**2. Mathematical formulation**

Non-Newtonian fluids can't be portrayed because of nonexistence of single constitutive connection among stress and rate of strain. In the current year, non-Newtonian fluids have turned out to be increasingly essential because of its mechanical applications. Truth be told, the enthusiasm for boundary layer flows of non-Newtonian fluid is expanding significantly because of its extensive number of functional applications in industry producing preparing and natural fluids. Maybe, a couple of principle illustrations identified with applications are plastic polymer, boring mud, optical fibers, paper generation, hot moving, metal turning, and cooling of metallic plates in a cooling shower and numerous others. Since no single non-Newtonian model predicts every one of the properties of non-Newtonian fluid along these lines examinations proposed different non-Newtonian fluid models. These models are essentially classified into three classifications specifically differential-, rate-, and fundamental-type fluids. In non-Newtonian fluid, shear stresses and rates of strain/disfigurement are not directly related. Such fluid underthought which does not comply with Newton's law is a straightforward non-Newtonian fluid model of respectful sort. In 1959, Casson displayed this model for the flow of viscoelastic fluids. This model has a more slow progress from Newtonian to the yield locale. This model is utilized by oil builds in the portrayal of bond slurry and is better to predict high shear-rate viscosities when just low and middle road shear-rate information are accessible. The Casson show is more exact at both high and low shear rates. Casson liquid has one of the kind attributes, which have wide application in sustenance handling, in metallurgy, in penetrating operation and bio-designing operations, and so on. The Casson show has been utilized as a part of different businesses to give more exact portrayal of high shearrate viscosities when just low and moderate shear-rate information are accessible [9]. Toward the starting Nadeem et al. [10] introduce the idea of Casson fluid and demonstrate over an exponentially extending sheet. Numerous examinations identified with viscoelastic proper-

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

The nanoparticles in by and large are made of metal oxides, metallic, carbon, or some other materials [18]. Standard fluid has weaker conductivity. This weaker conductivity can be enhanced incredibly with the utilization of nanoparticles. Truly, the Brownian movement factor of nanoparticles is base fluid and is essential toward this path. An extraordinary measure of warmth is delivered in warm exchangers and microelectromechanical procedures to lessen the framework execution. Fluid thermal conductivity is enhanced by nanoparticle expansion just to cool such modern procedures. The nanoparticles have shallow significance in natural and building applications like prescription, turbine sharp-edge cooling, plasma- and laser-cutting procedure, and so on. Sizeable examinations on nanofluids have been tended to in the writing. Buongiorno [19] has investigated the mechanisms of nanofluid by means of snapshot of nanoparticles in customary base fluid. Such instruments incorporate nanoparticle measure, Magnus effect, dormancy, molecule agglomeration, Brownian movement, thermophoresis, and volume portion. Here, we introduce some imperative scientists who have been accounted

for by considering the highlights of thermophoretic and Brownian movement [20–27].

In displaying the flow in permeable media, Darcy's law is a standout among the most prominent models. Particularly, flow in permeable media is exceptionally valuable in grain stockpiling, development of water in repositories, frameworks of groundwater, fermentation process, raw petroleum generation, and oil assets. In any case, it is by and large

ties of liquid are underthought [11–17].

Three-dimensional flow of Casson nanofluid filling permeable space by Darcy-Forchheimer connection is considered. Flow is bidirectional extending surface. Nanofluid model depicts the properties of Brownian dispersion and thermophoresis. Concurrent states of heat convective and heat source/sink are executed. We receive the Cartesian coordinate in such a way to the point that and pivot are picked along *x* and *y* ordinary to the stretchable surface. Let *Uw*(*x*) = *ax* and *Vw*(*y*) = *by* be the extending velocity along *x* and *y* directions separately. The surface temperature is directed by a convective heating procedure which is depicted by heat exchange coefficient *hf* and temperature of hot liquid *Tf* under the surface (see **Figure 1**). The boundary layer equations for flow underthought are

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \mathbf{0},\tag{1}$$

$$
\mu \frac{\partial u}{\partial x} + \upsilon \frac{\partial u}{\partial y} + \upsilon \nu \frac{\partial u}{\partial z} = \nu \left( 1 + \frac{1}{\beta} \right) \frac{\partial^2 u}{\partial z^2} - \frac{\nu}{K} u - F u^2 \tag{2}
$$

$$
\mu \frac{\partial v}{\partial x} + \upsilon \frac{\partial v}{\partial y} + \upsilon \upsilon \frac{\partial v}{\partial z} = \nu \left( \mathbf{1} + \frac{1}{\beta} \right) \frac{\partial^2 v}{\partial z^2} - \frac{\nu}{K} \upsilon - F \upsilon^2,\tag{3}
$$

**Figure 1.** Geometry of the problem.

The process.

$$
\mu \frac{\partial T}{\partial x} + \upsilon \frac{\partial T}{\partial y} + \upsilon \upsilon \frac{\partial T}{\partial z} = \alpha\_m \frac{\partial^2 T}{\partial z^2} + \frac{(\rho c)\_p}{(\rho c)\_p} \left( D\_\theta \left( \frac{\partial T}{\partial z} \frac{\partial C}{\partial z} \right) + \frac{D\_\tau}{T\_\text{\\_}} \left( \frac{\partial T}{\partial z} \right)^2 \right)
$$

$$
+ \frac{q\_0}{(\rho c)\_p} \{T - T\_\text{\\_}\} \tag{4}
$$

$$
\mu \frac{\partial \mathcal{C}}{\partial x} + \upsilon \frac{\partial \mathcal{C}}{\partial y} + \text{tr} \frac{\partial \mathcal{C}}{\partial z} = D\_y \left( \frac{\partial^2 \mathcal{C}}{\partial z^2} \right) + \frac{D\_y}{T\_-} \left( \frac{\partial^2 T}{\partial z^2} \right) \tag{5}
$$

*<sup>θ</sup>*(*ς*) <sup>=</sup> *<sup>T</sup>* <sup>−</sup> *<sup>T</sup>* \_\_\_\_\_<sup>∞</sup>

(<sup>1</sup> <sup>+</sup> 1\_

(<sup>1</sup> <sup>+</sup> \_\_1

*θ*'' + Pr((*f* + *g*) *θ*' + *Nb*'

Boundary conditions of Eq. (6) become

These dimensionless variables are given by

\_\_1 2

*<sup>ν</sup>* and Re*<sup>y</sup>* = *Vw*

*f*(0) = *g*(0) = 0, *f* '

*f* '

number are as follows:

Re*<sup>x</sup>*

where Re*<sup>x</sup>* = *Uw*

Re*<sup>x</sup>*

\_\_*x*

parameter.

*<sup>ϕ</sup>*'' <sup>+</sup> *Le* Pr(*<sup>f</sup>* <sup>+</sup> *<sup>g</sup>*) *<sup>ϕ</sup>*' <sup>+</sup> \_\_\_ *Nt*

Applying Eq. (7) in (1) is verified. and Eqs. (2)–(5) are

*Tf* − *T*<sup>∞</sup>

*<sup>β</sup>*) *g*''' + (*f* + *g*) *g*'' − *g* '

*β*)*f* ''' + (*f* + *g*)*f* '' − *f* ' 2

(0) = 1, *g*'

(∞) → 0, *g*'

In the above expressions, *λ* stands for porosity parameter, *Fr*

*λ* = \_\_\_*<sup>ν</sup> Ka*

*Cfx* <sup>=</sup> (<sup>1</sup> <sup>+</sup> 1\_

−\_\_1 2 , *Fr* <sup>=</sup> *<sup>C</sup>*\_\_\_*<sup>b</sup> K*\_\_1 2 , *α* = \_\_*<sup>b</sup>*

*Nt* <sup>=</sup> (*ρc*)*<sup>p</sup> DT*(*Tf* <sup>−</sup> *<sup>T</sup>*∞) \_\_\_\_\_\_\_\_\_\_\_\_ (*ρc*)*<sup>f</sup> ν T*<sup>∞</sup>

> *β*)*f* ″(0), ( *x*\_ *<sup>y</sup>*)Re*<sup>y</sup>* \_\_1 2

*N ux* = −*θ*′

*y*\_\_

, *<sup>ϕ</sup>*(*ς*) <sup>=</sup> *<sup>C</sup>* <sup>−</sup> *<sup>C</sup>* \_\_\_\_\_\_<sup>∞</sup> *Cw* − *C*<sup>∞</sup>

, *ς* = ( \_\_*a ν*) \_\_1 2

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

− *λ f* ' − *Fr f* '

<sup>2</sup> − *g*' − *Fr g* '

*ϕ*' + *Nt* ' 2

(0) = *α*, *θ*'

for ratio parameter, Pr for Prandtl number, *Le* for Schmidt number, *Bi* for Biot number, *Nt* for thermophoresis parameter, *Nb* for Brownian motion parameter, and *Q* heat source/sink

Dimensionless relations of skin friction coefficient, local Nusselt number, and local Sherwood

(0), and Re*<sup>x</sup>*

−\_\_1 2

*<sup>ν</sup>* depict the local Reynolds numbers.

*<sup>a</sup>*, Pr  <sup>=</sup> \_\_\_*<sup>ν</sup> αm*

, *Nb* <sup>=</sup> (*ρc*)*<sup>p</sup> DB <sup>C</sup>* \_\_\_\_\_\_\_\_\_<sup>∞</sup>

*Cfy* <sup>=</sup> *<sup>α</sup>*(<sup>1</sup> <sup>+</sup> 1\_

*S ux* = −*ϕ*′

2

*z* (7)

47

http://dx.doi.org/10.5772/intechopen.74170

= 0 (8)

<sup>2</sup> = 0 (9)

for Forchheimer parameter, *α*

, *Bi* <sup>=</sup> *hf* \_\_ *k* √ \_\_ \_\_*ν a* , 

, *<sup>Q</sup>* <sup>=</sup> *<sup>q</sup>*\_\_\_\_0 *aρcp*

) + Pr*Q*(*η*) = 0 (10)

*Nb <sup>θ</sup>*'' <sup>=</sup> <sup>0</sup> (11)

(0) = −*Bi*(1 − *θ*(0)), *ϕ*(0) = 1,

(∞) → 0, *θ*(∞) → 0, *φ*(∞) → 0 (12)

, *Le* = \_\_\_*<sup>ν</sup> DB*

(*ρc*)*<sup>f</sup> <sup>ν</sup>* .

*<sup>β</sup>*)*g*″(0),

(0),

and boundary conditions of the problem is

$$
\mu = ax,\\
v = by,\\
w = 0,\\
\text{C = C}\_w \text{ at } z = 0$$
 
$$u \to 0,\\
v \to 0,\\
T \to T\_\omega \text{ C } \to \text{C}\_\omega \text{ as } z \to \text{cs} \tag{6}$$

Here *u*, *v* and *w* represent as components of velocity in *x*, *y* and *z* directions, respectively; *<sup>ν</sup>* <sup>=</sup> *<sup>μ</sup>*\_\_ *ρf* stands for kinematic viscosity; *μ* for dynamic viscosity; *ρ<sup>f</sup>* for density of base liquid; *<sup>K</sup>* for permeability of porous medium; *<sup>F</sup>* <sup>=</sup> *<sup>C</sup>*\_\_\_*<sup>b</sup> xK*\_\_1 2 for nonuniform inertia coefficient of porous medium; *Cb* for drag coefficient; *T* for temperature; *α<sup>m</sup>* <sup>=</sup> \_\_\_\_ *<sup>K</sup>* (*c*)*<sup>f</sup>* for thermal diffusivity; *q*<sup>0</sup> volumetric rate of heat generation and absorption; *β* is the Casson parameter; *k* for thermal conductivity; (*c*)*<sup>p</sup>* for effective heat capacity of nanoparticles; (*c*)*<sup>f</sup>* for heat capacity of fluid; *C* for concentration; *DB* for Brownian diffusion coefficient; *DT* for thermophoretic diffusion coefficient; *T*∞ for ambient fluid temperature; *C*∞ for ambient fluid concentration; and *a* and *b* for positive constants.

Selecting similarity transformations are

$$u = \operatorname{axf}^{\cdot}(\boldsymbol{\varsigma}), \boldsymbol{v} = \operatorname{ayg}^{\cdot}(\boldsymbol{\varsigma}), \boldsymbol{w} = -(\boldsymbol{a}\boldsymbol{v})^{\underline{\underline{\boldsymbol{d}}}}(\boldsymbol{f} + \underline{\boldsymbol{g}}),$$

Darcy-Forchheimer Flow of Casson Nanofluid with Heat Source/Sink: A Three-Dimensional Study http://dx.doi.org/10.5772/intechopen.74170 47

$$\Theta(\mathfrak{z}) = \frac{T - T\_{\text{-}}}{T\_f - T\_{\text{-}}} \, \phi(\mathfrak{z}) = \frac{C - C\_{\text{-}}}{C\_{\text{-}} - C\_{\text{-}}} \, \mathfrak{z} = \left(\frac{a}{V}\right)^{\frac{1}{2}} \mathfrak{z} \tag{7}$$

Applying Eq. (7) in (1) is verified. and Eqs. (2)–(5) are

$$\left(1+\frac{1}{\beta}\right)f^{\circ}+\left(f+g\right)f^{\circ}-f^{\circ\circ}-\lambda f^{\circ}-F\_{r}f^{\circ\circ}=0\tag{8}$$

$$\left(1+\frac{1}{\beta}\right)g''' + \left(f+g\right)g' - g'^2 - \lambda g' - F\_r g'^2 = 0\tag{9}$$

$$
\boldsymbol{\Theta}^{\cdot} + \Pr\left( \left( \boldsymbol{f} + \boldsymbol{\mathcal{g}} \right) \boldsymbol{\Theta}^{\cdot} + \mathrm{N} \boldsymbol{b} \boldsymbol{\theta}^{\cdot} \boldsymbol{\phi}^{\cdot} + \mathrm{N} \boldsymbol{t} \boldsymbol{\theta}^{\cdot 2} \right) + \Pr\boldsymbol{Q} \boldsymbol{\theta}(\boldsymbol{\eta}) = \boldsymbol{0} \tag{10}
$$

$$
\phi^{\cdot \cdot} + L\varepsilon \Pr(\mathfrak{f} + \mathfrak{g})\,\phi^{\cdot} + \frac{\mathrm{Nt}}{\mathrm{Nb}}\,\Theta^{\cdot \cdot} = 0 \tag{11}
$$

Boundary conditions of Eq. (6) become

*<sup>ν</sup>* <sup>=</sup> *<sup>μ</sup>*\_\_ *ρf*

medium; *Cb*

ductivity; (*c*)*<sup>p</sup>*

for positive constants.

*u* \_\_\_ <sup>∂</sup>*<sup>T</sup>* <sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>v</sup>* \_\_\_ <sup>∂</sup>*<sup>T</sup>* ∂*y*

and boundary conditions of the problem is

*u* = *ax*, *v* = *by*, *w* = 0, −*k* \_\_\_ <sup>∂</sup>*<sup>T</sup>*

*<sup>K</sup>* for permeability of porous medium; *<sup>F</sup>* <sup>=</sup> *<sup>C</sup>*\_\_\_*<sup>b</sup>*

Selecting similarity transformations are

*u* = *axf* '

*u* \_\_\_ <sup>∂</sup>*<sup>C</sup>*

**Figure 1.** Geometry of the problem.

+ *w* \_\_\_ <sup>∂</sup>*<sup>T</sup>*

<sup>+</sup> *<sup>q</sup>* \_\_\_\_0

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>v</sup>* \_\_\_ <sup>∂</sup>*<sup>C</sup>* ∂*y*

stands for kinematic viscosity; *μ* for dynamic viscosity; *ρ<sup>f</sup>*

for drag coefficient; *T* for temperature; *α<sup>m</sup>* <sup>=</sup> \_\_\_\_ *<sup>K</sup>*

for effective heat capacity of nanoparticles; (*c*)*<sup>f</sup>*

(*ς*), *v* = *ayg*'

<sup>∂</sup>*<sup>z</sup>* <sup>=</sup> *<sup>α</sup><sup>m</sup>*

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

∂<sup>2</sup> \_\_\_*T* <sup>∂</sup> *<sup>z</sup>* <sup>2</sup> <sup>+</sup>

(*c*)*<sup>p</sup>*

+ *w* \_\_\_ <sup>∂</sup>*<sup>C</sup>*

(*c*) \_\_\_\_*<sup>p</sup>* (*c*)*f*(*DB*(

<sup>∂</sup>*<sup>z</sup>* <sup>=</sup> *DB*(

*u* → 0, *v* → 0, *T* → *T*∞, *C* → *C*<sup>∞</sup> as *z* → ∞ (6)

Here *u*, *v* and *w* represent as components of velocity in *x*, *y* and *z* directions, respectively;

*xK*\_\_1 2

metric rate of heat generation and absorption; *β* is the Casson parameter; *k* for thermal con-

*C* for concentration; *DB* for Brownian diffusion coefficient; *DT* for thermophoretic diffusion coefficient; *T*∞ for ambient fluid temperature; *C*∞ for ambient fluid concentration; and *a* and *b*

(*ς*), *w* = −(*a*)

∂<sup>2</sup> \_\_\_*C* <sup>∂</sup> *<sup>z</sup>* <sup>2</sup>) <sup>+</sup> \_\_\_ *DT <sup>T</sup>*<sup>∞</sup> ( ∂<sup>2</sup> \_\_\_*T*

\_\_\_ ∂*T* ∂*z* \_\_\_ ∂*C* <sup>∂</sup>*z*) <sup>+</sup> \_\_\_ *DT <sup>T</sup>*<sup>∞</sup> ( \_\_\_ ∂*T* <sup>∂</sup>*z*) 2 ) 

<sup>∂</sup>*<sup>z</sup>* <sup>=</sup> *hf*(*Tf* <sup>−</sup> *<sup>T</sup>*), *<sup>C</sup>* <sup>=</sup> *Cw* at *<sup>z</sup>* <sup>=</sup> <sup>0</sup>

(*c*)*<sup>f</sup>*

\_\_1 2(*f* + *g*),

(*T* − *T*∞ ) (4)

<sup>∂</sup> *<sup>z</sup>* <sup>2</sup>) (5)

for density of base liquid;

for heat capacity of fluid;

volu-

for nonuniform inertia coefficient of porous

for thermal diffusivity; *q*<sup>0</sup>

$$\begin{aligned} f(0) &= g(0) = 0, \; f'(0) = 1, \; g'(0) = \alpha, \; \theta'(0) = -Bi(1 - \theta(0)), \; \phi(0) = 1, \\\\ f(\circ \circ) &\to 0, \; g'(\circ \circ) \to 0, \; \theta(\circ \circ) \to 0, \; \phi(\circ \circ) \to 0 \end{aligned} \tag{12}$$

In the above expressions, *λ* stands for porosity parameter, *Fr* for Forchheimer parameter, *α* for ratio parameter, Pr for Prandtl number, *Le* for Schmidt number, *Bi* for Biot number, *Nt* for thermophoresis parameter, *Nb* for Brownian motion parameter, and *Q* heat source/sink parameter.

These dimensionless variables are given by

Mouse are given by

$$\lambda = \frac{\nu}{\text{K}a'} F\_r = \frac{\text{C}\_s}{\text{K}\_s^{\prime\prime}} \,\alpha = \frac{b}{\text{d}\prime} \,\text{Pr} = \frac{\nu}{\text{d}\prime} \,\text{L} \varepsilon = \frac{\nu}{\text{D}\_\text{g}} \,\text{Q} = \frac{q\_0}{\text{d}\rho \text{c}\_{\text{p}}} \,\text{Bi} = \frac{h\_\circ}{k} \sqrt{\frac{\text{v}}{\text{d}\prime}}.$$

$$\text{Nt} = \frac{\left(\rho \text{c}\right)\_\text{p} D\_r \left(T\_f - T\_\text{a}\right)}{\left(\rho \text{c}\right)\_\text{p} V \, T\_\text{a}}, \text{ N} \\ \text{b} = \frac{\left(\rho \text{c}\right)\_\text{p} D\_\text{b} \,\text{C}\_\text{a}}{\left(\rho \text{c}\right)\_\text{f} V}.$$

Dimensionless relations of skin friction coefficient, local Nusselt number, and local Sherwood number are as follows:

$$\begin{aligned} \mathrm{Re}\_{\boldsymbol{\chi}}^{\perp} \mathsf{C}\_{\circ \times} &= \left( 1 + \frac{1}{\beta} \right) \mathsf{f}'(\mathbf{0}), \left( \frac{\boldsymbol{\chi}}{\boldsymbol{\mathcal{Y}}} \right) \mathrm{Re}\_{\boldsymbol{\chi}}^{\perp} \mathsf{C}\_{\circ \times} = \alpha \left( 1 + \frac{1}{\beta} \right) \mathsf{g}'(\mathbf{0}), \\\\ \mathrm{Re}\_{\boldsymbol{\chi}}^{\perp} \mathsf{N} \, \boldsymbol{\upmu}\_{\boldsymbol{\chi}} &= -\mathsf{G}'(\mathbf{0}), \text{and } \mathrm{Re}\_{\boldsymbol{\chi}}^{\perp} \mathsf{S} \, \boldsymbol{\upmu}\_{\boldsymbol{\chi}} = -\mathsf{g}'(\mathbf{0}), \end{aligned}$$

where Re*<sup>x</sup>* = *Uw* \_\_*x <sup>ν</sup>* and Re*<sup>y</sup>* = *Vw y*\_\_ *<sup>ν</sup>* depict the local Reynolds numbers.
