**2. Materials and methods**

studied flow and heat transfer in a viscous fluid over a nonlinear stretching sheet without using the impact of viscous dissipation. Cortell [6] examined heat and fluid flow transportation over a nonlinear stretching sheet for two different types of thermal boundary conditions, prescribed surface temperature (PST) and constant surface temperature (CST). The influence of heat transfer on the stagnation point flow of a third-order fluid over a shrinking surface has been studied by Nadeem et al. [7]. Recently, Prasad et al. [8] examined the mixed convection heat transfer aspects with variable fluid flow properties over a non-linear stretching surface. Fluid heating and cooling are important in many industries such as power, manufacturing and transportation. Effective cooling techniques are greatly needed for cooling any sort of high energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have thermal conductivities up to three times higher than these fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal of a metal. Since last two decades, study of nanofluid has urged the researcher's attention due to their heat transportation rate. Nanofluid comes in existence when we add a small quantity of nano-sized 10<sup>9</sup> <sup>10</sup><sup>7</sup> particles to the base fluids. Low heat transportation fluids like fluorocarbons, glycol, deionized water, etc. have badly thermal conductivity and therefore deliberated necessary for heat transfer coefficient surrounded by heat transfer medium and surface. The nanoparticles are typically made up of metals (*Al*,*Cu*), nitrides (*AlN*, *SiN*), carbides (*Sic*), oxides (*Al*2*O*3), or nonmetals (carbon nanotubes, Graphite, etc.) and the base fluid (conductive fluid) is usually water or ethylene glycol. Also, it has been experimentally proved that rate of heat conduction of nanofluids is more than rate of heat conduction of the base fluids. The concept of nanofluid was initially proposed by Choi and Eastman [9] to indicate engineered colloids composed of nanoparticles dispersed in a base fluid. An MIT based comprehensive survey has been done by Buongiorno [10] for convective transportation in nanofluids by considering seven slip conditions that may produce a relative velocity within the base fluid and nanoparticles. Only two (Brownian motion and thermophoresis) out of these seven slip mechanisms were found to be important mechanisms. By adopting Buogiorno's model, Kuznetsov and Nield [11] explored the nanofluid boundary

*Heat Transfer - Design, Experimentation and Applications*

In recent years, MHD fluid flow has gained researchers attention due to its controllable heat transfer rate. Magnetohydrodynamics (MHD) effect also play and influential role in controlling the rate of cooling as well as segregation of molten metal's from various non-metallic impurities. Magnetohydrodynamic (MHD) fluid flow has enormous utilization in manufacturing processes, even in the industrial areas as well. The terminology "Magnetohydrodynamic" is combination of three elementary terms magneto that stands for magnetic field, hydro that stands for fluid/liquid and dynamics that stands for evolution of particles. The existence of external magnetic field gives rise to Lorentz drag force which acts on the fluid, so potentially altering the characteristics of fluid flow especially velocity, temperature and concentration. Grouping of electromagnetism Maxwell's equation and fluid mechanics Navier's stokes equations therefore provides Magnetohydrodynamic (MHD) relation [12, 13]. Hayat et al. [14] studied the MHD fluid flow transportation over stretching surfaces. Later, the influence of viscous and Ohmic dissipation (i.e. joule heating) in nanofluid has been presented by Hussain et al. [15]. Vajravelu and Canon [16] studied the flow behavior of fluid towards a non-linear stretching sheet. Further, Matin et al. [17] analyzed the entropy effect in MHD nanofluid flow over stretching surface. Shawky et al. [18] studied the Williamson nanofluid flow in porous medium and he acknowledged that enhancement in non-Newtonian parameter escalates skin friction

layer uniform convecting fluid flow.

**214**

In present analysis, 2-D incompressible fluid flow in MHD nanofluid over linear stretching sheet has been considered. Linear behavior generates flow and sheet is stretched in both direction of *x* axis with stretching velocity *uw* ¼ *ax*, where *a* and *x* denotes a constant and stretching surface coordinate respectively. *Tw* <sup>¼</sup> *<sup>T</sup>*<sup>∞</sup> <sup>þ</sup> *<sup>T</sup>*0*xm* at *y* ¼ 0, where *T*<sup>0</sup> refers to the positive constant, *T*<sup>∞</sup> refers to the ambient temperature attained and m refers to the physical parameter known as surface temperature parameter. Also, by introducing *m* ¼ 0, we have a special case of constant surface temperature (CST). **Figure 1** represents the physical model of the current study. The continuity, momentum, energy and concentration equations of the incompressible nanofluid boundary layer flow are as follows [10]

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial u}{\partial \mathbf{y}} = \mathbf{0} \tag{1}$$

$$
\mu \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} = \nu \frac{\partial^2 u}{\partial \mathbf{y}^2} - \frac{\sigma B^2}{\rho} u \tag{2}
$$

**Figure 1.** *Physical model and coordinate system.*

*Heat Transfer - Design, Experimentation and Applications*

$$
\mu \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} = \pi \left[ D\_B \frac{\partial \mathbf{C}}{\partial \mathbf{y}} \frac{\partial T}{\partial \mathbf{y}} + (D\_T / T\_{\text{os}}) \left( \frac{\partial T}{\partial \mathbf{y}} \right)^2 \right] \tag{3}
$$

$$
\mu \frac{\partial \mathbf{C}}{\partial \mathbf{x}} + \nu \frac{\partial \mathbf{C}}{\partial \mathbf{y}} = D\_B \frac{\partial^2 \mathbf{C}}{\partial \mathbf{y}^2} + \frac{D\_T}{T\_\infty} \frac{\partial^2 T}{\partial \mathbf{y}^2} \tag{4}
$$

*C f <sup>x</sup>* <sup>¼</sup> *<sup>τ</sup><sup>w</sup> ρu*<sup>2</sup> *w*

*DOI: http://dx.doi.org/10.5772/intechopen.95387*

the stretching surface serially given as:

number *Sc* and magnetic parameter *M* on *f*

*<sup>δ</sup>* ð Þ 0 , local Nusselt number�*θ*<sup>0</sup>

*Nb Nt Ec Sc M f*<sup>00</sup>

0.1 0.2 0.4 1.5 1.0 �0.62674

0.2 0.4 1.5 1.0 �0.69140

1.4 1.7

0.4 1.5 1.0 �0.69140

1.5 1.0 �0.69140

1.0 1.2

*<sup>δ</sup>* ð Þ 0 *, local Nusselt number* �*θ*<sup>0</sup>

*crucial fluid parameters Nb*, *Nt*, *Ec*, *Sc and M with fixed entries of Pr* ¼ 5*:*0 *and m* ¼ 1*:*0*.*

**3. Results and discussion**

00

temperature parameter *m* as 1.0.

*<sup>δ</sup>* ð Þ 0 , *qw* ¼ �*k T*ð Þ *<sup>w</sup>* � *T*<sup>∞</sup>

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet*

*<sup>τ</sup><sup>w</sup>* <sup>¼</sup> *<sup>μ</sup>ax*<sup>1</sup> 2 ffiffiffi *a ν* r *f* 00

coefficient *f*

0.05 0.10 0.15

**Table 1.**

**217**

0.1 0.2

0.3 0.4

0.1 0.2 0.0

*Values of skin friction coefficient f*<sup>00</sup>

0.1 0.2 0.4 1.1

0.1 0.2

0.1 0.2 0.4 1.5 0.8

, *Nux* <sup>¼</sup> *xqw*

*k T*ð Þ *<sup>w</sup>* � *T*<sup>∞</sup>

where *τw*, *qw* and *qm* are wall shear stress, local heat flux and local mass flux at

ffiffiffi *a ν* r *θ*0

Present study finds numerical solution of differential Eqs. (8)–(10) subjected to the boundary conditions (11) and (12) that are computed using RKF method by applying shooting technique. The main reason behind to solve the present problem

0

demonstrate the impact of fluid parameters *Nb*, *N t*, *Ec*, *Sc* and *M* on skin friction

by taking fixed entries of fluid parameters Prandtl number, *Pr* as 5.0 and surface

**Figure 2** manifests variation in fluid velocity against magnetic parameter *M* ð Þ 0*:*8, 1*:*0, 1*:*2 . This figure shows that existence of magnetic parameter *M* resists the fluid particle to move freely and main reason behind the resistance is that magnetic parameter *M* produces Lorentz force and this magnetism behavior can be adopted for controlling the fluid movement. Thus, enhancement in the value of

magnetic parameter *M* causes the declination of velocity distribution

are to determine the impact of prominent fluid parameters namely Eckert number *Ec*, thermophoresis *Nt*, Brownian motion parameter *Nb*, Schmidt

*and Shx* <sup>¼</sup> *xqm*

*<sup>δ</sup>*ð Þ 0 *and qm* ¼ �*DB*ð Þ *Cw* � *C*<sup>∞</sup>

*<sup>δ</sup>*ð Þ 0 , *θδ*ð Þ 0 and *ϕδ*ð Þ 0 . **Table 1**

*<sup>δ</sup>* ð Þ **0** �*θ*<sup>0</sup>

�0.69140 �0.74178

�0.69140 �0.69140

�0.69140 �0.69140

�0.69140 �0.69140

�0.65763 �0.69140 �0.72329

1.0 �0.69140 �0.69140 �0.69140

*<sup>δ</sup>*ð Þ 0 and local Sherwood number�*ϕ*<sup>0</sup>

*<sup>δ</sup>*ðÞ � **0** *ϕ*<sup>0</sup>

0.72369 0.78127 0.82246

0.82355 0.78127 0.74045

0.78127 0.73298 0.68790

2.18627 1.83840 1.48829

0.81074 0.78800 0.76889

0.95086 0.78127 0.61783

*<sup>δ</sup>*ð Þ 0 *and local Sherwood number* �*ϕ*<sup>0</sup>

*<sup>δ</sup>*ð Þ **0**

�0.41288 �0.55822 �0.66610

�1.77998 �0.55822 �0.15292

�0.55822 �0.95811 �1.28267

�3.19235 �2.54049 �1.88419

�0.78101 �0.60993 �0.46155

�0.85564 �0.55882 �0.27004

*<sup>δ</sup>*ð Þ 0 *for*

*DB*ð Þ *Cw* � *C*<sup>∞</sup>

(14)

ffiffiffi *a ν* r *ϕ*0 *<sup>δ</sup>*ð Þ 0

(15)

*<sup>δ</sup>*ð Þ 0

Boundary conditions are given as:

$$\mu = \mu\_w, \upsilon = 0, T = T\_w, \mathcal{C} = \mathcal{C}\_w \text{ at } \mathcal{y} = \mathbf{0} \tag{5}$$

$$
\mu \to 0, T \to T\_{\approx}, C \to \mathcal{C}\_{\approx} \text{ as } \mathcal{Y} \to \infty \tag{6}
$$

Here horizontal and vertical velocities are represented by *u* and *v*, respectively. Also *<sup>ν</sup>* denotes kinematic viscosity, *<sup>ρ</sup>* is the density of fluid, *<sup>τ</sup>* <sup>¼</sup> ð Þ *<sup>ρ</sup><sup>c</sup> <sup>p</sup>* ð Þ *ρc <sup>f</sup>* defines a proportion of heat capacities, *DT* reflects thermophoretic diffusion coefficient, *B* is the magnetic field intensity, *DB* denotes Brownian diffusion coefficient, *σ* represents electrical conductivity

The fundamental Eqs. (1)-(4) with boundary conditions (5) and (6) are transformed using similarity variables

$$\begin{aligned} u &= a \text{x} f'\_{\delta}(\xi), v = -\sqrt{a\nu} f\_{\delta}(\xi) \\ \phi\_{\delta}(\xi) &= \frac{C - C\_{\infty}}{C\_{w} - C\_{\infty}}, \theta\_{\delta}(\xi) = \frac{T - T\_{\infty}}{T\_{w} - T\_{\infty}}, \xi = \mathcal{y} \sqrt{\frac{a}{\nu}} \end{aligned} \tag{7}$$

Inserting Eq. (7) into Eqs. (2)–(4), the governing Eqs. (1)–(4) takes the form

$$f\_{\delta}^{\prime\prime} + f\_{\delta} f\_{\delta}^{\prime} - \mathbf{M} f\_{\delta}^{\prime} - \left. f\_{\delta}^{\prime 2} \right| = \mathbf{0} \tag{8}$$

$$\frac{1}{Pr}\theta\_\delta'' + f\_\delta \theta\_\delta' - f\_\delta' \theta\_\delta + \text{Nb}\theta\_\delta' \phi\_\delta' + \text{Nt}\theta\_\delta^{\dagger^2} + \text{Ec}\, f\_\delta''^2 + \text{MSc}\, f\_\delta'^2 = \mathbf{0} \tag{9}$$

$$
\delta \phi''\_{\delta} + \frac{1}{2} \text{Scf}\_{\delta} \phi'\_{\delta} + \frac{\text{Nt}}{\text{Nb}} \theta''\_{\delta} = 0 \tag{10}
$$

The relevant boundary conditions are reduced to

$$f\_{\delta}(\xi) = 0, f\_{\delta}^{'}(\xi) = \mathbf{1}, \theta\_{\delta}(\xi) = \mathbf{1} \text{ and } \phi\_{\delta}(\xi) = \mathbf{1} \text{ at } \xi = \mathbf{0} \tag{11}$$

$$f'\_{\delta}(\xi) \to 0, \phi\_{\delta}(\xi) \to 0 \text{ and } \theta\_{\delta}(\xi) \to 0 \text{ as } \xi \to \infty \tag{12}$$

where prime denotes derivative with respect to *ξ* and the key crucial parameters are defined by:

$$\begin{aligned} M &= \frac{\sigma B^2}{a}, Nt = \frac{(\rho c)\_p D\_T (T\_w - T\_\infty)}{(\rho c)\_f \nu T\_\infty}, \text{Sc} = \frac{\nu}{D\_B}, Pr = \frac{\nu}{a}, \\\ Ec &= \frac{u\_w^2}{C\_p (T\_w - T\_\infty)} \text{ and } Nb = \frac{(\rho c)\_p D\_B (C\_w - C\_\infty)}{(\rho c)\_f \nu} \end{aligned} \tag{13}$$

Here *M* is the magnetic parameter, *Nt* is the thermophoresis parameter, *Sc* is the Schmidt number, *Pr* is Prandtl number, *Ec* is the Eckert number and *Nb* is the Brownian motion parameter. Also, the physical quantities of interest skin friction coefficient, local Nusselt number and local Sherwood number are respectively defined as:

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet DOI: http://dx.doi.org/10.5772/intechopen.95387*

$$\text{C}f\_{\text{x}} = \frac{\text{\tau}\_{w}}{\rho u\_{w}^{2}}, \text{Nu}\_{\text{x}} = \frac{\text{\chi}q\_{w}}{k(T\_{w} - T\_{\text{os}})} \text{ and } \text{Sh}\_{\text{x}} = \frac{\text{\chi}q\_{m}}{D\_{B}(\text{C}\_{w} - \text{C}\_{\text{os}})} \tag{14}$$

where *τw*, *qw* and *qm* are wall shear stress, local heat flux and local mass flux at the stretching surface serially given as:

$$\tau\_w = \mu a \mathbf{x}^\dagger \sqrt{\frac{a}{\nu}} f''\_\delta(\mathbf{0}),\\q\_w = -k(T\_w - T\_\infty) \sqrt{\frac{a}{\nu}} \theta'\_\delta(\mathbf{0}) \, and \; q\_m = -D\_B(\mathbf{C}\_w - \mathbf{C}\_\infty) \sqrt{\frac{a}{\nu}} \phi'\_\delta(\mathbf{0}) \tag{15}$$

#### **3. Results and discussion**

*u ∂T ∂x* þ *v ∂T*

Boundary conditions are given as:

sents electrical conductivity

1 *Pr <sup>θ</sup>*<sup>00</sup>

are defined by:

defined as:

**216**

transformed using similarity variables

*<sup>δ</sup>* þ *f <sup>δ</sup>θ*<sup>0</sup>

*f <sup>δ</sup>*ð Þ¼ *ξ* 0, *f*

*f* 0

*<sup>M</sup>* <sup>¼</sup> *<sup>σ</sup>B*<sup>2</sup>

*Ec* <sup>¼</sup> *<sup>u</sup>*<sup>2</sup>

*u* ¼ *ax f*<sup>0</sup>

*ϕδ*ð Þ¼ *<sup>ξ</sup> <sup>C</sup>* � *<sup>C</sup>*<sup>∞</sup>

*f* 000 *<sup>δ</sup>* þ *f <sup>δ</sup> f* 00 *<sup>δ</sup>* � *M f*<sup>0</sup>

*ϕ*00 *<sup>δ</sup>* þ 1 2 *Sc f <sup>δ</sup>ϕ*<sup>0</sup> *<sup>δ</sup>* þ *Nt Nb <sup>θ</sup>*<sup>00</sup>

The relevant boundary conditions are reduced to

0

*w Cp*ð Þ *Tw* � *T*<sup>∞</sup>

*<sup>δ</sup>* � *f* 0

*Cw* � *C*<sup>∞</sup>

*<sup>δ</sup>θδ* þ *Nbθ*<sup>0</sup>

*<sup>a</sup>* , *Nt* <sup>¼</sup> ð Þ *<sup>ρ</sup><sup>c</sup> <sup>p</sup>DT*ð Þ *Tw* � *<sup>T</sup>*<sup>∞</sup>

ð Þ *ρc <sup>f</sup> νT*<sup>∞</sup>

Schmidt number, *Pr* is Prandtl number, *Ec* is the Eckert number and *Nb* is the Brownian motion parameter. Also, the physical quantities of interest skin friction coefficient, local Nusselt number and local Sherwood number are respectively

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>τ</sup> DB*

Also *<sup>ν</sup>* denotes kinematic viscosity, *<sup>ρ</sup>* is the density of fluid, *<sup>τ</sup>* <sup>¼</sup> ð Þ *<sup>ρ</sup><sup>c</sup> <sup>p</sup>*

*u ∂C ∂x* þ *v ∂C <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *DB*

*Heat Transfer - Design, Experimentation and Applications*

*∂C ∂y ∂T ∂y*

> *∂*2 *C ∂y*<sup>2</sup> þ

Here horizontal and vertical velocities are represented by *u* and *v*, respectively.

proportion of heat capacities, *DT* reflects thermophoretic diffusion coefficient, *B* is the magnetic field intensity, *DB* denotes Brownian diffusion coefficient, *σ* repre-

The fundamental Eqs. (1)-(4) with boundary conditions (5) and (6) are

*<sup>δ</sup>*ð Þ*<sup>ξ</sup>* , *<sup>v</sup>* ¼ � ffiffiffiffiffi

*δϕ*0

*<sup>a</sup><sup>ν</sup>* <sup>p</sup> *<sup>f</sup> <sup>δ</sup>*ð Þ*<sup>ξ</sup>*

*<sup>δ</sup>* � *f* 0 *δ*

*Tw* � *T*<sup>∞</sup>

, *ξ* ¼ *y*

*<sup>δ</sup>*ð Þ¼ *ξ* 1, *θδ*ð Þ¼ *ξ* 1 *and ϕδ*ð Þ¼ *ξ* 1 *at ξ* ¼ 0 (11)

, *Sc* <sup>¼</sup> *<sup>ν</sup> DB*

ð Þ *ρc <sup>f</sup> ν*

*and Nb* <sup>¼</sup> ð Þ *<sup>ρ</sup><sup>c</sup> <sup>p</sup>DB*ð Þ *Cw* � *<sup>C</sup>*<sup>∞</sup>

*<sup>δ</sup>*ð Þ!*ξ* 0, *ϕδ*ð Þ!*ξ* 0 *and θδ*ð Þ!*ξ* 0 *as ξ* ! ∞ (12)

ffiffiffi *a ν*

<sup>2</sup> <sup>¼</sup> <sup>0</sup> (8)

*δ*

*<sup>δ</sup>* ¼ 0 (10)

, *Pr* <sup>¼</sup> *<sup>ν</sup> α* ,

<sup>2</sup> <sup>þ</sup> *MSc f*<sup>0</sup>

, *θδ*ð Þ¼ *<sup>ξ</sup> <sup>T</sup>* � *<sup>T</sup>*<sup>∞</sup>

Inserting Eq. (7) into Eqs. (2)–(4), the governing Eqs. (1)–(4) takes the form

*<sup>δ</sup>* þ *Ntθ*<sup>0</sup> 2 *<sup>δ</sup>* þ *Ec f*<sup>00</sup> *δ*

where prime denotes derivative with respect to *ξ* and the key crucial parameters

Here *M* is the magnetic parameter, *Nt* is the thermophoresis parameter, *Sc* is the

þ ð Þ *DT=T*<sup>∞</sup>

*DT T*<sup>∞</sup>

*u* ¼ *uw*, *v* ¼ 0, *T* ¼ *Tw*,*C* ¼ *Cw at y* ¼ 0 (5) *u* ! 0, *T* ! *T*∞,*C* ! *C*<sup>∞</sup> *as y* ! ∞ (6)

*∂*2 *T*

� �<sup>2</sup> " #

*∂T ∂y*

*<sup>∂</sup>y*<sup>2</sup> (4)

ð Þ *ρc <sup>f</sup>*

defines a

r (7)

<sup>2</sup> <sup>¼</sup> 0 (9)

(13)

(3)

Present study finds numerical solution of differential Eqs. (8)–(10) subjected to the boundary conditions (11) and (12) that are computed using RKF method by applying shooting technique. The main reason behind to solve the present problem are to determine the impact of prominent fluid parameters namely Eckert number *Ec*, thermophoresis *Nt*, Brownian motion parameter *Nb*, Schmidt number *Sc* and magnetic parameter *M* on *f* 0 *<sup>δ</sup>*ð Þ 0 , *θδ*ð Þ 0 and *ϕδ*ð Þ 0 . **Table 1** demonstrate the impact of fluid parameters *Nb*, *N t*, *Ec*, *Sc* and *M* on skin friction coefficient *f* 00 *<sup>δ</sup>* ð Þ 0 , local Nusselt number�*θ*<sup>0</sup> *<sup>δ</sup>*ð Þ 0 and local Sherwood number�*ϕ*<sup>0</sup> *<sup>δ</sup>*ð Þ 0 by taking fixed entries of fluid parameters Prandtl number, *Pr* as 5.0 and surface temperature parameter *m* as 1.0.

**Figure 2** manifests variation in fluid velocity against magnetic parameter *M* ð Þ 0*:*8, 1*:*0, 1*:*2 . This figure shows that existence of magnetic parameter *M* resists the fluid particle to move freely and main reason behind the resistance is that magnetic parameter *M* produces Lorentz force and this magnetism behavior can be adopted for controlling the fluid movement. Thus, enhancement in the value of magnetic parameter *M* causes the declination of velocity distribution


#### **Table 1.**

*Values of skin friction coefficient f*<sup>00</sup> *<sup>δ</sup>* ð Þ 0 *, local Nusselt number* �*θ*<sup>0</sup> *<sup>δ</sup>*ð Þ 0 *and local Sherwood number* �*ϕ*<sup>0</sup> *<sup>δ</sup>*ð Þ 0 *for crucial fluid parameters Nb*, *Nt*, *Ec*, *Sc and M with fixed entries of Pr* ¼ 5*:*0 *and m* ¼ 1*:*0*.*

**Figure 2.** *Impact of magnetic parameter M on velocity profile f*<sup>0</sup> *<sup>δ</sup>*ð Þ*ξ :*

conduction of nanoparticles. Thus, width of boundary layer enhances due to reallocation of ultrafine particles from hotter to colder part and hence, temperature enhances for higher thermophoresis parameter *Nt* that can be seen in **Figure 4**. **Figure 5** demonstrate fluid temperature variation against Eckert number *Ec* ð Þ 0*:*0, 0*:*1, 0*:*2 *:* A dimensionless quantity *Ec* is the fraction of advective transportation and heat dissipation potential. As Eckert number *Ec* enhances, thermal buoyancy effect raises that results in increasing temperature and that is the main reason behind the conversion of kinetic energy into thermal energy. Hence, fluid temperature enhances because of this conversion effect. Consequently, declination

in Nusselt number *Nux* is noticed that can be seen via **Table 1**.

*Impact of thermophoresis parameter Nt on temperature profile θδ*ð Þ*ξ .*

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet*

*DOI: http://dx.doi.org/10.5772/intechopen.95387*

**Figure 4.**

**Figure 5.**

**219**

*Impact of Eckert number Ec on temperature profile θδ*ð Þ*ξ .*

**Figure 3.** *Impact of Brownian motion parameter Nb on temperature profile θδ*ð Þ*ξ .*

**Figure 3** examines temperature distribution variation against the fluid parameter Brownian motion parameter *Nb* ð Þ 0*:*05, 0*:*10, 0*:*15 . The striking of atoms or molecules of the fluid particles with each other will create an arbitrary motion called Brownian motion of suspended (pendulous) particles and that will enhances width of boundary layer. Hence, fluid temperature increases for higher Brownian motion parameter *Nb* and in consequence local Nusselt number decreases.

**Figure 4** deliberates the impact of fluid temperature under the consequence of thermophoresis parameter *Nt* ð Þ 0*:*2, 0*:*3, 0*:*4 . Temperature gradient falls down for higher values of thermophoresis parameter *Nt* that result in reduction of

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet DOI: http://dx.doi.org/10.5772/intechopen.95387*

**Figure 4.** *Impact of thermophoresis parameter Nt on temperature profile θδ*ð Þ*ξ .*

conduction of nanoparticles. Thus, width of boundary layer enhances due to reallocation of ultrafine particles from hotter to colder part and hence, temperature enhances for higher thermophoresis parameter *Nt* that can be seen in **Figure 4**.

**Figure 5** demonstrate fluid temperature variation against Eckert number *Ec* ð Þ 0*:*0, 0*:*1, 0*:*2 *:* A dimensionless quantity *Ec* is the fraction of advective transportation and heat dissipation potential. As Eckert number *Ec* enhances, thermal buoyancy effect raises that results in increasing temperature and that is the main reason behind the conversion of kinetic energy into thermal energy. Hence, fluid temperature enhances because of this conversion effect. Consequently, declination in Nusselt number *Nux* is noticed that can be seen via **Table 1**.

**Figure 5.** *Impact of Eckert number Ec on temperature profile θδ*ð Þ*ξ .*

**Figure 3** examines temperature distribution variation against the fluid parame-

*<sup>δ</sup>*ð Þ*ξ :*

**Figure 4** deliberates the impact of fluid temperature under the consequence of thermophoresis parameter *Nt* ð Þ 0*:*2, 0*:*3, 0*:*4 . Temperature gradient falls down for

ter Brownian motion parameter *Nb* ð Þ 0*:*05, 0*:*10, 0*:*15 . The striking of atoms or molecules of the fluid particles with each other will create an arbitrary motion called Brownian motion of suspended (pendulous) particles and that will enhances width of boundary layer. Hence, fluid temperature increases for higher Brownian motion

parameter *Nb* and in consequence local Nusselt number decreases.

*Impact of Brownian motion parameter Nb on temperature profile θδ*ð Þ*ξ .*

**Figure 2.**

**Figure 3.**

**218**

*Impact of magnetic parameter M on velocity profile f*<sup>0</sup>

*Heat Transfer - Design, Experimentation and Applications*

higher values of thermophoresis parameter *Nt* that result in reduction of

**Figure 6.** *Impact of magnetic parameter M on temperature profile θδ*ð Þ*ξ .*

**Figure 6** reflects variation of temperature distribution against magnetic parameter *M*. With an increases in magnetic parameter *M*, velocity profile decreases because of generation of Lorentz force that consequently intensify the boundary thickness and rate of heat transportation and hence fluid temperature enhances as shown via **Figure 6**.

**Figure 8** portraits variation for nanoparticle volume fraction *ϕδ*ð Þ*ξ* against thermophoresis parameter *Nt* ð Þ 0*:*2, 0*:*3, 0*:*4 . This graph shows that with an increase in thermophoresis parameter, nanoparticle concentration increases. Basically, in case of thermophoresis force applied by a particle on the other particle will generates the movement of particles from hotter to colder part and hence fluid moves from hotter to colder region and hence intensification in the nanoparticle volume

*Impact of thermophoresis parameter Nt on Concentration profile ϕδ*ð Þ*ξ .*

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet*

*DOI: http://dx.doi.org/10.5772/intechopen.95387*

**Figure 9** portraits the impact of Schmidt number *Sc* ð Þ 1*:*1, 1*:*4, 1*:*7 on profile of nanoparticle concentration. Intensification in the value of physical parameter Sc,

fraction is observed via **Figure 8**.

*Impact of Schmidt number Sc on Concentration profile ϕδ*ð Þ*ξ .*

**Figure 8.**

**Figure 9.**

**221**

**Figure 7** manifests the impact of Brownian motion parameter *Nb* ð Þ 0*:*05, 0*:*10, 0*:*15 on nanoparticle concentration *ϕδ*ð Þ*ξ :* With an increase in the value of Brownian motion parameter *Nb*, fluid particles collides with each other with higher speed which results in increase in the nanoparticle concentration and consequently, local Sherwood number reduces as depicted in the **Table 1**.

**Figure 7.** *Impact of Brownian motion parameter Nb on Concentration profile ϕδ*ð Þ*ξ .*

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet DOI: http://dx.doi.org/10.5772/intechopen.95387*

**Figure 8.** *Impact of thermophoresis parameter Nt on Concentration profile ϕδ*ð Þ*ξ .*

**Figure 8** portraits variation for nanoparticle volume fraction *ϕδ*ð Þ*ξ* against thermophoresis parameter *Nt* ð Þ 0*:*2, 0*:*3, 0*:*4 . This graph shows that with an increase in thermophoresis parameter, nanoparticle concentration increases. Basically, in case of thermophoresis force applied by a particle on the other particle will generates the movement of particles from hotter to colder part and hence fluid moves from hotter to colder region and hence intensification in the nanoparticle volume fraction is observed via **Figure 8**.

**Figure 9** portraits the impact of Schmidt number *Sc* ð Þ 1*:*1, 1*:*4, 1*:*7 on profile of nanoparticle concentration. Intensification in the value of physical parameter Sc,

**Figure 9.** *Impact of Schmidt number Sc on Concentration profile ϕδ*ð Þ*ξ .*

**Figure 6** reflects variation of temperature distribution against magnetic param-

eter *M*. With an increases in magnetic parameter *M*, velocity profile decreases because of generation of Lorentz force that consequently intensify the boundary thickness and rate of heat transportation and hence fluid temperature enhances as

*Nb* ð Þ 0*:*05, 0*:*10, 0*:*15 on nanoparticle concentration *ϕδ*ð Þ*ξ :* With an increase in the value of Brownian motion parameter *Nb*, fluid particles collides with each other with higher speed which results in increase in the nanoparticle concentration and

**Figure 7** manifests the impact of Brownian motion parameter

*Impact of Brownian motion parameter Nb on Concentration profile ϕδ*ð Þ*ξ .*

*Impact of magnetic parameter M on temperature profile θδ*ð Þ*ξ .*

*Heat Transfer - Design, Experimentation and Applications*

consequently, local Sherwood number reduces as depicted in the **Table 1**.

shown via **Figure 6**.

**Figure 6.**

**Figure 7.**

**220**

*m* Surface temperature parameter

*Heat Transfer in a MHD Nanofluid Over a Stretching Sheet*

*Cw* Nanoparticle volume fraction

*Nt* Thermophoresis parameter *T*<sup>∞</sup> Ambient temperature attained

*DB* Brownian diffusion coefficient *Nb* Brownian motion parameter

*Tw* Temperature at the sheet

*u* Horizontal velocity *M* Magnetic parameter

*ν* Kinematic viscosity *β* Casson fluid parameter *σ* Electrical conductivity *ξ* Similarity variable *α<sup>m</sup>* Thermal diffusivity *τ* Ratio of heat capacities *θδ* Non-dimensional temperature

Kishanlal Public College, Rewari, Haryana, India

provided the original work is properly cited.

\*Address all correspondence to: vikaspoply@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*DT* Thermophoresis diffusion coefficient

*C*<sup>∞</sup> Ambient nanoparticle volume fraction

*ϕδ* Non-dimensional nanoparticle concentration

*Shx* Sheerwood number *v* Vertical velocity

*T* Temperature *uw* Stretching velocity

*qw* Hass flux

**Greek symbols**

**Author details**

Vikas Poply

**223**

*Sc* Schmidt number *C* Concentration

*DOI: http://dx.doi.org/10.5772/intechopen.95387*

*qm* Mass flux *Pr* Prandtl number *Nux* Nusselt number

**Figure 10.** *Impact of magnetic parameter M on Concentration profile ϕδ*ð Þ*ξ .*

declination in mass diffusivity is observed. Due to this effect nanoparticle concentration decreases.

**Figure 10** reflects the variation for nanoparticle concentration *ϕδ*ð Þ*ξ* against the magnetic parameter *M* ð Þ 0*:*8, 1*:*0, 1*:*2 . With increase in magnetic parameter *M*, rate of mass transportation decreases that consequently increase nanoparticle concentration and hence reduction in the value of local Sherwood number is notice as seen in **Table 1**.
