**4. Physical quantities**

In this study, the skin friction *τ*ð Þ*t* and Nusselt number *Nu* for the flow of Newtonian nanofluid in non-coaxal rotation are also analyzed. Their dimensional form is expressed as

$$
\pi(t) = -\mu\_{\eta f} \frac{\partial F}{\partial \mathbf{z}}\bigg|\_{\mathbf{z}=\mathbf{0}} \tag{22}
$$

$$Nu = -k\_{\eta^f} \frac{\partial T}{\partial \mathbf{z}}\Big|\_{\mathbf{z}=\mathbf{0}} \tag{23}$$

Incorporating Eqs. (22) and (23) with the nanofluid model Eq. (4), dimensionless variables Eq. (5) and solutions Eqs. (18) and (19), the following dimensionless skin friction and Nusselt number form as

$$\begin{split} \pi(\mathbf{t}) &= -\frac{\mathbf{1}}{\left(\mathbf{1} - \boldsymbol{\phi}\right)^{2.5}} \frac{\partial \mathbf{F}}{\partial \mathbf{z}}\bigg|\_{\mathbf{z}=0}, \\ &= -\frac{\mathbf{1}}{\left(\mathbf{1} - \boldsymbol{\phi}\right)^{2.5}} (\pi\_1(\mathbf{t}) - \pi\_2(\mathbf{t}) - \pi\_3(\mathbf{t}) + \pi\_4(\mathbf{t}) - \pi\_5(\mathbf{t}) + \pi\_6(\mathbf{t})), \end{split} \tag{24}$$
 
$$\mathbf{N}u = -\frac{k\_{\rm nf}}{k\_f} \frac{\partial T}{\partial \mathbf{z}}\bigg|\_{\mathbf{z}=0} = \lambda \sqrt{\frac{a\_1}{\pi t}}, \tag{25}$$

where

<sup>τ</sup>1ðÞ¼ <sup>t</sup> <sup>U</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 p erfc ffiffiffiffiffiffiffi d4t � � <sup>p</sup> � <sup>U</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 <sup>p</sup> � <sup>U</sup> 2 ffiffiffiffiffi ϕ1 πt r exp ð Þ �d4t , <sup>τ</sup>2ðÞ¼ <sup>t</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 p erfc ffiffiffiffiffiffiffi d4t � � <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffi ϕ1d4 <sup>p</sup> � ffiffiffiffiffi ϕ1 πt r exp ð Þ �d4t , τ3ðÞ¼ t a4 ffiffiffiffiffiffiffiffiffiffi ϕ1d4 p erfc ffiffiffiffiffiffiffi d4t � � <sup>p</sup> � a4 ffiffiffiffiffiffiffiffiffiffi ϕ1d4 <sup>p</sup> � a4 ffiffiffiffiffi ϕ1 πt r exp ð Þ �d4t , τ4ðÞ¼ t a4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ1ð Þ a3 þ d4 <sup>p</sup> exp að Þ 3t erfc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ a3 þ d4 t � � <sup>p</sup> � a4 ffiffiffiffiffi ϕ1 πt r exp ð Þ �d4t � a4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ1ð Þ a3 þ d4 <sup>p</sup> exp að Þ 3t , τ5ðÞ¼� t a4 ffiffiffiffi a1 πt r , *τ*6ðÞ¼ *t a*<sup>4</sup> ffiffiffiffiffiffiffiffiffi *a*1*a*<sup>3</sup> <sup>p</sup> *exp a*ð Þ <sup>3</sup>*<sup>t</sup> erfc* ffiffiffiffiffiffi *<sup>a</sup>*3*<sup>t</sup>* <sup>p</sup> ð Þ� *a*<sup>4</sup> ffiffiffiffiffiffiffiffiffi *a*1*a*<sup>3</sup> <sup>p</sup> *exp a*ð Þ� <sup>3</sup>*<sup>t</sup> <sup>a</sup>*<sup>4</sup> ffiffiffiffi *a*1 *πt* r , (26) with *<sup>τ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>τ</sup>* ffiffiffiffi *ν f* p *=μ <sup>f</sup>* Ω 3 2*ℓ*.

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*
