An analytical investigation of the mixed convective Casson fluid flow past a yawed cylinder with heat transfer analysis

• Shahid Khan , Mahmoud M. Selim , Khaled A. Gepreel , Asad Ullah , Ikramullah , Muhammad Ayaz , Wali Khan Mashwani and Emel Khan
From the journal Open Physics

Abstract

The hydrodynamic flow of an incompressible and isotropic Casson fluid through a yawed cylinder is investigated by employing continuity, momentum, and energy equations satisfying suitable boundary conditions. The density variation is governed by Boussinesq approximation. The model equations consisting of coupled partial differential equations (PDEs) are transformed by applying non-similar transformation relations. The set of transformed PDEs is solved using the analytical technique of homotopy analysis method (HAM). The impacts of varying yaw angle, and mixed convection and Casson parameters over fluid velocity (chordwise and spanwise components), its temperature, Nusselt number, and skin friction coefficients are investigated and explained through various graphs. It is found that the enhancing yaw angle, Casson parameter, and convection parameter augment the fluid velocity, heat transfer rate, and skin friction and reduce the fluid temperature. The agreement of present and published results justifies the application of HAM in modeling the mixed convective Casson fluid flow past a yawed cylinder.

1 Introduction

In the current work, the laminar and incompressible Casson fluid through a yawed cylinder is analytically investigated by employing continuity, momentum, and energy equations. HAM is applied for the solution of governing equations. The novelty of the current study is the analytical investigation of the Casson fluid motion past a yawed cylinder. The structure of the article goes as follows: the geometry and mathematical modeling of the study are given in Section 2. The HAM technique used to solve the set of governing equations is elaborated in Section 3. The results are discussed through displayed plots in Section 4. The comparison of the present and published results is presented in Section 5. The main findings of the undertaken study are outlined in Section 6.

2 Problem modeling

Consider an incompressible, isotropic, and mixed convective Casson fluid motion near a yaw cylinder as depicted in Figure 1. The angle θ varies in the interval 0 ° 6 0 ° for the inclusion of the buoyancy forces. Here, z -axis is assumed along the cylinder surface, while the impact of boundary layer motion is considered along y -direction. Here, θ = 0 ° represents the normal, whereas θ = 9 0 ° represents the horizontal configurations of the yaw cylinder, respectively. Furthermore, θ greater than 6 0 ° is not considered for ignoring the effects of stagnation point flow. The streaming velocity U e ˜ U ˜ = 2 cos x R is chosen such that U ˜ W = sin θ , where W and U ˜ are the fixed velocities along z and x directions, respectively. The free stream flow is considered in the x z -plane. The spanwise free streaming is chosen as W e W = cos θ . The symbol g represents the gravitational acceleration acting vertically downward.

Figure 1

Flow geometry.

For isotropic and incompressible Casson fluid motion, the Cauchy tensor is represented as follows [33,34,35]:

(1) τ i j = 2 e i j μ b + p y 2 ϕ for ϕ > ϕ c , 2 e i j μ b + p y 2 ϕ c for ϕ < ϕ c ,

where e i j is the ( i , j ) th component of the deformation rate, ϕ c is the non-Newtonian model critical value, μ b is the plastic dynamic fluid viscosity, p y represents the yield stress, and ϕ = e i j e i j .

The flow velocities u , v , and w are chosen along x , y , and z directions, respectively. The ambient temperature T and the cylinder wall surface temperature T w are chosen such that T < T w . The Boussinesq approximation is considered for the density variation. The basic continuity, momentum, and energy conservation laws for the present Casson fluid motion take the following form:

(2) u x + v y = 0 ,

(3) u u x + v u y = ν 1 + 1 γ 2 u y 2 + U e ˜ d U e ˜ d x + g β T ( T T ) sin ( θ ) ,

(4) u w x + v w y = ν 1 + 1 γ 2 w y 2 + g β T ( T T ) cos ( θ ) ,

(5) u T x + v T y = α ρ C p 2 T y 2 .

The corresponding boundary restrictions are given as follows:

(6) u = 0 , v = 0 , w = 0 , T = T w , at y = 0 u U e ˜ , w W e , T T , at y .

In the above equations, ρ is the density, β T is the expansion coefficient, ν is the kinematic viscosity, T is the temperature of the fluid, α is the thermal conductivity, γ = μ b ( 2 ϕ c ) 0.5 p y is the Casson fluid parameter, and C p is the constant pressure specific heat. Using the non-similar transformations relations as in the following:

(7) ξ = 1 3 0 x U ˜ e U ˜ d x R , η = U ˜ e U ˜ U ˜ R 2 ν ξ 0.5 y R , ψ ( x , y ) = U ˜ R 2 ξ ν U ˜ R f ( ξ , η ) , u = ψ y , v = ψ x , Θ = T T T w T , x ¯ = x R and F = f η .

Using these non-similar transformations (7) in equations (2)–(6), we obtained the transformed equations and boundary restrictions as follows:

(8) 1 + 1 γ F η η + 1 3 ( f + 2 ξ f ξ ) F η + Ω ( ξ ) 3 ( 1 F 2 ) 2 3 ξ F F ξ + λ i l ( ξ ) Θ sin ( θ ) = 0 ,

(9) 1 + 1 γ S η η + 1 3 ( f + 2 ξ f ξ ) S η 2 3 ξ F S ξ + λ i I ( ξ ) Θ sin ( θ ) = 0 ,

(10) Θ η η + P r 3 ( f + 2 ξ f ξ ) Θ η 2 3 ξ P r F Θ ξ = 0 ,

(11) F = 0 , S = 0 , Θ = 1 , at η = 0 F 1 , S 1 , Θ 0 , at η .

Here, Pr = ρ C p ν k is the Prandtl number, f and ψ are non-dimensional and dimensional stream functions, respectively, λ i = g β T ( T w T ) R U ˜ e 2 is the mixed convection variable, F and S are the chordwise and spanwise dimensionless velocities, respectively. Furthermore, Θ is the dimensionless temperature, and the quantities Ω ( ξ ) , l ( ξ ) , and I ( ξ ) are defined as follows:

Ω ( ξ ) = 2 ξ U ˜ e d U ˜ e d ξ , l ( ξ ) = ξ 4 cos 3 x ¯ , I ( ξ ) = ξ 2 cos 2 x ¯ .

The transformed velocity components are given as follows:

(12) u = U ˜ e F ( ξ , η ) , v = U ˜ 3 2 ν ξ U ˜ 0.5 f ξ cos ( x ¯ ) + 2 f ξ cos ( x ¯ ) 3 η F tan ( x ¯ ) η ξ cos ( x ¯ ) F , w = W e S ( ξ , η ) .

The relation between ξ and x ¯ can be established with the following transformation:

(13) ξ ξ = ε ( x ¯ ) x ¯ ,

where ε ( x ¯ ) = tan x ¯ .

Now, ξ ( x ¯ ) , Ω ( x ¯ ) , l ( x ¯ ) , and I ( x ¯ ) as functions of x ¯ are given by the following equations.

(14) ξ ( x ¯ ) = 2 3 sin x ¯ , Ω ( x ¯ ) = tan 2 x ¯ , l ( x ¯ ) = sin x ¯ 6 cos 3 x ¯ and I ( x ¯ ) = sin x ¯ 3 cos 2 x ¯ .

Using equations (13) and (14) in equations (8)–(11), we get:

(15) 1 + 1 γ F η η + 1 3 ( 2 ε ( x ¯ ) f x ¯ + f ) F η + Ω ( x ¯ ) 3 ( 1 f 2 ) 2 3 ε ( x ¯ ) F F x ¯ + λ i l ( x ¯ ) Θ sin ( θ ) = 0 ,

(16) 1 + 1 γ S η η + 1 3 ( 2 ε ( x ¯ ) f x ¯ + f ) S η 2 3 ε ( x ¯ ) F S x ¯ + λ i I ( x ¯ ) Θ sin ( θ ) = 0 ,

(17) Θ η η + Pr 3 ( 2 ε ( x ¯ ) f x ¯ + f ) Θ η 2 3 ε ( x ¯ ) Pr F Θ x ¯ = 0 .

The boundary constraints are as follows:

(18) F ( x ¯ , 0 ) = 0 , S ( x ¯ , 0 ) = 0 , Θ ( x ¯ , 0 ) = 1 , at η = 0 , F ( x ¯ , 0 ) 1 , S ( x ¯ , 0 ) 1 , Θ ( x ¯ , 0 ) 0 , as η .

The relations for chordwise and spanwise skin frictions, and the Nusselt number are, respectively, given as follows:

(19) Re 0.5 C f = 4 3 1 + 1 γ cos 2 x ¯ sin x ¯ F η ( x ¯ , 0 ) ,

(20) Re 0.5 C f * = 2 3 1 + 1 γ cot ( θ ) cos x ¯ sin x ¯ S η ( x ¯ , 0 ) ,

(21) Re 0.5 Nu = 3 cos x ¯ sin x ¯ Θ η ( x ¯ , 0 ) .

3 Solution by HAM

In engineering sciences, majority of the problems are nonlinear in nature and are difficult to solve with the existing numerical methods. Since 1992, after the work of Liao [36], analytical methods are widely used to handle such complex problems. The HAM can solve both nonlinear ODEs and partial differential equations (PDEs). If the similarity solution exists, then the original problem reduces to the nonlinear ODEs. Solution of such problems can be seen in refs. [37,38,39]. On the other hand, if the similarity solution does not exist, then the original problem can not reduce to ODEs. Thus, the transformed problem retains its PDEs nature. Such problems are not widely seen in the literature. The reason behind this rareness is the solution of coupled non-linear PDEs, which is always a tough task for the researchers [40]. The mechanism for such complex problems is explained by Liao in “Homotopy Analysis Method in non-linear differential equations” [41]. In this method, the transformed problem retains its PDEs nature. The basic mechanism of HAM depends on the following functional relation:

(22) Ψ ˜ : X ˆ × [ 0 , 1 ] Y ˆ .

Here, Ψ ˜ [ x ˆ , 0 ] = ζ 1 ( x ˆ ) and Ψ ˜ [ x ˆ , 1 ] = ζ 2 ( x ˆ ) hold x ˆ X ˆ , while, ζ 1 ( x ˆ ) and ζ 2 ( x ˆ ) are the elements of X ˆ and Y ˆ spaces. The set of equations (15)–(18) are solved through HAM by choosing the non-linear operators N F ¯ ( F ¯ ) , N S ¯ ( S ¯ ) , and N Θ ¯ ( Θ ¯ ) with the corresponding initial guesses F 0 , S 0 , and Θ 0 .

4 Results and discussion

Here, we explain the variation of various aspects of Casson fluid flow with varying yaw angle θ , Casson fluid, and mixed convection parameters ( γ , λ i ) by depicting plots for chordwise ( F ) and spanwise ( S ) velocity components, dimensionless temperature, skin friction coefficients, and Nusselt number.

Figure 2(a and b) depicts the dependence of the velocity F ( η ) (chordwise direction) on varying θ (yaw angle) at different mixed convection parameter λ i values. Figure 2(a) is plotted for λ i = 2 , whereas Figure 2(b) is plotted for λ i = 10 . The values of θ used are θ = 1 5 ° , 3 0 ° , 4 5 ° , 6 0 ° . Figure 2(a) shows that the velocity displays a descending behavior with the rising θ . Figure 2(b) displays that the velocity in the chordwise direction shows an augmenting trend with the rising θ . The enhancement in the chordwise fluid velocity ( F ( η ) ) with the rising θ due to aiding buoyancy flow ( λ i = 10 ) is due to the more tilting of the cylinder, which develops a pressure gradient and causes the fluid to move faster. Whereas in the case of opposing buoyancy flow ( λ i = 2 ), the velocity in the chordwise direction drops with the enhancing yaw angle θ .

Figure 2

Influence of θ on (a) F ( η ) for λ i = 2 and (b) F ( η ) for λ i = 10 .

Figure 3 displays the dependence of F ( η ) on varying Casson parameter ( γ ) and mixed convection parameter λ i values. Figure 3(a) is plotted for λ i = 2 , whereas Figure 3(b) is plotted for λ i = 10 . The different values of γ used in displaying Figure 3 are γ = 2 , 4 , 6 , 8 . It is clear from Figure 3(a) that the chordwise velocity reduces with the augmenting values of λ i . Figure 3(b) displays that adding buoyancy ( λ i = 10 ) results in an enhancement in the chordwise fluid velocity of the Casson fluid. Therefore, it can be concluded that aiding buoyancy to the Casson fluid flow enhances the spanwise velocity with the rising Casson fluid parameter (higher γ means approaching toward Newtonian fluid character).

Figure 3

Impact of γ on (a) F ( η ) for λ i = 2 and (b) F ( η ) for λ i = 10 .

The behavior of the spanwise velocity component ( S ( η ) ) with the changing θ (yaw angle) and mixed convection parameter λ i is depicted in Figure 4. Figure 4(a) is depicted for λ i = 2 , while Figure 4(b) is plotted for λ i = 10 , respectively. It is apparent from Figure 4(a) that the spanwise velocity augments with the opposing buoyancy. From Figure 4(b) we see that spanwise velocity S profiles drop with the enhancing values of θ . Thus, the chordwise velocity drops with the increasing yaw angle due to aiding buoyancy flow. This means that in case of aiding buoyancy flow, the chordwise velocity component drops with the augmenting values of the yaw angle, whereas the opposing buoyancy flow shows an opposite impact on the chordwise velocity component.

Figure 4

Impact of θ on (a) S ( η ) for λ i = 2 and (b) S ( η ) for λ i = 10 .

The effects of changing Casson parameter ( γ ) and the mixed convection parameter λ i on S ( η ) (the spanwise velocity component) are depicted in Figure 5. Figure 5(a) is depicted for λ i = 2 , whereas Figure 5(b) is plotted for λ i = 10 , respectively. The first graph shows that S ( η ) varies directly with the rising η at fixed γ . The S ( η ) profiles shift to higher values with the augmenting γ up to η = 6.0 and then overlap with one another. The dependence of S ( η ) on the varying γ for λ i = 10 is displayed in Figure 5(b). The behavior of S ( η ) in this case with the augmenting θ looks to be completely opposite to that of Figure 5(a). It is therefore concluded that aiding buoyancy to the Casson fluid flow depreciates the spanwise fluid velocity, while augments it in case of opposing buoyancy with increasing Casson fluid parameter ( γ ) values.

Figure 5

Impact of γ on (a) S ( η ) , for λ i = 2 and (b) S ( η ) for λ i = 10 .

The variation of the dimensionless temperature ( Θ ( η ) ) with the changing values of θ and mixed convection parameter λ i is shown in Figure 6. Figure 6(a) is plotted for λ i = 2 , while Figure 6(b) is depicted for λ i = 10 , respectively. The values of θ used are θ = 1 5 ° , 3 0 ° , 4 5 ° , 6 0 ° . Figure 6(a) shows that with increasing θ , the Θ ( η ) profiles shift to higher values. Figure 6(b) displays the functional dependence of Θ ( η ) on η with changing θ for the higher value of the mixed convection parameter λ i . We see that the fluid temperature Θ ( η ) reduces with the augmenting values of the yaw angle. The reducing behavior of the fluid temperature for the rising θ with aiding buoyancy flow (larger λ i ) is due to the enhancing fluid velocity, which carries away more heat from the cylinder surface and thus causing the Casson fluid temperature to drop.

Figure 6

Impact of θ on (a) Θ ( η ) , for λ i = 2 and (b) Θ ( η ) for λ i = 10 .

The variations of the skin friction in the chordwise direction with changing yaw angle θ and mixed convection parameter λ i are shown in Figure 7. Figure 7(a) is plotted for λ i = 2 and Pr = 7.0 , whereas Figure 7(b) is drawn for λ i = 10 and Pr = 7.0 , respectively. It is clear that the skin friction in the chordwise direction enhances with the increasing yaw angle at higher value of mixed convection parameter λ i . This enhancement in the skin friction is due to the higher Casson fluid velocity with the augmenting yaw angle at higher λ i (aiding buoyancy flow) as observed in Figure 2(b).

Figure 7

Dependence of chordwise skin friction for (a) λ i = 2 and (b) λ i = 10 with varying θ at Pr = 7.0 .

The impact of varying yaw angle θ and mixed convection flow parameter λ i on the spanwise skin friction is shown in Figure 8. Figure 8(a) is plotted for λ i = 2 and Pr = 7.0 , whereas Figure 8(b) is drawn for λ i = 10 and Pr = 7.0 , respectively. From these figures, it is clear that the spanwise skin friction drops with the augmenting values of yaw angle for both opposing and aiding buoyancy flows. The skin friction drops with the aiding buoyancy as shown in Figure 8(b), which is due to the slip experienced by the Casson fluid in the spanwise direction, although the spanwise velocity augments with the aiding buoyancy flow as given in Figure 4(a).

Figure 8

Dependence of spanwise skin friction for (a) λ i = 2 and (b) λ i = 10 with the varying yaw angle at Pr = 7.0 .

Figure 9 depicts the variation of the Nusselt number with changing values of the yaw angle θ for the aiding buoyancy flow ( λ i = 10 ). It is clear that the thermal energy transfer flow augments with the rising yaw angle. This enhancement in the thermal energy transfer with the rising yaw angle is due to the augmenting Casson fluid flow velocity, which carries away more heat from the cylinder surface, and causing the Nusselt number to augment.

Figure 9

Heat transfer enhancement when λ i = 10 and Pr = 7.0 .

The dependence of chordwise velocity F and spanwise velocity S on the non-similarity variable x ¯ is depicted in Figure 10. The values of the other parameters are taken as λ i = 10 and Pr = 7.0 . It is apparent from the figures, that both velocities enhance with augmenting x ¯ , that is moving away from the origin of the coordinate system. The higher value of x ¯ corresponds to the more tilting configuration of the yawed cylinder, which causes to enhance the Casson fluid velocities in both chordwise and spanwise directions.

Figure 10

Dependence of (a) chordwise velocity and (b) spanwise velocity on x ¯ for λ i = 10 and Pr = 7.0 .

The variation of the dimensionless temperature ( Θ ) with varying x ¯ is displayed in Figure 11. The values of the other parameters are taken as λ i = 10 and Pr = 7.0 . It is observed that the temperature drops as one moves away from the origin (with rising x ¯ ). The drop in the temperature with the augmenting x ¯ is due to the enhancing fluid velocities ( F , S ) as observed in Figure 10.

Figure 11

Temperature variation with x ¯ , when λ i = 10 and Pr = 7.0 .

5 Validation of HAM

The comparison of the results achieved in this study with the results obtained in ref. [27] is displayed in Table 1. In Table 1, the skin friction (spanwise direction) and Nusselt number are computed with changing values of x ¯ and λ i . The first and second columns show the variation in x ¯ and λ i , respectively. The third and fourth columns represent the numerical values of skin friction and Nusselt number obtained in ref. [27]. The last two columns show the results of skin friction and Nusselt number computed in the current investigation. Table 1 shows the accuracy of our currently employed technique HAM (analytical), which is in a complete agreement with the previously used technique (numerical).

Table 1

Variations in skin friction and Nusselt number for θ = 0 ° and Pr = 0.7

x ¯ λ i Re 0.5 C f ref. [28] Re 0.5 Nu ref. [28] Present ( Re 0.5 C f ) Present ( Re 0.5 Nu )
0 0 1.3282 0.5853 1.32823 0.58532
0 1 4.9664 0.8220 4.96641 0.82204
0 2 7.7120 0.9304 7.71201 0.93043
1 0 1.9169 0.8667 1.91694 0.86672
1 1 5.2578 1.0618 5.25782 1.06183
1 2 7.8864 1.1686 7.88644 1.16862
2 0 2.3974 1.0965 2.39742 1.09653
2 1 5.6995 1.2713 5.69952 1.27133
2 2 8.3556 1.3743 8.35561 1.37433

6 Conclusions

The outcomes of the current investigations are outlined in this section. The hydrodynamic flow of an incompressible and isotropic Casson fluid through a yawed cylinder is analytically examined. The governing equations are simplified by using non-similar transformation relations. The standard analytical procedure of HAM is used to solve the developed system of equations. The influence of varying strength of yaw angle, mixed convection, and Casson fluid parameters over the velocity components (chordwise and spanwise), dimensionless temperature, skin friction coefficients, and heat transfer rate are investigated and explained with the help of different graphs. During this study, the following results are concluded:

• The enhancing yaw angle and Casson parameter augment the chordwise velocity F ( η ) at higher mixed convection parameter λ i (aiding buoyancy flow), while opposite variation is observed for smaller λ i (opposing buoyancy flow).

• The spanwise velocity S ( η ) enhances with the rising yaw angle and Casson parameter for opposing buoyancy flow, while drops with aiding buoyancy flow.

• The Casson fluid temperature drops with the enhancing yaw angle with aiding buoyancy flow.

• The chordwise skin friction augments, while the spanwise skin friction drops with the rising yaw angle at higher λ i .

• The increasing yaw angle enhances the heat energy transfer rate.

• The comparison of the obtained and published results shows an outstanding agreement that justifies the accuracy of the applied technique.

Abbreviations

The below mentioned parameters and abbreviations with their possible dimensions are used in this article.

β T

expansion of thermal coefficient 1 K

μ b

dynamic plastic viscosity k g m s

μ b

dynamic plastic viscosity k g m s

T

fluid temperature K

μ b

dynamic plastic viscosity k g m s

γ

Casson fluid parameter k g m s

α

thermal conductivity W m K

C p

specific heat J kg K

condition at infinity

0

reference condition

x , y , and z

coordinates (m)

Pr

Prandtl number

λ i

mixed convective parameter

Θ

dimensionless temperature

C f

skin friction

R e

local Reynolds number

F

chordwise dimensionless velocity

S

spanwise dimensionless velocity

Ψ

stream function

ρ

density ( kg m 3 )

t

time (s)

U ˜

constant velocity along x -direction m s

W ˜

constant velocity along z -direction m s

x ¯

dimensionless distance along the surface

W e

velocity at the edge of the boundary layer m s

U e

potential flow velocity at the edge of the boundary layer m s

η , ζ

transformed coordinates

θ

yawed angle (Degree)

Acknowledgments

The research was supported by the Taif University Researchers Supporting Project number (TURSP-2020/16), Taif University, Taif, Saudi Arabia.

1. Funding information: The authors received financial support from the Taif University Researchers Supporting Project Number (TURSP-2020/16), Taif University, Taif, Saudi Arabia.

2. Conflict of interest: The authors declare no conflict of interest.

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