**3. Results and discussion**

The correct arrangements do not appear to be achievable for an entire arrangement of Eqs. (8)–(11) with proper limit conditions given in Eq. (12) in light of the nonlinear shape. This reality compels one to get the arrangement of the issue numerically. Suitable likeness change is received to change the overseeing incomplete differential conditions into an arrangement of non-straight customary differential conditions. The resultant limit esteem issue is understood numerically utilizing an efficient fourth-order Runge-Kutta method alongside shooting method (see Ramesh and Gireesha [27]). Facilitate the union examination is available in **Table 1**. For the verification of precision of the connected numerical plan, an examination of the present outcomes compared to the −(<sup>1</sup> <sup>+</sup> \_\_1 *<sup>β</sup>*) *f* ″(0) and −(<sup>1</sup> <sup>+</sup> \_\_1 *β*) *g*″(0) (nonappearance of Forchheimer parameter, porosity parameter) with the accessible distributed consequences of Ahmad and Nazar [16] and Nadeem et al. [15] is made and exhibited in **Table 2**, demonstrates a great understanding in this manner give confidence that the numerical outcomes got are precise.

This section is fundamentally arranged to depict the effect of different correlated physical parameters on velocity profiles *f* ′ (*ς*), *g*′ (*ς*), temperature profile *θ*(*ς*), nanoparticle part *ϕ*(*ς*), skin friction coefficient, and the local Nusselt and local Sherwood number through **Figures 2**–**16**. Give the first focus on the impact of extending parameter (*α*) on velocity profile as shown in **Figure 2**. It is noticed that for expanding benefits of extending parameter *α*, it decreases the

*β c* **= 0** *c* **= 0.5** *c* **= 1.0**

1.545869 [15, 16]

1.197425 [15, 16]

1.093641 [15, 16]

**Table 2.** Current numerical values and validation for friction factor −(<sup>1</sup> <sup>+</sup> \_\_1

*<sup>β</sup>*) *<sup>f</sup> ″***(0) <sup>−</sup>**(**<sup>1</sup> <sup>+</sup> \_\_1**

**(0) <sup>−</sup>**(**<sup>1</sup> <sup>+</sup> \_\_1**

for *α* = 0, exhibit wonders lessen the instance of two dimensional linear stretching, while for

(*ς*) fluctuates for different benefits of extending parameter *α*. It see that

*<sup>β</sup>*) *f* ″(0).

*<sup>β</sup>*) *g″*

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

0.657899 [15, 16]

0.509606 [15, 16]

0.465437 [15, 16]

**(0) <sup>−</sup>**(**<sup>1</sup> <sup>+</sup> \_\_1**

*<sup>β</sup>*) *g″* **(0)** 49

1.659891 [15, 16]

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1.285746 [15, 16]

1.174307 [15, 16]

speed *f* ′

(*ς*), while *g*′

**<sup>−</sup>**(**<sup>1</sup> <sup>+</sup> \_\_1** *<sup>β</sup>*) *f″*

[15, 16]

[15, 16]

[15, 16]

1 1.414213

5 1.095445

1000 1.000499

**Figure 2.** Influence of *α* on velocity field.


**Table 1.** Convergence analysis of the present work.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 49


**Table 2.** Current numerical values and validation for friction factor −(<sup>1</sup> <sup>+</sup> \_\_1 *<sup>β</sup>*) *f* ″(0).

friction coefficient, and the local Nusselt and local Sherwood number through **Figures 2**–**16**. Give the first focus on the impact of extending parameter (*α*) on velocity profile as shown in **Figure 2**. It is noticed that for expanding benefits of extending parameter *α*, it decreases the speed *f* ′ (*ς*), while *g*′ (*ς*) fluctuates for different benefits of extending parameter *α*. It see that for *α* = 0, exhibit wonders lessen the instance of two dimensional linear stretching, while for

**Figure 2.** Influence of *α* on velocity field.

*ς f(ς) f′*

that the numerical outcomes got are precise.

parameters on velocity profiles *f* ′

**3. Results and discussion**

**Table 1.** Convergence analysis of the present work.

 0.000000 1.000000 0.763674 0.695167 0.456344 0.365768 1.008110 0.202536 0.166242 1.146071 0.088680 0.073585 1.206291 0.038589 0.032175 1.232460 0.016746 0.013992 1.243809 0.007258 0.006070 1.248727 0.003144 0.002630 1.250857 0.001362 0.001139 1.251780 0.000590 0.000493 1.252179 0.000255 0.000213 1.252352 0.000111 0.000092 1.252427 0.000047 0.000040 1.252460 0.000020 0.000017 1.252474 0.000008 0.000007 1.252480 0.000006 0.000003 1.252482 0.000001 0.000001 1.252483 0.000000 0.000000

(*ς*), *g*′

plan, an examination of the present outcomes compared to the −(<sup>1</sup> <sup>+</sup> \_\_1

48 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

*(ς) −f″*

The correct arrangements do not appear to be achievable for an entire arrangement of Eqs. (8)–(11) with proper limit conditions given in Eq. (12) in light of the nonlinear shape. This reality compels one to get the arrangement of the issue numerically. Suitable likeness change is received to change the overseeing incomplete differential conditions into an arrangement of non-straight customary differential conditions. The resultant limit esteem issue is understood numerically utilizing an efficient fourth-order Runge-Kutta method alongside shooting method (see Ramesh and Gireesha [27]). Facilitate the union examination is available in **Table 1**. For the verification of precision of the connected numerical

*g*″(0) (nonappearance of Forchheimer parameter, porosity parameter) with the accessible distributed consequences of Ahmad and Nazar [16] and Nadeem et al. [15] is made and exhibited in **Table 2**, demonstrates a great understanding in this manner give confidence

This section is fundamentally arranged to depict the effect of different correlated physical

*(ς)*

(*ς*), temperature profile *θ*(*ς*), nanoparticle part *ϕ*(*ς*), skin

*<sup>β</sup>*) *f*

″(0) and −(<sup>1</sup> <sup>+</sup> \_\_1

*β*)

**Figure 3.** Influence of *Fr* on velocity fields.

**Figure 5.** Influence of *λ* on velocity field.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

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51

**Figure 6.** Influence of *λ* on temperature and concentration fields.

**Figure 4.** Influence of *Fr* on temperature and concentration fields.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 51

**Figure 5.** Influence of *λ* on velocity field.

**Figure 3.** Influence of *Fr* on velocity fields.

50 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

**Figure 4.** Influence of *Fr* on temperature and concentration fields.

**Figure 6.** Influence of *λ* on temperature and concentration fields.

**Figure 9.** Influence of Pr on temperature fields.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

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53

**Figure 10.** Influence of *Nb* on temperature and concentration fields.

**Figure 7.** Influence of *β* on velocity fields.

**Figure 8.** Influence of *β* on temperature and concentration fields.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 53

**Figure 9.** Influence of Pr on temperature fields.

**Figure 7.** Influence of *β* on velocity fields.

52 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

**Figure 8.** Influence of *β* on temperature and concentration fields.

**Figure 10.** Influence of *Nb* on temperature and concentration fields.

**Figure 11.** Influence of *Nt* on temperature and concentration fields.

**Figure 13.** Influence of *Q* on temperature field.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

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55

**Figure 14.** Influence of *Bi* on temperature and concentration fields.

**Figure 12.** Influence of *Le* on temperature and concentration fields.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 55

**Figure 13.** Influence of *Q* on temperature field.

**Figure 11.** Influence of *Nt* on temperature and concentration fields.

54 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

**Figure 12.** Influence of *Le* on temperature and concentration fields.

**Figure 14.** Influence of *Bi* on temperature and concentration fields.

**Figure 15.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *β* for different values of *Fr*, *λ*.

*α* = 1, sheet will extended along the two bearings with a similar proportion (axisymmetric case), and third and last case identify with extending proportion parameter *α* other than 0 or

(*ς*), *g*′

(*ς*) for particular estimations of *λ*. Both *f* ′

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

(*ς*). It is seen that with the influence of *β* infers an abatement in the yield worry

(*ς*) is a diminishing capacity of *Fr*. **Figure 4** features the impact of *Fr* on *θ*(*ς*), *ϕ*(*ς*). Of course *θ*(*ς*), *ϕ*(*ς*) and related thickness of boundary layer are higher when *Fr* increments. **Figure 5**

layer thickness decay for larger *λ*. Physically, nearness of permeable media is to upgrade the protection from smooth movement which makes decay in fluid speed and thickness of energy boundary layer. **Figure 6** depicts the impact of *λ* on *θ*(*ς*), *ϕ*(*ς*). It is discovered that bigger *λ*

**Figure 7** showed the impacts of non-Newtonian Casson fluid parameter (*β*) on velocity pro-

of the Casson fluid. This successfully encourages flow of the fluid, i.e., quickens the boundary layer flow near the extending surface, as appeared in **Figure 7**. In addition, it is found that with substantial estimations of *β*, the fluid is nearer to the Newtonian fluid. Truth be told, expanding estimations of the Casson fluid parameter *β* upgrade both temperature and

The variety in dimensionless temperature profile *θ*(*ς*) with expanding estimations of generalized Prandtl number Pr is appeared through **Figure 9**. The temperature profile diminishes

(*ς*) is plotted in **Figure 3**. Obviously, *f* ′

(*ς*), *g*′

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(*ς*),

57

(*ς*) and related

1; at that point, the flow conduct along both directions will be extraordinary.

Attributes of Forchheimer parameter (*Fr*) on *f* ′

nanoparticle division which is shown in **Figure 8**.

(*ς*), *g*′

**Figure 17.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *Nt* for different values of *Nb*.

illustrates the varieties in *f* ′

causes an addition in *θ*(*ς*), *ϕ*(*ς*).

*g*′

files *f* ′

(*ς*), *g*′

**Figure 16.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *Bi* for different values of *Q*.

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 57

**Figure 17.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *Nt* for different values of *Nb*.

**Figure 15.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *β* for different values of *Fr*, *λ*.

56 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

**Figure 16.** Variation of −*θ* ′(0) and −*ϕ* ′(0) with *Bi* for different values of *Q*.

*α* = 1, sheet will extended along the two bearings with a similar proportion (axisymmetric case), and third and last case identify with extending proportion parameter *α* other than 0 or 1; at that point, the flow conduct along both directions will be extraordinary.

Attributes of Forchheimer parameter (*Fr*) on *f* ′ (*ς*), *g*′ (*ς*) is plotted in **Figure 3**. Obviously, *f* ′ (*ς*), *g*′ (*ς*) is a diminishing capacity of *Fr*. **Figure 4** features the impact of *Fr* on *θ*(*ς*), *ϕ*(*ς*). Of course *θ*(*ς*), *ϕ*(*ς*) and related thickness of boundary layer are higher when *Fr* increments. **Figure 5** illustrates the varieties in *f* ′ (*ς*), *g*′ (*ς*) for particular estimations of *λ*. Both *f* ′ (*ς*), *g*′ (*ς*) and related layer thickness decay for larger *λ*. Physically, nearness of permeable media is to upgrade the protection from smooth movement which makes decay in fluid speed and thickness of energy boundary layer. **Figure 6** depicts the impact of *λ* on *θ*(*ς*), *ϕ*(*ς*). It is discovered that bigger *λ* causes an addition in *θ*(*ς*), *ϕ*(*ς*).

**Figure 7** showed the impacts of non-Newtonian Casson fluid parameter (*β*) on velocity profiles *f* ′ (*ς*), *g*′ (*ς*). It is seen that with the influence of *β* infers an abatement in the yield worry of the Casson fluid. This successfully encourages flow of the fluid, i.e., quickens the boundary layer flow near the extending surface, as appeared in **Figure 7**. In addition, it is found that with substantial estimations of *β*, the fluid is nearer to the Newtonian fluid. Truth be told, expanding estimations of the Casson fluid parameter *β* upgrade both temperature and nanoparticle division which is shown in **Figure 8**.

The variety in dimensionless temperature profile *θ*(*ς*) with expanding estimations of generalized Prandtl number Pr is appeared through **Figure 9**. The temperature profile diminishes with an expansion in the estimations of Prandtl number Pr, as Prandtl number is the proportion of energy diffusivity to thermal diffusivity. So, an expanding estimation of Prandtl number Pr infers a moderate rate of thermal dissemination which thus lessens the thermal boundary layer thickness. It can be directly seen that Prandtl number has more noticeable impact on Newtonian liquid when contrasted with non-Newtonian liquid.

**4. Conclusions**

there are no noteworthy changes in −*θ*′

*Bi −θ′*

thermal boundary layers.

**Author details**

Karnataka, India

Gosikere Kenchappa Ramesh

Address all correspondence to: gkrmaths@gmail.com

Department of Mathematics, School of Engineering, Presidency University, Bengaluru,

**Table 3.** Computational values of local Nusselt number and local Sherwood number for several values of *Bi*.

Three-dimensional flow of Casson nanoliquid within the sight of Darcy-Forchheimer connection, uniform warmth source/sink, and convective type boundary condition is considered. Numerical plan prompts the arrangements of physical marvel. From this investigation, we analyzed that the expanding Casson parameter compares to bring down velocity and higher temperature fields. The nearness of *Fr* and *λ* caused a lessening in velocity and increasing speed on temperature and nanoparticle portion. The bigger Biot number improved the temperature and nanoparticle division. Additionally, for vast estimations of Biot number,

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study

(0) and −*ϕ*′

0.1 0.084997 1.521377 0.5 0.239615 1.532624 2 0.324388 1.568759 5 0.343057 1.582440 10 0.349174 1.587619 50 0.353999 1.591990 100 0.354597 1.592550 500 0.355075 1.593001 1000 0.355134 1.593057 5000 0.355182 1.593102 10,000 0.355188 1.593108 100,000 0.355193 1.593113 1,000,000 0.355194 1.593113 5,000,000 0.355194 1.593114

exchange issues, heat sink parameter controls the relative thickening of the force and the

**(0) −***ϕ***′**

(0), which are available in **Table 3**. In heat

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59

**(0)**

**Figure 10** displays the temperature *θ*(*ς*) and the nanoparticle division *ϕ*(*ς*) for variable estimations of Brownian movement *Nb*. The fluid velocity is found to increment with expanding *Nb*, while in nanoparticle fraction decreases as *Nb* expansion which consequently improves the nanoparticle's concentration at the sheet. This might be because of the way the Brownian movement parameter diminishes the mass exchange of a nanofluid. The diagram of thermophoresis parameter *Nt* on the temperature *θ*(*ς*) and the nanoparticle part *ϕ*(*ς*) profiles is portrayed in **Figure 11**. From these plots, it is seen that the impact of expanding estimations of *Nt* is to build the temperature and nanoparticle fraction.

**Figure 12** shows the impact of Lewis number *Le* on temperature *θ*(*ς*) and the nanoparticle portion *ϕ*(*ς*) profiles. It is take note of that the temperature of the liquid increments however nanoparticle portion of the fluid diminishes with increment in *Le*. Physically truth that the bigger estimations of Lewis number makes the mass diffusivity littler, subsequently it diminishes the fixation field. The impacts of heat source/sink parameter *Q* can be found in **Figure 13**. For *Q* > 0 (heat source), it can be seen that the thermal boundary layer produces the vitality, and this causes the temperature in the thermal boundary layer increments with increment in *Q*. Though *Q* < 0 (heat sink) prompts diminish in the thermal boundary layer. *Q* = 0 speaks to the nonattendance of heat source/sink.

Impacts of the Biot number (*Bi*) on temperature are shown in **Figure 14**. Physically, the Biot number is communicated as the convection at the surface of the body to the conduction inside the surface of the body. At the point when thermal angle is connected at first glance, the proportion representing the temperature inside a body fluctuates significantly, while the body heats or cools over a period. Regularly, for uniform temperature field inside the surface, we consider *Bi* < < 1. In any case, *Bi* > > 1 portrays that temperature field inside the surface is not uniform. In **Figure 14**, we have examined the impacts of Biot number *Bi* on the temperature and nanoparticle portion profiles in two ways. The first one is the situation.

when *Bi* < 1. It is seen from **Figure 14** that for the littlest estimations of the Biot number *Bi* < 1, the variety of temperature inside the body is slight and can sensibly be approximated as being uniform. While in the second case, *Bi* > 1 delineates that the temperature inside the body is not performing a uniform conduct (see **Figure 14**).

The impact of physical parameter on nearby Nusselt −*θ*′ (0) and Sherwood number −*ϕ*′ (0) is displayed in **Figure 15**. We can see through **Figure 15** that non-Darcy-Forchheimer connection creates the low heat and mass at the divider when contrasted with the Darcy-Forchheimer connection. Thus, it is seen with an expansion of the two reasons for speeding up in the *λ* and *Fr*. From **Figure 16**, the expanding estimations of the heat source/sink parameter (*Q*) improve the local Nusselt number −*θ*′ (0) and decrease the local Sherwood number −*ϕ*′ (0) with *Bi* . A similar conduct is likewise found for the variety in *Nt* and *Nb* which is portrayed in **Figure 17**.
