒀𝒔(𝑠)

|𝒀𝒔(𝑡) − 𝒀𝒔(𝑠)|
 

Unit vector in align with the x axis 

A 
𝐴 =

𝑅𝑒

2
(

𝑡 − 𝑠

𝑅𝑒𝑆𝑙
)

1/2

(
𝒀𝒔(𝑡) − 𝒀𝒔(𝑠)

𝑡 − 𝑠
) 

𝑼𝒑(𝑡) Particle velocity 

𝑼𝒔(𝑡) = 𝑼𝒑(𝑡) − 𝑼∞(𝑡) Stream velocity 

𝑼∞(𝑡)  Uniform velocity 

Fext External force 

𝜹 idem tensor of order 2 

mp Particle mass 

Ys Integrated particle displacement 

 Couple stress parameter 

1. Introduction 

Solutions containing long-chain polymers, biological fluids, suspensions and colloidal fluids, liquid 

crystals, viscoelastic materials and other complex geometry structures and the complex fluids can 

exhibit couple stress behaviour [3 - 7]. The entangled polymer chains introduce microscale interactions 

that affect the overall rheological properties of the fluid[3]. Biological fluids, such as synovial fluid in 

joints, can exhibit non-Newtonian behaviour with microstructure effects. Blood is a complex fluid with 

suspended particles, cells, and plasma constituents. The interactions between these components at the 

microscale can be significant. In the context of blood flow, the micro-rotations can represent the 

rotation of individual blood cells or the effects of cell-cell interactions. The additional terms in the 



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constitutive equations of couple stress fluid account for the microscale contributions to the stress 

tensor. Understanding blood flow as a couple stress fluid can have implications for studying hemo -  

dynamics, especially in micro vessels and regions with complex geometries. It may provide insights 

into phenomena such as margination (concentration of blood cells near vessel walls) and rheological 

changes under pathological conditions [8-17]. The presence of cells, proteins, and other 

microstructures contributes to the overall rheological response. Fluids containing suspended particles 

or colloids interact between particles and the surrounding fluid introduce microscale effects that 

influence the overall flow characteristics. Liquid crystal fluids, which are composed of anisotropic 

molecules due to the alignment and rotation of the molecules in response to flow exhibits couple stress 

behaviour. The couple stress fluids exhibit more complex flow patterns and vortical structures 

compared to Newtonian fluids, particularly in regions with high shear rates[17]. The microstructural 

effects in couple stress fluids contribute to a viscoelastic response, meaning that these fluids may 

exhibit both viscous and elastic behaviour. There are various engineering applications of couple stress 

fluids [ 19-43] obtained by understanding the rheological behaviour of couple stress fluids, including 

biomedical engineering, polymer processing, and the design of devices involving microfluidics. The 

study of couple stress fluids contributes to the development of materials with tailored rheological 

properties for specific applications, such as enhanced lubrication or controlled drug delivery in medical 

devices. Hence, the couple stress fluids provide a more refined description of fluid behaviour by 

considering microstructural effects. The study of these fluids is essential in understanding and 

predicting the flow characteristics of complex fluids encountered in various natural and engineered 

systems. The couple stress fluids are a kind of micropolar fluids and also a non – Newtonian fluid that 

accounts for the effects of microstructure in the fluid. In the couple stress fluid model additional 

internal micro – rotations and micro – deformations are considered. This microstructure introduces an 

additional complexity to fluid dynamics and its behaviour is characterized by a couple stress tensor. 

This tensor describes the internal moments or torques within the fluid. Couple stress fluids incorporate 

microscale effects within the fluid. These microstructures represent the inherent rotational or 

deformational aspects at the microscopic level. The distinguishing feature of couple stress fluids is the 

inclusion of a couple stress tensor in the stress-strain relationship. This tensor describes the internal 

moments or torques within the fluid. In addition to the conservation of mass and momentum equations, 

couple stress fluids require additional equations governing the evolution of microstructure, introducing 

new parameters related to micro-rotation. These governing equations was first modelled by V K Stokes 

and is given by [1]. 

Understanding Sphere Flow in Couple Stress Fluids at Low Reynolds Number 

The behaviour of fluids can be described using a concept known as couple stress fluids, which take 

into account the microstructure and the intermolecular interactions within the fluid. In couple stress 

fluids, the stress at a point is not only dependent on the rate of strain but also on the rate of rotation at 

that point. This unique characteristic of couple stress fluids leads to deviations from the predictions of 

classical fluid mechanics. When considering low Reynolds number in fluid mechanics, the inertial 

forces are negligible compared to viscous forces, resulting in a laminar flow regime. This has 

significant implications for the behaviour of fluids around objects such as spheres, as the flow patterns 

and drag forces experienced by the sphere are greatly influenced by the low Reynolds number 

conditions. 



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Governing Equations 

In the study of couple stress fluids, the governing equations play a crucial role in understanding the 

behaviour of such fluids, especially at low Reynolds numbers. The specific equations that apply to low 

Reynolds number flow in couple stress fluids include the momentum balance equation, which 

incorporates the effects of couple stresses, and the angular momentum balance equation, which 

accounts for the microstructure of the fluid. These equations, when applied to low Reynolds number 

flow, offer insights into the intricate interplay between viscous forces, couple stresses, and inertial 

forces, providing a comprehensive framework for analysing the dynamics of such fluids in the presence 

of a sphere. 

In the study of couple stress fluids, it is crucial to derive and understand the governing equations that 

describe their behaviour, particularly at low Reynolds numbers. The low Reynolds number regime is 

characterized by the dominance of viscous forces over inertial forces, making it essential to account 

for the effects of microstructural phenomena such as couple stresses. Two fundamental equations that 

play a key role in describing low Reynolds number flow in couple stress fluids are the momentum 

balance equation and the angular momentum balance equation. 

The motivation and objective of the research 

In this work the viscous force and the inertial force is taken care of by the particle Reynolds number 

and the couple stresses are taken care by the couple stress parameter given by Stokes [1]. This work is 

a first attempt to combine the hydrodynamic force expression given by Lovalenti and Brady [2] and 

the drag expression for the spherical particle in couple stress fluid given by Stokes [1]. The motivation 

for this work is that the expression for long time limit is given by Lovalenti and Brady [2] and the drag 

expression for various radius are given by Stokes [50, 1]. Until now the attempt to combine these were 

not found in the literature. Hence the curious question answered here is what is the hydrodynamic force 

over a sphere at low Reynolds numbers in the presence of couple stresses. 

Formulation of the problem 

The Lovalenti and Brady (1993) formalism for the hydrodynamic force on a rigid sphere undergoing 

arbitrary time – dependent motion in an arbitrary time dependent uniform flow field at small Reynolds 

numbers is given by the expression as 

𝐹𝐻(𝑡) =
4𝜋

3
𝑅𝑒𝑆𝑙�̇�∞(𝑡) − 6𝜋𝑈𝑠(𝑡) −

2𝜋

3
𝑅𝑒𝑆𝑙�̇�𝑠(𝑡) 

               +
3

8
(

𝑅𝑒𝑆𝑙

𝜋
)

1
2⁄

{∫ [
2

3
𝐹𝑠

𝐻||(𝑡) − {
1

|𝐴|2
(

𝜋
1

2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))} 𝐹𝑠

𝐻||(𝑠)
𝑡

−∞

 

               +
2

3
𝐹𝑠

𝐻⊥(𝑡) − {𝑒𝑥𝑝(−|𝐴|2) −
1

2|𝐴|2
(

𝜋
1

2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))} 𝐹𝑠

𝐻⊥] 

                ×
2𝑑𝑠

(𝑡 − 𝑠)
3

2⁄
} + 𝑜(𝑅𝑒).                                                                                           (1) 

Here, 𝑈𝑠 = 𝑈𝑝 − 𝑈∞is the slip velocity of the fluid. 𝑈𝑝 is the velocity of the particle. Us has been non-

dimensionalized by Uc. The acceleration terms �̇�𝑠 and �̇�∞are non - dimensionalized by Uc/c, where 



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c is the characteristic timescale. 𝑈∞ is the velocity of the fluid as |𝑟| → ∞. Re is the Reynolds number, 

defined as Re = aUc/ based on a characteristic slip velocity, Uc, a denotes the characteristic particle 

dimension and   is the kinematic viscosity of the fluid. 𝐹𝑠

𝐻||  =  −6𝜋𝑈𝑠. 𝑝𝑝 and𝐹𝑠
𝐻⊥  =  −6𝜋𝑈𝑠. (𝛿 −

𝑝𝑝), where 𝛿 is the idem tensor of order 2 and unit vector𝑝 =
𝑌𝑠(𝑡)−𝑌𝑠(𝑠)

|𝑌𝑠(𝑡)−𝑌𝑠(𝑠)|
, here 𝑌𝑠(𝑡) − 𝑌𝑠(𝑠) is the 

integrated displacement of the particle relative to the fluid from time s to the current time t. Sl is the 

Strouhal number and 𝐴 =  
𝑅𝑒

2
(

𝑡−𝑠

𝑅𝑒𝑆𝑙
)

1
2⁄

(
𝑌𝑠(𝑡)−𝑌𝑠(𝑠)

𝑡−𝑠
). FH is scaled by aUc. 

We use equation (1) to obtain the equation governing the unidirectional motion of a sphere in a 

quiescent fluid, starting with zero velocity at time t = 0, with Us = Up where Up is the velocity of the 

particle, scaled with respect to the size of the particle and the frequency of the external periodic force, 

, i.e., we take Uc=a and 𝑈∞ = 0. Under these conditions, equation (3.2.1) reduces to 

𝐹𝐻(𝑡) = −6𝜋𝑈𝑝(𝑡) −
2𝜋

3
𝑅𝑒𝑆𝑙�̇�𝑝(𝑡) 

              +
3

8
(

𝑅𝑒𝑆𝑙

𝜋
)

1
2⁄

∫ [
−8𝜋𝑈𝑝(𝑡)𝑑𝑠

(𝑡 − 𝑠)
3

2⁄

𝑡

0

         

             − {
1

|𝐴|2
(

𝜋
1

2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))}

−12𝜋𝑈𝑝(𝑠)𝑑𝑠

(𝑡 − 𝑠)
3

2⁄
]                                   (2) 

We note that the integral in equation (2) contains a singularity at s = t. In order to take account of this 

singularity, the integral was split into the intervals [0, 𝑡 − 휀] and [𝑡 − 휀, 𝑡] for a small positive휀. Thus 

we get two expressions for the integral.   

Where, 𝐴 =  
𝑅𝑒

2
(

𝑡−𝑠

𝑅𝑒𝑆𝑙
)

1
2⁄

(
𝑌𝑠(𝑡)−𝑌𝑠(𝑠)

𝑡−𝑠
)  

That is we get,   

𝐹𝐻(𝑡) = −6𝜋𝑈𝑝(𝑡) −
2𝜋

3
𝑅𝑒𝑆𝑙�̇�𝑝(𝑡) +

3

8
(

𝑅𝑒𝑆𝑙

𝜋
)

1
2⁄

(𝑃 + 𝑄)      (3) 

 where 

𝑃 = ∫ [
−8𝜋𝑈𝑝(𝑡)𝑑𝑠

(𝑡−𝑠)
3

2⁄
− {

1

|𝐴|2 (
𝜋

1
2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))}

−12𝜋𝑈𝑝(𝑠)𝑑𝑠

(𝑡−𝑠)
3

2⁄
]

𝑡−𝜀

0
                    (3a) and 

𝑄 = ∫ [
−8𝜋𝑈𝑝(𝑡)𝑑𝑠

(𝑡−𝑠)
3

2⁄
− {

1

|𝐴|2 (
𝜋

1
2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))}

−12𝜋𝑈𝑝(𝑠)𝑑𝑠

(𝑡−𝑠)
3

2⁄
]

𝑡

𝑡−𝜀
                    (3b)                  

Denoting the components of FH(t),Up(t), �̇�𝑝, A, P and Q along the direction of the force field by FH(t), 

Up(t), �̇�𝑝, A, P and Q  and then transforming the integral with respect to A, we get 



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𝑄 = ∫
8𝜋𝑈𝑝

2(𝑡) 𝑅𝑒 𝑑 𝐴

(𝑅𝑒 𝑆 𝑙)
1

2⁄ 𝐴2

𝑐√𝜀

0

− ∫
1

|𝐴2|

𝑐√𝜀

0

(
√𝜋

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴2|))

12𝜋𝑈𝑝
2(𝑡) 𝑅𝑒 𝑑 𝐴

(𝑅𝑒 𝑆 𝑙)
1

2⁄ 𝐴2
                  (3c) 

where 𝑐 =
𝑅𝑒𝑈𝑝(𝑡)

2√𝑅𝑒𝑆𝑙
. 

Note that, Q vanishes as 휀  tends to zero; i.e., as s→t, 
1

|𝐴2|
(

√𝜋

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴2|)) →

2

3
.  

Hence the two singular terms cancel each other as s→t, and thus we obtain an expression for the 

hydrodynamic force on a couple stressed sphere in a quiescent fluid as: 

𝐹𝐻(𝑡) = −6𝜋𝑈𝑝(𝑡)(2 + 𝛼) −
2𝜋

3
𝑅𝑒𝑆𝑙�̇�𝑝(𝑡) 

+
3

8
(

𝑅𝑒𝑆𝑙

𝜋
)

1
2⁄

{∫ [{
1

|𝐴|2
(

𝜋
1

2⁄

2|𝐴|
𝑒𝑟𝑓(|𝐴|) − 𝑒𝑥𝑝(−|𝐴|2))}

12𝜋𝑈𝑝(𝑠)𝑑𝑠

(𝑡 − 𝑠)
3

2⁄
]

𝑡−𝜀

0

 

+16𝜋𝑈𝑝(𝑡) [
1

√𝑡
−

1

√𝜀
]}                                                                                                    (4)       

Here  is the couple stress parameter, equation included from the works of Stokes [1]. 

 

Method used to validate the expression: 

The program to solve the integral were written using GNU OCTAVE and the results of Lovalenti and 

Brady [1] and Stokes’ [2] were reproduced. Hence, we claim that the programs are robust and valid.  

Results: 

The difference between the inclusion of couple stresses is that there is a decay in the drag ratio when 

the radius of sphere increases as seen in figure (1), where as in the absence of couple stress the 

Reynolds number is attained eventually at large times as in figure (2). Further the inclusion of both 

leads to the convergence of the curves of the first figures. That is all the curves of the figure(1) 

irrespective of the  value merge with the last curve which is due to the inclusion of resistance to the 

change in the change in the motion of the fluid. 

Conclusion: 

This work is the combination of expressions given by V K Stokes and Lovalenti and Brady, in which 

we extended the couple stress fluid hydrodynamic force at zero Reynolds numbers to small and finite 

Reynolds numbers. It is observed that the combination of both couple stress parameters and low 

Reynolds numbers gives linearly decaying trajectory at large times. This is an expected result as there 

is a fore – aft symmetry in couple stresses and as the resistance to change in motion is applied on the 

couple forces of the fluids, they tend to nullify their effects and tend to zero. The authors in spite of 

performing the numerical experiments several times the expression for hydrodynamic force gave only 

the linear trend at large times. 



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Figure 1: The hydrodynamic force versus time 

at zero Reynolds numbers. 

Figure 2: The hydrodynamic force with 

respect to time at zero couple stress 

parameter at small Reynolds numbers 

showing self – similar patterns 

 

Figure 3: the combination of both Re and 

couple stress parameter, showing the linear 

decay of hydrodynamic force at large times. 

Figure 4: the combination of both Re and 

couple stress parameter, showing the linear 

decay of hydrodynamic force at large times, 

for Re < . 

 

Figure 5: the combination of both Re and couple stress parameter, showing the linear decay of 

hydrodynamic force at large times, Re > . 



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Acknowledgement: 

The authors wish to acknowledge the support and encouragement given by the Principals, Head, 

Colleagues and Management of the Institutes BMSCE and BMSIT&M. 

References: 

[1] Couple-Stress Fluids by V.K. Stokes (1966) 

[2] The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number, 

P M Lovalenti and J F Brady, 1993, J. Fluid Mech.  

[3] Effect of couple stresses on the rheology and dynamics of linear Maxwell viscoelastic fluids Physics of Fluids 32, 

013108 (2020) 

[4] Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with 

Slip Effect by Srinivasacharyulu et al. (2023):  

[5] Couple-Stress Fluid Improves Dynamic Response of Gear-Pair System Supported by Journal Bearings by Sui et al. 

(2012) 

[6] Elastohydrodynamic Lubrication Theory of Line Contacts: Couple Stress Fluid Model by Gupta and Kumar (1997) 

[7] A Review on Applications of Couple Stress Fluids by Marinca et al. (2013) 

[8] Blood flow as a couple stress fluid: Effects on hemorheology in stenosed microchannels by Hayat et al. (2012) 

[9] Flow of a couple stress fluid representing blood through stenotic vessels with a peripheral layer by Misra and Pandey 

(2013) 

[10] A new constitutive model for blood flow based on the couple stress theory by Gursel (2004) 

[11] Blood Microrheology in Stenosis Using Couple Stress Fluid Model by Misra et al. (2010) 

[12] A Theoretical Model for Blood Flow in Microchannels Based on Couple Stress Theory by Hayat et al. (2010) 

[13] Numerical Simulation of Blood Flow in Microchannels Using Couple Stress Fluid Model by Hayat et al. (2012) 

[14] Non-Newtonian Hemodynamics in Stenosed Arteries by Marinca et al. (2011) 

[15] A Mathematical Model for Blood Flow Through Arterial Stenosis Based on Couple Stress Theory by Hayat et al. 

(2009) 

[16] Blood Flow Through a Stenosed Artery with a Peripheral Plasma Layer by Hayat et al. (2011):  

[17] Blood Microrheology in Channels with Contractions and Expansions Using Couple Stress Fluid Model by Misra et 

al. (2009) 

[18] Numerical Simulation of Flow of a Couple Stress Fluid Past a Sphere by Rashidi et al. (2013) 

[19] Natural Convection Flow of a Couple Stress Fluid in a Square Cavity by Haq et al. (2013) 

[20] Boundary layer flow of a couple stress fluid over a porous stretching sheet by Mukhopadhyay (2012) 

[21] Flows of a Couple Stress Fluid Past a Circular Cylinder by Hayat et al. (2010) 

[22] Mixed Convection Flow of a Couple Stress Fluid over a Vertical Porous Plate in the Presence of Heat Source/Sink 

by Mukhopadhyay (2011):  

[23] Heat Transfer Characteristics of MHD Couple Stress Fluid Flow over a Permeable Stretching Sheet with 

Suction/Injection by Pal and Mandal (2019):  

[24] Steady MHD Couple Stress Fluid Flow and Heat Transfer over a Stretching Vertical Plate with Suction/Injection by 

Pal and Mandal (2018) 

[25] Peristaltic Transport of a Couple Stress Fluid in a Channel with Variable Viscosity by Srinivas and Rani (2014):  

[26] Numerical Simulation of MHD Couple Stress Fluid Flow Past a Sphere with Heat Transfer by Khan et al. (2018):  

[27] MHD couple stress fluid flow past a permeable stretching sheet with heat transfer by Mukhopadhyay (2017) 

[28] Boundary layer flow of a couple stress fluid over a porous stretching sheet by Mukhopadhyay (2012) 



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[29] Flow and heat transfer characteristics of a couple stress fluid over an exponentially stretching sheet with variable 

thermal conductivity by Mukhopadhyay (2018) 

[30] Flows of a couple stress fluid due to a continuously stretching porous sheet with heat transfer by Sharma and Kumar 

(2013) 

[31] Flows of a Couple Stress Fluid Past a Sphere by Wang (2012) 

[32] Heat transfer analysis of couple stress fluid flow past a porous stretching plate with convective boundary condition 

by Mukhopadhyay (2014)  

[33] Heat Transfer Analysis of MHD Couple Stress Fluid Flow over a Permeable Stretching Sheet with Suction/Injection 

by Mukhopadhyay (2013):  

[34] Instability of flow of a couple stress fluid through a porous medium by Srinivas and Raju (2016):  

[35] Mixed convection flow of a couple stress fluid over a heated vertical plate with uniform surface heat flux by 

Mukhopadhyay (2015) 

[36] Mixed Convection Flow of a Couple Stress Fluid over a Vertical Plate with Internal Heat Generation by Khan et al. 

(2015) 

[37] Natural convection flow of a couple stress fluid along a vertical wavy surface by Rahman and Khan (2016) 

[38] Natural Convection Flow of a Couple Stress Fluid confined in a Horizontal Porous Layer with Internal Heat 

Generation by Singh and Kumar (2017) 

[39] Numerical Simulation of Couple Stress Fluid Flow Past a Circular Cylinder by Li et al. (2013):  

[40] Numerical Simulation of Flow of a Couple Stress Fluid Past a Sphere by Mandal and Kumar (2004): 

[41] Numerical simulation of unsteady flow of a couple stress fluid past a circular cylinder by Mandal and Kumar (2007) 

[42] Onset of Convection in a Couple Stress Fluid Layer Heated from Below by Chandrasekhar (1964):  

[43] Slip Flow of a Couple Stress Fluid over a Stretching Sheet with Suction or Blowing by Mukhopadhyay (2015) 

[44] Stability of couple stress fluid flows by Straughan (2004):  

[45] Elastohydrodynamic Lubrication Theory of Line Contacts: Couple Stress Fluid Model by Gupta and Kumar (1997) 

[46] Analytical Solutions of Incompressible Couple Stress Fluid Flows by A.J. Chamkha et al. (2017). 

[47] Comparative study of generalized Couette flow of couple stress fluid using optimal homotopy asymptotic method and 

new iterative method by M.A. Khan et al. (2021) 

[48] Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with 

Slip Effect by M. Ijaz et al. (2023).  

[49] Flow of Two Immiscible Couple Stress Fluids between Two Permeable Beds by N.S. Akbar et al. (2018). 

[50] Stokes’ Couple Stress Fluids: Mathematical Descriptions and Applications by A.C. McBride (2016).  

[51] Theory of Elasticity for Microstructured Solids by R.A. Toupin (1964).  

[52] Flows of Anisotropic Fluids by G. Leal (1976).