8346


FACTA UNIVERSITATIS  

Series: Mechanical Engineering  

https://doi.org/10.22190/FUME211228019H 

© 2020 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

STUDY OF DOUBLE SLIP BOUNDARY CONDITION ON THE 

OSCILLATORY FLOW OF DUSTY FERROFLUID CONFINED IN 

A PERMEABLE CHANNEL 

Jafar Hasnain1, Hina Ghias Satti1, Mariam Sheikh2, Zaheer Abbas3 

1Bahria University Islamabad Campus, Pakistan 
2University of Sialkot, Pakistan 

3The Islamia University of Bahawalpur, Pakistan 

Abstract. The effects of slips and porosity on the channel walls in the flow of a heat-

absorbing/generating dusty ferrofluid streaming through a porous medium are 

investigated in this article. The channel is upright and subjected to a transverse 

magnetic flux along with thermal radiation. Kerosene with magnetite is used as the 

base fluid. The basic equations of the channel flow, which seem dimensional, are 

redesigned in a dimensionless manner utilizing non-dimensional variables. The 

variable separable method approach is used to solve the obtained equations 

analytically. The graphs demonstrate the behavior of these parameters on the flow 

fields, skin friction, and heat transmission rate, and are explained briefly. Results 

reveal that the flow velocity for heat-generating fluids is greater than the heat-

absorbing liquids. The fluid velocity upsurges with the improved values of the velocity 

slip parameter. The heat-generating dusty liquid has a higher heat transmission rate as 

compared to heat-absorbing dusty liquid.  

Key words: Ferro-fluid, Dust particles, Oscillatory flow, Suction/injection, Heat 

absorption/generation 

1. INTRODUCTION 

Many researchers have concentrated in recent decades on integrating nano or 

micrometer-sized artefacts in the base liquids because of the increased applications in 

areas such as lubrication, vibration damping, pressure seals for compressors and blowers, 

oscillation damping, seals and cooling of loudspeakers, chemical reactors, and several 

others. The combination of milli or micrometer-sized particles (dust particles) in the base 

                                                           
Received: December 28, 2021 / Accepted April 12, 2022 

Corresponding author: Jafar Hasnain 

Bahria University, Shangrilla Road, E-8, Islamabad, Pakistan  

E-mail: jafar.buic@bahria.edu.pk 



2 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

liquids creates dusty fluid. At the same time, nanofluid is formed by combining 

nanometer-sized particles into the base fluid. Magnetic nanofluid or ferrofluid is a 

subcategory of nanofluids built on tiny magnetic particles with only one magnetic domain. 

When there is no magnetic field in the proximity of these particles, they act similarly to 

ordinary metallic particles. Sunil et al. [1] reported that for very high values of the 

magnetic parameter, ferrofluid reacts like a usual fluid while examining the influence of 

dust particles on ferrofluid in the presence of the magnetic field. Sekar and Raju [2] 

analyzed the stability of Soret effects on dusty ferrofluid. They reported that the system 

was considered unstable due to the absorption of the porous media and dust particle 

parameters. In the existence of heat and magnetic field, a numerical result of dusty 

nanofluid flow was found by Sulochana and Sundeep [3] along with a stretching/shrinking 

cylinder. They noticed that the rise in dust particles' volume fraction deteriorates the heat 

transfer rate. Majeed et al. [4] discussed the flow, including dust and ferro particles with 

porous media over an extending sheet with heat. They concluded that if the values of the 

ferromagnetic parameter and Eckert number increase, so do the temperature profiles for 

both the ferrofluid and the dust phase. Gireesha et al. [5] considered Hall's effects, 

including magnetic field and heat generation/absorption on two-phase dusty-nanofluid 

with the observation that on the thermal distribution, Hall and magnetic parameters have 

opposite effects. Raizah [6] analyzed a hybrid nanofluid having dust particles moving 

through an enclosure containing two heated fins. He claimed that at the center of the top 

sheet, there are the lowest values for the local Nusselt number. Hatami and Jing [7] used 

an asymmetric wavy channel to describe the nanofluid flow. They concluded that the 

Brownian motion parameter affects the concentration of nanomaterials. Azam et al. [8] 

studied the axisymmetric flow of a non-Newtonian nanofluid in the presence of activation 

energy and reported that as the activation energy parameter was increased, the 

concentration of nanoparticles increased. Kaneez et al. [9] reported the numerical result 

of the flow of dust- concentrated micropolar fluid having hybrid nano-particles and 

presented the result that the parameter of the dust fleck association performs a crucial part 

in the fluid temperature field. The impact of activation energy on cross nanofluid in 

axisymmetric flow was explored numerically by Azam et al. [10]. One of the significant 

findings was that the concentration of nanoparticles increases as the activation energy 

increases. Nanjundappa et al. [11] theoretically investigated the flow of dusty ferrofluid 

flow with the perception that in comparison to very small magnetic intensity and 

ferromagnetic surfaces, strong paramagnetic surfaces have a greater steadying impact on 

ferro-thermal-convection. Mousavi el al. [12] found the dual solution for the flow of a 

hybrid nanofluid over a thin needle with thermal radiation, noting that the first solution 

was always thinner than the second. In a cross nanofluid with non-linear radiation, Azam 

et al. [13] considered gyrotatic microorganisms. They concluded that an increase in 

Lewis's number lowered the microorganism field. 

The fluid field boundary conditions play an essential role in measuring fluid flows in 

various industrial and medical systems. The dearth of the no-slip state is very obvious in 

polymer melts that often experience microscopic wall slippage. Slip state, either velocity 

or thermal, is essential in cleaning interval cavities and artificial heart valves, as well as 

shear skin, hysteresis effect, and spurt. Hayat et al. [14] highlighted the influence of MHD 

and slip in a channel on nanofluid flow and obtained an analytical solution. Kamel et al. 

[15] worked on the two-phase stream of dusty liquid in a planar channel with slip 



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 3 

boundary condition. The findings suggested that the condition of slip has a significant 

impact under certain concentration ranges. Guria [16] applied the periodic suction on the 

vertical channel with radiation and slip boundary condition and found that the slip effect 

has been noticed to minimize the velocity. Panaseti and Georgiou [17] discussed 

viscoplastic fluid flow in the channel in the slip regime with the observation that there has 

huge spike in the mild slip region. Slip parameter reduces the radial velocity was reported 

by Kashyap et al. [18] while studying the flow of UCM fluid past a channel having a 

chemical reaction and velocity slip boundary condition. Saleem et al. [19] investigated the 

peristaltic flow of MHD Jeffery fluid over a tapered channel with velocity second slip and 

concluded that a rise in slip parameters decreases the size of the trapping bolus.  

The heat generation/absorption investigation is carried out under a variety of 

conditions. Fluids go through chemical reactions that are either exothermic or 

endothermic in these situations. When heat energy is obtained from radioactive waste and 

water is heated in a solar collector, heat generation/absorption procedures occur. It is also 

used in packed-bed reactors, radiative cooling of molten steel, semi-conductor wafers, 

food storage, and underground disposal of nuclear waste, as well as electronic devices. 

The generation or absorption of heat determines both the fluid flow and the heat transfer 

rate in the moving fluids. Malik et al. [20] discussed the impact of thermal conductivity 

and heat generation/absorption on Williamson fluid flow over a stretching cylinder. They 

concluded that the heat source increases the temperature profile, whereas the heat sink 

shows the opposite behavior. Pandey and Kumar [21] reported that the temperature 

profiles are expanded according to each heat generation/absorption parameter value while 

analyzing MHD nanofluid flow through convergent/divergent channels under heat 

generation/absorption. With heat generation/absorption, Jha et al. [22] investigated fluid 

flow through the rotating channel. They found out that fluid’s temperature decreases with 

the rise in heat absorption while increasing with heat generation. Mishra et al. [23] 

unveiled the effects of heat generation/absorption through stretching/shrinking channels 

on MHD nanofluid flow. The highest heat generation/absorption parameter values in the 

shrinking convergent channel caused a spike in velocity and thermal profiles. Prakesh et 

al. [24] performed an analysis of the MHD flow of viscous fluid within the vertical 

channel along with heat generation/absorption and variable properties. They reported that 

the heat transmission rate drops in the left plate with the incorporation of heat 

generation/absorption and increases in the right plate. Sobamowo [25] studied the 

temperature of a moving porous fin as a function of internal heat generation and found 

that the temperature of the fin decreases as internal heat generation increases. Azam et al. 

[26] numerically investigated the flow of cross nanofluid under the influence of heat 

source and heat sink with the observation that heat source increases the temperature of the 

fluid. 

The utility of more than one phase streams analysis, in which solid particles are 

dispersed in a base liquid, is immense. Strong fuel rocket nozzles, crude oil cleaning, 

nuclear reactor cooling, hydro cyclones, electrical gadgets, cryopreservation, and cancer 

treatments are descriptions of such flows. Suction/injection of fluid through borders is 

critical in many areas and functional aspects such as aerodynamics, space sciences, film 

cooling, and boundary layer monitoring. Taking these applications into consideration, this 

study aims to evaluate the effect of dusty-ferrofluid oscillatory flow in a channel. Porous 



4 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

media, heat generation/absorption, the velocity slip boundary state, and thermal jump are 

also taken into account. 

2. PROBLEM DEVELOPMENT 

The laminar flow is considered under the impact of radiative heat flux (RHF) in a 

permeable media, saturated with incompressible dusty fluid. The medium is drawn along 

the x-axis and the y-axis is kept perpendicular to the walls of the channel (Fig. 1). The 

magnetic field strength B0 is parallel to the y-axis. The fluid temperature on both walls is 

different. Tf is the initial fluid temperature. T0 is the left wall temperature whereas Tw is the 

temperature at the right wall. Reynolds number Re is considered very small and hence the 

induced magnetic field becomes insignificant. The fluid is drawn with velocity v0 from the 

left wall and injected from the right wall. With these assumptions and using Boussinesq 

approximation, the flow equations are ([24], [27], [28]) 

 

Fig. 1 Geometry of the channel 

    20 00 0 0
1

,
n fnf

t y x nf yy p nf

nf nf n f

N K
u v u P u u u u B u g T T

K
         


 

  
 (1) 

    0 ,p p
t

u K u u   (2) 

 
     

 
*

0

1
,

nf

t o y yy y

p p p
nf nf nf

k Q
T v T T q T T

C C C  
      (3) 



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 5 

with the initial and surface constraints  

 

   

   

   

* *

2 2

* *

1 0 1

0, at , , 0 ,

, 0, at , , ,

, 0, at , 0, ,

p f

y p w y

y p y

u u T T y t y

u u u T T T y t a t

u u u T T T y t t

   

    

    

 

 

 (4) 

where Tw= T0+( Tf- T0) e
i ω t. In the above expressions u, up are velocity components for 

fluid and dust particles in the x-direction, P is pressure, K is porous media permeability, 

K0 (=6πρυD) is Stokes resistance, D is the average radius of a dust particle, g is 

gravitational constant, γ1*, γ2* are slip constants for velocity, α1*, α2* are slip constants for 

temperature, Q* is a heat source/sink and a is the width of the channel. The subscripts x, y, 

t represent the derivatives with respect to x, y, and t respectively. 

Assuming fluid to be thin optically and have low density relatively, the expression for 

RHF q is given as [29] 

 
2

0
4 ( ).

y
q T T   (5) 

where ξ is the coefficient of mean radiative absorption. 

The expressions for effective values viscosity μnf, density ρnf, thermal diffusivity αnf, 

thermal expansion βnf, specific heat capacitance (ρC)nf, thermal conductivity knf and 

electrical conductivity σnf  for ferrofluid are expressed as [9]  

 

 
 

 

 
 

    

 
 

   

2.5

*

*

* *

, 1 , ,
1

2 2
, 1 ,

2

3 1
1 , 1 , .

2 1

nf f

nf nf f s nf

p
nf

s f f snf

p p
nf s

f s f f s

nf s
nf f s

f f

k

C

k k k kk
C C

k k k k k


     

 


  



  
    

   

    


  
  

  


     

  

 (6) 

Here ϕ is the nanomaterial volume fraction. The subscripts nf, f, s stands for nanofluid, 

fluid and nanoparticles respectively. The thermophysical properties for nanomaterials and 

base fluid are given in Table 1. 

Table 1 Thermo-physical properties of NFs and nanoparticles ([30], [31]) 

Liquids and 

nanoparticles 3( / )Kg m


 

1 1
( )

p
C

Jkg K
 

 

1 1
( )

k

W m K
 

 

5

1

*10

( )K






 

( / )S m


 

Kerosene 783 2090 0.145 99 6*10-10 

Magnetite 5180 670 9.7 0.5 25000 

Copper 8933 385 401 1.67 59.6*106 

Silver 10500 235 429 1.89 6.3*107 

 



6 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

The forthcoming non-dimensional quantities are used. 

 

0

0

2

0 0

2 2

0

, , , , , ,

, , , Re , Pr ,

p

p

f

f f f p

f f f f

uT Tx y u tU
x y u t u

a a U T T a U

CN K aK Ua
Da M l

ka K a



  

  


     



    

  

 

22 2 * 2
02 *

2 2 * * * *
020 1 2 1 2

1 2 1 2

(T T )4
, , Gr , ,

, , , , , .

f f

f f f f f

f

f f f

g aa Q a a P
N Q P

k k U U

a Bv a
S H

a a a a



  

    
   

  


   


     

 (7) 

The flow equations along with the surface constraints in non-dimensional form can be 

written as  

 
2 2

* 4

3

1 1 2 1 1 2 1 1

1 1
Re ,

t x y yy p

H Q s l l
u P S u u u u Gr Q

Q Q Q Q Q Q Q Q

 
         

 
  (8) 

 Re ( ) ,p t pM u u u   (9) 

 
2

5 5 6
Re Pr Pr ,

t y yy
Q S Q Q N Q         (10) 

with surface constraints as 

 

   

   

   

2 2

1 1

0, 1 at , , 0 ,

, 0, at , 1, ,

, 0, at , 0, .

p

i t

y p y

y p y

u u y t y

u u u e y t t

u u u y t t

   

     

   





   

   

 (11) 

The bars are discarded for simplicity. The parameters H, Gr, l, M, N, P*, Pr, Q, s, S, α1, 

α2, γ1, γ2 and λ are Hartman number, Garshof number, particle concentration, particle 

mass, radiation, dimensionless pressure, Prandtl number, heat source parameter, shape 

factor for porous zone, a parameter of suction, slip parameters for velocity and 

temperature and oscillation amplitude for pressure respectively. 

3. METHOD OF SOLUTION 

The solution of the Eqs. (8) - (10) is obtained for pure oscillatory flow by letting  

 
   

       

*

0

0 0

, , ,

, , , .

i t i t

x

i t i t

p p

P e u y t u y e

u y t u y e y t y e

  

 

 

 



 
 (12) 

 

 



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 7 

Substituting the values from Eq. (12) into Eqs. (8-11) we obtain: 

     20 0 8 0 0 3
7 1

1
,

yy y
u S u Q u Gr Q

Q Q
    


  (13) 

 
 

0
0

,
1 Re

p

u
u

i M


 
 (14) 

     26 0 5 0 9 0Pr 0,yy yQ S Q Q      (15) 

with  

 

   

   

   

0

0

0

0 0

0 2 0 0 2 0

0 1 0 0 1 0

0, 1 at , , 0 ,

, 0, at , 1, ,

, 0, at , 0, .

p

i t

y p y

y p y

u u y t y

u u u e y t t

u u u y t t

   

     

   





   

   

 (16) 

On solving equation (15) with constraints (16), the expression for fluid temperature is 

obtained as  

    11 121 2, ,
Q y Q yi t

y t e C e C e 


  (17) 

The expressions for fluid and dusty particles velocities are obtained after solving Eqs. 

(13) and (14) respectively along with the constraints (16) and given as 

    17 18 11 123 4 19 20 21, ,
Q y Q y Q y Q yi t

u y t e C e C e Q Q e Q e    


 (18) 

    17 18 11 123 4 19 20 21
1

, .
1 Re

Q y Q y Q y Q yi t

p
u y t e C e C e Q Q e Q e

i M
    






 (19) 

The quantities of interest from the engineering point of view are surface friction Cf and 

Nusselt number Nu, which are expressed as  

  17 18 11 123 17 4 18 20 11 21 12
1

,
f Q y Q y Q y Q yi t

f y
y

nf nf

a
C u e C Q e C Q e Q Q e Q Q e

U 
     




 
 (20) 

where τf = ρnf υf uy at y=1 represents the shear stress of fluid at the right wall of the 

channel. 

and  
 

 11 121 11 2 12
0,1

0

,
Q y Q yi tw

y
y

nf f

aq
Nu e C Q e C Q e

k T T 
     




  (25) 

where qw=-knf Ty at y=1 represents the heat flux at the right wall of the channel. 

 

 

 

 



8 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

The constants that appear in the above expressions are given below: 

2.5

1 2 3 4 5 6

22
24

7 1 2 8 9

1 2 1

2 2 2 2

10 6 9 10 6 9

10 5 11 12 13

6 6

( )
1 , (1 ) , 1 , , 1 , ,

( )

Re
, Re , Pr Re,

1 Re

Q 4 Q 4
Q Pr , Q , Q ,

2 2

nf p s nfs s

f f f p f f

C K
Q Q Q Q Q Q

C K

H Q L i M
Q Q Q Q i Q N Q i

Q Q Q Q i M

Q Q Q Q Q Q
Q Q

Q Q

   
    

   

 
 





            

 
        

 

     
  

11 12

1 11

14

14 1 12 15 2 11 16 2 12 1

14 15 13 16

2 2 2 2 2 2

7 7 8 7 7 7 8 713

1 17 18

14 15 13 16

1 ,

Q
1 , (1 ), (1 ), ,

Q Q Q Q

4 Q Q 4 Q QQ
, , ,

Q Q Q Q 2 2

Q Q

Q

Q Q Q e Q Q e Q C

Q S Q S Q S Q S
C Q Q



  

 

      


     
  



 

 

17

18 11 12

1 3 7 2 3 7

19 20 21 22 1 172 2 2 2 2

1 8 11 7 8 7 11 12 7 8 7 12

23 1 18 24 19 20 1 11 21 1 12 25 2 17

26 2 18 27 19 20 2 11 21

, , , 1 ,

1 , Q (1 ) (1 Q ), Q (1 Q ),

Q (1 Q ), Q (1 Q ) Q (

Q

Q Q Q

C GrQ Q C GrQ Q
Q Q Q Q Q

Q Q Q Q Q Q Q S Q Q Q Q Q S

Q Q Q Q Q Q e

e Q Q e e




   

 

 
    

   

        

     
2 12

24 26 23 27 22 27 24 25

3 4

23 25 22 26 23 25 22 26

1 Q ),

.
Q Q Q Q Q Q Q Q

C C
Q Q Q Q Q Q Q Q



 
 

 
 

4. RESULTS AND DISCUSSION 

The expressions for the velocity of both fluid and dust particles and temperature of 

fluid are obtained by solving flow equations with the surface constraints analytically using 

the method of separating variables. Graphs are plotted to express the effects of the 

physical parameters on the velocity and temperature fields. 

Figure 2 is plotted to present the behavior of fluid velocity u and particle velocity up 

with the variation in S for both heat absorbing and generating liquids. The figure shows 

that both u and up are increasing with a rise in suction parameter S (>0) however u 

decreases with a rise in injection parameter S (<0). The physical fact is that cold fluid 

particles are injected into the porous channel through the channel's hot wall, and these 

heated fluid particles are then removed. As the temperature in the channel drops, the 

convection circulation weakens. As a result, the velocity decreases. The flow is parabolic 

for higher values of the suction parameter while it shifts towards the right wall for 

increasing injection. The flow field for heat-generating liquid is higher than the heat-

absorbing liquid.  



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 9 

  

Fig. 2 Variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for S 

Fig. 3 displays the variation in u and up with γ1, γ2 for two values of Q. This figure 

suggests that both the slip parameters have an increasing impact on both flow velocities. 

However, the fluid and dust particle velocities for a heat-absorbing case are less than the 

heat-generating case. 

When velocities of fluid and dusty particles are graphed through Fig. 4 to analyze the 

impact of ϕ, it is found that the ϕ has a decreasing effect on the velocities for heat-

generating liquid. While it has an increasing effect on the case of heat-absorbing liquid. 

 

Fig. 3 Variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for γ1, γ2 

 

Fig. 4 Variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for ϕ 



10 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

A comparative analysis is shown in Fig. 5(a, b) for the velocities of fluid and dust 

particles by dispersing different kinds of nanoparticles namely magnetite (magnetic 

particles) and copper and silver (non-magnetic particles). These figures show that the 

velocities of liquid and dust particles are lower when magnetic particles are immersed in 

the fluid as compared to non-magnetic particles. However, the velocities are higher when 

silver nanoparticles are used in the liquid. The reason for the lower value of velocities for 

the magnetic nanoparticles is the Lorentz force which arises due to the magnetic field 

affecting the magnetic particles as compared to non-magnetic particles. 

 

Fig. 5 Comparison of velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for different 

nanoparticles 

Fig. 6 is plotted to show the impacts of ϕ, α1, α2 and S on fluid temperature θ. The 

decreasing impact of ϕ for both heat absorbing and heat-generating liquids can be noticed 

in Fig. 6(a) with the higher temperature for the heat-generating liquid. The effects of slip 

parameters α1, α2 on the temperature of the dusty liquid can see in Fig. 6(b) which shows 

an increasing behavior with the rise in α1, α2 for the heat-generating liquid. The variation 

is reversed near the right wall for heat-absorbing liquid. Q rises, the fluid becomes more 

heated, and so the velocity and temperature rise. The temperature of both heat 

generating/absorbing dusty liquids increase when suction is increased while it reduces 

with a raise in injection (see Fig. 6(c)). In all of the figures above, the magnitude of the 

fluid's velocity and temperature in the case of heat generation is greater than that of heat 

absorption. This finding is consistent with expectations: as Q rises, the fluid becomes 

more heated, and so the velocity and temperature rise. 

Fig. 7 exhibits the variation in Cf at the right wall for both heat-absorbing and 

generating fluids. Surface friction versus S and γ2 are presented in Fig. 7(a) and 7(b) 

respectively. It can be observed that an increase in ϕ raises the magnitude of surface 

friction for the heat-absorbing case while it decreases for the heat-generating liquid. The 

surface friction increases with suction at the wall whilst it reduces with a rise in injection. 

(Fig. 7(a)). The magnitude of surface friction shows a decreasing trend for the heat-

generating liquid with the rise in ϕ and γ2. 



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 11 

 

 

Fig. 6 Change in fluid temperature θ(y,t) for (a) ϕ (b) α1, α2 (c) S 

 

Fig. 7 Surface friction Cf at right wall for ϕ (a) versus S (b) versus γ2 

Heat transmission rates versus S  and γ2 at the right wall for different values of ϕ are 

presented in Fig. 8. An increase in ϕ and S (<0) reduce the heat transmission rate while S 

(>0) has an enhancing impact on the rate of heat transfer for both heat-absorbing and 

generating liquids (Fig. 8(a)). Fig. 8(b) shows a slight increase in the transmission rate for 

heat-absorbing liquid with α2. However, a significant improvement in the heat 

transmission rate is noticed for heat-generating liquid. 



12 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

 

Fig. 8 Heat transfer rate Nu at right wall for ϕ (a) versus S (b) versus α2 

Fig. 9 is plotted to present the oscillatory behavior for the flow fields for the case of 

heat-generating fluids with variation in suction/injection parameter. It can be seen the 

oscillation is higher for the case of suction for all flow fields. 

Fig. 10 presents the agreeable results of the present study with the results of already 

published literature for limiting values of parameters S=ϕ=γ1=γ2 =α1=α2=Q=0. 

  

 

Fig. 9 Oscillatory behavior of (a) velocities of liquid u(y,t) (b) dust particle up(y,t) (c) 

fluid temperature θ(y,t) versus t for S  



 Study of Double Slip Boundary Condition on the Oscillatory Flow of Dusty Ferrofluid Confined … 13 

  

 

Fig. 10 Comparison of present results with existing literature for (a) velocities of liquid 

u(y,t) (b) dust particle up(y,t) (c) fluid temperature θ(y,t)  

5. CONCLUDING REMARKS 

The velocity and thermal slip effects on the oscillatory flow of a ferrofluid in the 

presence of dusty particles are studied. The heat source/sink is taken into account when 

investigating heat transfer features. For the flow query, an exact solution is sought. The 

numerical values are graphed to demonstrate the effect of the physical parameters 

concerned. The results are summarized below: 

 When the suction parameter is increased, both the fluid and particle velocities 

increase; but, when the injection parameter is increased, the fluid velocity 

decreases. 

 The fluid velocity of the heat-generating liquid is greater than that of the heat-

absorbing liquid. 

 Velocity slip parameters have a growing influence on both flow velocities. 

 The temperature of the dusty fluid rises as the thermal slip parameters of the heat-

generating fluid increase. 

 The temperature of both heat generating/absorbing dusty liquids rises as suction 

rises, while it falls as injection rises. 

 For heat-generating liquid, there is a significant increase in heat transmission rate. 

 Oscillation in the flow fields is higher for the case of suction. 

 Velocities of liquid and dust particles are higher when non-magnetic nanoparticles 

are dispersed compared to magnetic nanoparticles. 



14 J. HASNAIN, H. G. SATTI, M. SHEIKH, Z. ABBAS 

 
Acknowledgements: We are thankful to the reviewers for their encouraging comments and 

constructive suggestions to improve the quality of the manuscript. The authors appreciate the 

financial support from HEC Pakistan through SRGP-2109. 

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