Convective Mass and Heat ransferT of Sakiadis Flow of

Malaysian Journal of Mathematical Sciences 15(1): 125�136 (2021)

MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES

Journal homepage: http://einspem.upm.edu.my/journal

Convective Mass and Heat Transfer of Sakiadis

Flow of Magnetohydrodynamic Casson Fluid

Over a Horizontal Surface Employing

Cattaneo-Christov Heat Flux Model

Asmadi, M. S.1, Siri, Z. ∗1, and Kasmani, R. M.2

1Institute of Mathematical Sciences, Faculty of Science, University

of Malaya, 50603, Kuala Lumpur, Malaysia2Mathematics Division, Center for Foundation Studies in

Sciences, University of Malaya, 50603, Kuala Lumpur, Malaysia

E-mail: [email protected]∗ Corresponding author

Received: 21 January 2020

Accepted: 19 November 2020

ABSTRACT

Convective mass and heat transfer study of Sakiadis �ow of magneto-

hydrodynamic Casson �uid past a horizontal plate is produced in this

article by employing the Cattaneo-Christov heat �ux model. The gov-

erning equations are transformed into a set of ordinary di�erential equa-

tion by applying some similarity variables. A standard second-order and

a variation of third-order �nite di�erence method are employed to �nd

the e�ect of the magnetic force on the �uid system using a local similar-

ity approach. The variation in the boundary layer with the parameters

present in the problem is discussed in detail in this paper. Several dimen-

sional forms of the system are provided and the heat map of the thermal

boundary layer with di�erent parameters are investigated further in this

article. Based on the analysis, the higher the Casson parameter in the

Asmadi, M. S., Siri, Z. & Kasmani, R. M.

system, the lower the velocity of the �uid while the temperature and the

concentration increase.

Keywords: Magnetohydrodynamic, Casson �uid, Heat and mass trans-

fer, Cattaneo-Christov heat �ux model, Sakiadis �ow

1. Introduction

There are two types of �uids in our daily life; Newtonian and non-Newtonian�uids. The non-Newtonian �uids attracted a lot of interests and have beenongoing research among scienti�c communities due to its useful application inmany aspects of the industry. Toothpaste, ketchup, saliva and human blood areseveral examples of the said �uid. Since the non-Newtonian �uid model consistsof complex characteristics, there does not exist a single equation to constituteall the characteristics of the �uid. A typical non-Newtonian �uid model mayonly capture one or two aspects of the �uid, so researchers will choose the modelthat suits the most to their research analysis. Because of these limitations,several researchers developed non-Newtonian �uid models to capture di�erentcharacteristics of the model and are divided into several categories. Some �uidmodel is more useful for certain application than others. Casson �uid capturesthe aspect of the yield stress of the �uid which was proposed by Casson (1959)in his study on the �ow of the pigment oil suspensions of the printing ink. Itis also the most common rheological model used in engineering and industriessince this area deals with a lot of �uids with the yield stress characteristics.Several researches prior to the present study have been done by Mukhopadhyay(2013), Arthur et al. (2015), Animasaun et al. (2016), Malik et al. (2016), Rajuet al. (2017) and Mythili et al. (2017).

Mass and heat transfer exists in many instances in industrial processes suchas heat conduction in mobile smartphones, human tissues, and water heater.Heat transfer is a phenomenon in which the di�erences in temperature causesenergy transfer between some medium. The idea of the study of heat transfercharacteristics has been revolved around the well-known Fourier Law of heatconduction for quite some time. This simplistic law may be used for a simplesystem but is not suitable for a more complex set up. The downside of thislaw is a small disturbance will a�ect the �uid instantly due to the parabolictype energy equation that the law produced. Cattaneo (1948) modi�ed thelaw since he thought the law is somewhat unrealistic physically from the abovestandpoint of view. So, he includes a relaxation time for heat �ux, whichwas further re�ned by Christov (2009) by employing the Oldroyd-B's upper

126 Malaysian Journal of Mathematical Sciences

MHD Casson Fluid Flow Over A Surface Using Cattaneo-Christov Heat Flux Model

convective derivatives. This new heat �ux model is a generalization over theFourier Law of heat conduction. The study of several types of �uids governingby Cattaneo-Christov heat �ux model has been done by researchers such as Hanet al. (2014), Abbasi et al. (2015), Khan et al. (2015), Hayat et al. (016a),Waqaset al. (2016), Sui et al. (2016), Mushtaq et al. (2016) and Vinod et al. (2017).

The objective of this article is to analyse numerically the mass and heattransfer rate in Magnetohydrodynamic (MHD) Casson �uid with Sakiadis �ow.The heat transfer is analysed using the Cattaneo-Christov heat �ux model asopposed to the classical Fourier Law. The resulting velocity, temperature andconcentration gradient pro�le are computed using the �nite di�erence method(FDM). The dimensionless parameter associated with the problems are variedaccordingly to see the e�ect of variation, and the results are discussed in thegraphical form. The novelty of this paper is the usage of Cattaneo-Christovheat �ux model in investigating the behavior of the �uid �ow as well as theheat and mass transfer of MHD Casson �uid employing Sakiadis �ow past a�at plate. Previous researches uses the Fourier law of heat conduction, which isa simple heat transfer model. Since the application of heat transfer knowledgeis vastly applied in the industries and manufacturing processes, the model istoo simple to capture the behavior of the �uid. By including the relaxationtime for heat �ux, the �uid behavior can be observed in detail.

2. Mathematical Formulation

The laminar, two-dimensional set up of MHD Casson �uid over a semi-in�nite positive plane at y = 0 is considered as shown in Figure 1. An assump-tion is made by assuming the plate has a constant temperature Tw, constantconcentration Cw, ambient �uid temperature T∞ and ambient concentrationC∞. For the purpose of this problem, the �uid is incompressible and electricallyconducting.

𝒚

𝒙

Boundary Layer

Plate

Casson fluid

𝟎

𝒗

𝒖

𝑩𝟎 (Magnetic Field)

Figure 1: Geometry representation of the �uid

Malaysian Journal of Mathematical Sciences 127


The rheological equation for a non-Newtonian �uid is de�ned as,

τ = τ0 + µα∗, (1)

where τ is the Cauchy stress tensor, µ is the dynamic viscosity of the �uidunder consideration and α∗ is the shear rate. Equivalently, for Casson �uid,Eq.(1) can be expanded as,

τij =

2

(µB +

py√2π

)eij , π > πc

2

(µB +

py√2πc

)eij , π < πc

(2)

where π = eijeji with eij is the (i, j)th component of the �uid deformation

rate, py =µB√2π

βis the yield stress of the Casson �uid, µB is the dynamic

viscosity of the Casson �uid,πc is the critical value of π and β is the Casson�uid parameter.

The governing equations of steady Sakiadis �ow of an incompressible MHDCasson �uid are:

∂u

∂x+∂v

∂y= 0, (3)

u∂u

∂x+ v

∂u

∂y= ν

(1 +

1

β

)∂2u

∂y2− σB2

0

ρu, (4)

u∂T

∂x+v

∂T

∂y+λ2

u2∂2T

∂x2+ v2

∂2T

∂y2+ 2uv

∂2T

∂x∂y+

(u∂u

∂x+ v

∂u

∂y

)∂T

∂x+

(u∂v

∂x+ v

∂v

∂y

)∂T

∂y

= α∂2T

∂y2. (5)

u∂C

∂x+ v

∂C

∂y= D

∂2C

∂y2, (6)

where x is the coordinate along the semi-in�nite plane, y is the coordinatenormal to the surface of the plate, u and v are the velocity components along thex and y axis respectively, ν is the �uid kinematic viscosity, ρ is the density of the�uid, σ is the electrical conductivity of the �uid, B0 is the magnetic �eld in thedirection of normal to the horizontal surface, β is the Casson parameter, Cp isthe heat capacity, T is the local temperature of the �uid, D is the mass di�usioncoe�cient and C the local mass concentration. The boundary conditions for



Sakiadis �ow are,

u = U, v = 0, T = Tw, C = Cw at y = 0,

u→ 0, T → T∞, C → C∞ as y →∞.(7)

To solve Eqs.(4), (5) and (6), the following similarity transformation is used,

η = y

√U

νx, u = Uf ′(η), v = −1

2

√Uν

x(f − ηf ′),

θ =T − T∞Tw − T∞

, φ =C − C∞Cw − C∞

.

(8)

By the above transformation, Eq.(3) is satis�ed unconditionally. Using trans-formation (8), Eqs.(4), (5) and (6) are then transformed into the following:(

1 +1

β

)f ′′′ +

1

2ff ′′ −Mf ′ = 0, (9)

1

Prθ′′ +

1

2fθ′ − γ

2

(3ff ′θ′ + f2θ′′

)= 0, (10)

φ′′ +Sc

2fφ′ = 0. (11)

By using the same transformation, the boundary conditions (7) are reducedinto the following:

f(0) = 0, f ′(0) = 0, f → 1 as η →∞,

θ(0) = 1, θ → 0 as η →∞,

φ(0) = 1, φ→ 0 as η →∞.

(12)

where β =µB√2πc

pyis the Casson parameter, M =

σB20x

Uρis the magnetic �eld

parameter or Hartmann number, Pr =ν

αis the Prandtl number, γ =

λ2U

2xis

the local Deborah number for temperature and Sc =ν

Dis the Schmidt number.

Since the parameters M and γ contains the function of x, the local similaritysolutions are used instead, and the local solution found can be used to see thee�ect of parameters on the system.



3. Results and Discussion

To solve the nonlinear boundary value problem (9), a newly developed �nitedi�erence method (FDM) that was proposed by Pandey (2017) was used. ForEqs. (10) and (11) the method which presented by Burden and Faires (2011)was employed. All computations were done with a tolerance of ε = 10−5.

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f’(η

)

β = 0.5, 1.0, 1.5, 2.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f′(η)

M = 0.0, 0.4, 0.8, 1.2

Figure 2: The �uid velocity gradient for several values of: (a) β when M = 0.5 (left); (b) M whenβ = 0.5 (right); with Pr = Sc = 1 and γ = 0.5

Figure 2a shows the �uid velocity pro�le for some Casson parameter. As βincreases, the �uid velocity decreases. This means as the �uid tends to behavelike the Newtonian �uid, i.e. β →∞, the magnitude of �uid velocity decreases.Consequently, this produces a thinner momentum boundary layer of the �uid.This also means that the Newtonian-behaviour �uid will produce a thinnermomentum boundary layer than Casson-behaviour �uid. Figure 2b presentsthe �uid velocity pro�le for some Hartmann number. From Figure 2b, the �uidvelocity decreases as M increases signi�cantly. As M increases, the velocityof the Casson �uid is disturbed by the Lorenz force present in the �uid andthus helps the �uid to move faster, and in turns, produce a thinner momentumboundary layer. It can be concluded that the higher the Hartmann number,the thinner the momentum boundary layer.

The e�ect of Casson parameter on the �uid temperature gradient is shownin Figure 3a. Based on the Figure 3a, as β increases, the �uid temperaturegradient increases insigni�cantly. This means as the �uid behaves more likeNewtonian �uid (as β →∞), the thermal boundary layer becomes thicker andproduce a lower rate of the heat dissipation in the system. It can be concludedthat the �uid that behaves like Casson behaviour has a higher heat dissipationthan Newtonian-like �uid. Figure 3b presents the e�ect of Hartmann numberon the �uid temperature. In Figure 3b, as M increases, the Casson �uid tem-perature gradient increases. Since the presence of the magnetic �eld in the



0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

β = 0.5, 1.0, 1.5, 2.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

M = 0.0, 0.4, 0.8, 1.2

Figure 3: The �uid temperature gradient for several values of: (a) β when M = 0.5(left); (b) Mwhen β = 0.5(right); with Pr = Sc = 1 and γ = 0.5

system produces Lorenz force, this causes the said force to disturb the heatdissipation of the �uid by retarding the rate of the heat transfer in the �uid.This causes the thermal boundary layer to become thicker as M increases.

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η) Pr = 0.8, 1.0, 1.2, 1.4

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ(η)

γ = 0.0, 0.2, 0.4, 0.5, 0.6

Figure 4: The �uid temperature gradient for several values of: (a) Pr when γ = 0.5 (left); (b) γwhen Pr= 1 (right); with β = M = 0.5 and Sc= 0.5

Figure 4a presents the e�ect of Prandtl number on the �uid temperature.As the value of Pr increases, the temperature gradient of the �uid decreases.As Pr increases, the momentum di�usivity increases and dominates the ther-mal di�usivity. The �uid velocity is high enough to help the heat transfer ofthe �uid. This, makes the heat dissipation rate faster and makes the boundarylayer to become thinner. It can be seen that Figure 4b shows the e�ect of thelocal Deborah number, γ on the �uid temperature gradient. The temperaturegradient of the system increases at �rst, but later at η ≈ 2, the temperaturegradient drops when the value of gamma rises. From this observation, it can benoted that the heat transfer relaxation time of the �uid increases, which pro-duces a thinner boundary layer of the �uid. This illustrates that the dissipationof heat occurs at a faster rate.



0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

β = 0.5, 1.0, 1.5, 2.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

M = 0.0, 0.4, 0.8, 1.2

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ(η)

Sc = 1, 2, 3, 4, 6

Figure 5: The �uid concentration gradient for several values of: (a) β when M = 0.5 and Sc = 1(top, left); (b) M when β = 0.5 and Sc = 1 (top, right); (c) Sc when β = M = 0.5 (bottom); withPr = 1 and γ = 0.5

The e�ect of Casson parameter to the �uid concentration gradient is shownin Figure 5a. As β rises, the concentration gradient increases insigni�cantly.This means as the �uid employs more of the characteristics of a Newtonian �uid(as β →∞), the concentration boundary layer to become thicker and producea low mass transfer rate. It can be concluded that the �uid that behaves likeCasson behaviour has a higher mass transfer rate than Newtonian-like �uid.Figure 5b shows the e�ect of the magnetic force on the concentration gradientof the �uid. Since magnetic �eld strength in the system is represented bythe Hartmann number in the system, as M increases, the �uid concentrationgradient increases. This is because the presence of the magnetic �elds throughLorenz force retards the mass transfer rate of the �uid by applying oppositeforce. This causes the concentration boundary layer becomes thicker as Mincreases. The e�ect of the Schmidt number onto the �uid mass distributionis shown in Figure 5c. It can be seen from Figure 5c that as Sc increases,the concentration gradient decrease. It means that the higher ratio of �uidmomentum di�usivity and �uid mass di�usivity in the system resulted in athinner boundary layer of mass transfer.



Table 1: The physical properties for selected �uid

Fluid Pr T∞(◦C) ν (cm2/s)

Gaseous ammonia 1.5-2 25 0.145

The dimensional form for some type of system is presented for a betterunderstanding of the results produced earlier. Table 1 presents the physicalproperties of �uid used to demonstrate the system in a real-world situation.Figure 6 showed the heat map of gaseous ammonia at a constant T∞, Tw, U∞,M , γ and Sc, with β = 0.5 for Figure 6a and β = 2.0 for Figure 6b. It can beshown from Figure 6 that the thermal boundary layer for β = 0.5 is thinnerthan β = 2.0. Based on the Figure 6, it can be concluded that as the Cassonparameter increases, it produce a thicker thermal boundary layer, which meansthe heat dissipation rate decreases. This result strengthens the result found inFigure 2a.

Horizontal Position (cm)

Ver

tical

Pos

ition

(cm

)

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45°C

30

40

50

60

70

80

90


Ver

tical

Pos

ition

(cm

)

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45°C

30

40

50

60

70

80

90

Figure 6: The heat map of gaseous ammonia with: (a) β = 0.5; (b) β = 2.0; at T∞ = 25◦C withU∞ = 100 cm/s, Tw = 100◦C, M= γ = 0.5 and Sc= 1


Ver

tical

Pos

ition

(cm

)

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45°C

30

40

50

60

70

80

90


Ver

tical

Pos

ition

(cm

)

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45°C

30

40

50

60

70

80

90

Figure 7: The heat map of gaseous ammonia with: (a) γ = 0.0; (b) γ = 0.6; at T∞ = 25◦C withU∞ = 100 cm/s, Tw = 100◦C, M= β = 0.5 and Sc= 1



Figures 7 presented the heat map of Sakiadis �ow for gaseous ammoniaat a constant T∞, Tw, U∞, β, M and Sc, with γ = 0.0 for Figure 7a andγ = 0.6 for Figure 7b. Noted that γ = 0.0 corresponds to the classical simplisticFourier law heat transfer model. Based on the heat map shown in Figures7a and 7b, the thermal boundary layer for γ = 0.0 is thicker than that forγ = 0.6. This means that when the �uid behaves like Fourier law model, theheat dissipation is not as e�cient as Cattaneo-Christov heat �ux model. Inconclusion, Cattaneo-Christov heat �ux model increase the e�ciency of heatdissipation in the system. This observation is in match with the analysis fromFigure 4b.

4. Conclusion

This article discusses the convective mass and heat transfer of Sakiadis�uid �ow of Magnetohydrodynamic Casson �uid over a horizontal surface byemploying Cattaneo-Christov heat �ux model. The summary of the work islisted below:

1. The higher Casson parameter causes the velocity pro�le to decrease butvice versa for the temperature and the concentration pro�le.

2. As the Hartmann number increases, the lower the �uid velocity in thesystem but the temperature and concentration of the �uid increase.

3. Increasing the local Deborah number for temperature produce the lowertemperature gradient of the �uid.

4. The higher the Prandtl number, the lower the temperature gradient ofthe �uid.

5. For a higher Schmidt number, the �uid concentration gradient in thesystem decreases.

6. The boundary layer thickness decreases with the increase of the localDeborah number for temperature but produce a thicker boundary layerwith the increase of the Casson parameter.

Acknowledgement

The authors express their gratitude towards the Ministry of EducationMalaysia and University of Malaya for the support given �nancially through



Fundamental Research Grant Scheme FP043-2017A and University of MalayaResearch Grant RG397-17AFR.

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