Ordinary Differential Equations of Boundary Layer of Nanofluid’s Natural Convection

Abstract

The partial differential equations of natural convection boundary layer with conventional fluid’s flow are equivalently transformed to the related ordinary differential equations. A feasibility is demonstrated to describe the mass, momentum and energy conservation of nanofluid’s natural convection by using those of nanofluid’s natural convection. By an innovative similarity transformation, the partial differential equations of nanofluid’s natural convection boundary layer are equivalently transformed to the related ordinary differential equations. On this basis, the two-dimensional ordinary differential equations of nanofluid’s natural convection are determined for extensive exploration of heat transfer of nanofluid’s natural convection.

Appendix: Similarity Transformation of (4.12)–(4.14)

4.1.1 Similarity Transformation for (4.1)

Equation (4.12) is initially changed into

$$\rho_{pf} \left( {\frac{{\partial w_{x} }}{\partial x} + \frac{{\partial w_{y} }}{\partial y}} \right) + w_{x} \frac{{\partial \rho_{pf} }}{\partial x} + w_{y} \frac{{\partial \rho_{pf} }}{\partial y} = 0$$

(a)

With (4.15) we have

$$\frac{{\partial w_{x} }}{\partial x} = \left[ {2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]\frac{{dW_{x} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial x} + \frac{1}{2}x^{{ - \frac{1}{2}}} \left[ {2\sqrt {g\,\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]W_{x}$$

Where

$$\begin{aligned} \frac{{\partial \eta_{pf} }}{\partial x} & = \frac{\partial }{\partial x}\left[ {\frac{y}{x}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} } \right] = \frac{\partial }{\partial x}\left[ {y\left( {\frac{1}{4}\frac{{g\cos \alpha \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|x^{ - 1} }}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/4} } \right] \\ & = - \frac{1}{4}\left[ {y\left( {\frac{1}{4}\frac{{g\cos \alpha \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|x^{3} }}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/4} } \right]x^{ - 2} = - \frac{1}{4}x^{ - 1} \eta_{pf} \\ \end{aligned}$$

(b)

Then,

$$\begin{aligned} \frac{{\partial w_{x} }}{\partial x} & = \left[ {2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]\frac{{dW_{x} }}{{d\eta_{pf} }}\left( { - \frac{1}{4}x^{ - 1} \eta_{pf} } \right) \\ & \quad + \frac{1}{2}x^{{ - \frac{1}{2}}} \left[ {2\sqrt {g\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]W_{x} \\ & = - \frac{1}{2}\left[ {\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }} + \left[ {\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} } \right]W_{x} \\ & = \sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {W_{x} - \frac{1}{2}\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) \\ \end{aligned}$$

(c)

With (4.7) we have

$$\begin{aligned} \frac{{\partial w_{y} }}{\partial y} & = \left[ {2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{ - 1/4} } \right]\frac{{dW_{y} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial y} \\ & = \left[ {2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{ - 1/4} } \right]\frac{{dW_{y} }}{{d\eta_{pf} }}\frac{1}{x}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} \\ & = 2\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{dW_{y} }}{{d\eta_{pf} }} \\ \end{aligned}$$

(d)

While,

$$\frac{{\partial \rho_{pf} }}{\partial x} = \frac{{d\rho_{pf} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial x} = - \frac{1}{4}x^{ - 1} \eta_{pf} \frac{{d\rho_{pf} }}{{d\eta_{pf} }}$$

(e)

$$\frac{{\partial \rho_{pf} }}{\partial y} = \frac{{d\rho_{pf} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial y} = \frac{1}{x}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} \frac{{d\rho_{pf} }}{{d\eta_{pf} }}$$

(f)

By using (a)–(f), Eg. (4.12) can be changed to

$$\begin{aligned} & \rho_{pf} \left[ {\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {W_{x} - \frac{1}{2}\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) + 2\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{dW_{y} }}{{d\eta_{pf} }}} \right] \\ & \quad + 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} W_{x} \left( { - \frac{1}{4}x^{ - 1} \eta_{pf} \frac{{d\rho_{pf} }}{{d\eta_{pf} }}} \right) \\ & \quad + 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{{ - \frac{1}{4}}} W_{y} \frac{1}{x}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} \frac{{d\rho_{pf} }}{{d\eta_{pf} }} = 0 \\ \end{aligned}$$

The above equation is divided by $\left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \sqrt {\frac{g \cdot \,\cos \alpha }{x}}$ and transformed into

$$\rho_{pf} \left[ {\left( {W_{x} - \frac{1}{2}\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) + 2\frac{{dW_{y} }}{{d\eta_{pf} }}} \right] + 2W_{x} \left( { - \frac{1}{4}\eta_{pf} \frac{{d\rho_{pf} }}{{d\eta_{pf} }}} \right) + 2W_{y} \frac{{d\rho_{pf} }}{{d\eta_{pf} }} = 0$$

or

$$2W_{x} - \eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }} + 4\frac{{dW_{y} }}{{d\eta_{pf} }} - \frac{1}{{\rho_{pf} }}\frac{{d\rho_{pf} }}{{d\eta_{pf} }}\left( {\eta_{pf} W_{x} - 4W_{y} } \right) = 0$$

(4.23)

4.1.2 Similarity Transformation for (4.13)

Equation (4.13) can be rewritten as

$$\rho_{pf} \left( {w_{x} \frac{{\partial w_{x} }}{\partial x} + w_{y} \frac{{\partial w_{x} }}{\partial y}} \right) = \mu_{pf} \frac{{\partial^{2} w_{x} }}{{\partial y^{2} }} + \frac{{\partial w_{x} }}{\partial y}\frac{{\partial \mu_{pf} }}{\partial y} + g \cdot \cos \alpha \left| {\rho_{\infty ,pf} - \rho_{pf} } \right|$$

(g)

Where

$$\frac{{\partial w_{x} }}{\partial y} = 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{dW_{x} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial y}$$

While,

$$\frac{{\partial \eta_{pf} }}{\partial y} = x^{ - 1} \left( {\frac{1}{4}Gr_{x,\infty,pf} } \right)^{1/4}$$

Then,

$$\frac{{\partial w_{x} }}{\partial y} = 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{dW_{x} }}{{d\eta_{pf} }}x^{ - 1} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4}$$

(h)

$$\begin{aligned} \frac{{\partial^{2} w_{x} }}{{\partial y^{2} }} & = 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }}x^{ - 1} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} \frac{{\partial \eta_{pf} }}{\partial y} \\ & = 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }}x^{ - 1} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} \\ & = 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }}x^{ - 2} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/2} \\ \end{aligned}$$

(i)

$$\frac{{\partial \mu_{pf} }}{\partial y} = \frac{{d\mu_{pf} }}{{d\eta_{pf} }}\frac{{\partial \eta_{pf} }}{\partial y} = \frac{{d\mu_{pf} }}{{d\eta_{pf} }}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1}$$

(j)

Using the above related equations, (g) becomes

$$\begin{aligned} & \rho _{{pf}} \left[ {2\sqrt {gx\cos \alpha } \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} W_{x} \sqrt {\frac{{g\,\cos \alpha }}{x}} \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} \left( {W_{x} - \frac{1}{2}\eta _{{pf}} \frac{{dW_{x} }}{{d\eta _{{pf}} }}} \right)} \right. \\ & \quad + 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} \left( {\frac{1}{4}Gr_{{x,\infty ,pf}} } \right)^{{ - \frac{1}{4}}} W_{y} 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} \\ & \left. {\quad \times \frac{{dW_{x} }}{{d\eta _{{pf}} }}x^{{ - 1}} \left( {\frac{1}{4}Gr_{{x,\infty ,pf}} } \right)^{{1/4}} } \right] \\ & = 2\mu _{{pf}} \sqrt {gx\cos \alpha } \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} \frac{{d^{2} W_{x} }}{{d\eta _{{pf}}^{2} }}x^{{ - 2}} \left( {\frac{1}{4}Gr_{{x,\infty ,pf}} } \right)^{{1/2}} \\ & \quad + 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho _{{\infty ,pf}} }}{{\rho _{{w,pf}} }} - 1} \right|^{{1/2}} \frac{{dW_{x} }}{{d\eta _{{pf}} }}x^{{ - 1}} \left( {\frac{1}{4}Gr_{{x,\infty ,pf}} } \right)^{{1/4}} \frac{{d\mu _{{pf}} }}{{d\eta _{{pf}} }}\left( {\frac{1}{4}Gr_{{x,\infty ,pf}} } \right)^{{1/4}} x^{{ - 1}} \\ & \quad + g \cdot \cos \alpha \left| {\rho _{{\infty ,pf}} - \rho _{{pf}} } \right| \\ \end{aligned}$$

With consideration of the definition of local Grashof number $Gr_{x,\infty ,pf}$, the above equation is divided by $\rho_{pf} g\cos \alpha \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|$, and becomes the following one:

$$\begin{aligned} & 2W_{x} \left( {W_{x} - \frac{1}{2}\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) + 2W_{y} \left( {2\frac{{dW_{x} }}{{d\eta_{pf} }}} \right) = 2\nu_{pf} \frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }}\left( {\frac{1}{4}\frac{1}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} \\ & \quad + 2\frac{1}{{\rho_{pf} }}\frac{{dW_{x} }}{{d\eta_{pf} }}\frac{{d\mu_{pf} }}{{d\eta_{pf} }}\left( {\frac{1}{4}\frac{1}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} + \frac{{\frac{{\rho_{\infty ,pf} }}{{\rho_{pf} }} - 1}}{{\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1}} \\ \end{aligned}$$

or

$$\begin{aligned} & 2W_{x} \left( {W_{x} - \frac{1}{2}\eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) + 4W_{y} \frac{{dW_{x} }}{{d\eta_{pf} }} = \frac{{\nu_{pf} }}{{\nu_{\infty ,pf} }}\frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }} \\ & \quad + \frac{1}{{\rho_{pf} }}\frac{{dW_{x} }}{{d\eta_{pf} }}\frac{{d\mu_{pf} }}{{d\eta_{pf} }}\frac{1}{{\nu_{\infty ,pf}^{{}} }} + \frac{{\frac{{\rho_{\infty ,pf} }}{{\rho_{pf} }} - 1}}{{\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1}} \\ \end{aligned}$$

The above equation is divided by $\frac{{\nu_{pf} }}{{\nu_{\infty ,pf} }}$ and simplified to

$$\frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\left( {W_{x} \left( {2W_{x} - \eta_{pf} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) + 4W_{y} \frac{{dW_{x} }}{{d\eta_{pf} }}} \right) = \frac{{d^{2} W_{x} }}{{d\eta_{pf}^{2} }} + \frac{1}{{\mu_{pf} }}\frac{{d\mu_{pf} }}{{d\eta_{pf} }}\frac{{dW_{x} }}{{d\eta_{pf} }} + \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\frac{{\frac{{\rho_{\infty ,pf} }}{{\rho_{pf} }} - 1}}{{\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1}}$$

(4.24)

4.1.3 Similarity Transformation for (4.14)

Equation (4.14) is rewritten as

$$\rho_{pf} c_{{p_{pf} }} \left( {w_{x} \frac{\partial T}{\partial x} + w_{y} \frac{\partial T}{\partial y}} \right) + \rho_{pf} T\left( {w_{x} \frac{{\partial c_{{p_{pf} }} }}{\partial x} + w_{y} \frac{{\partial c_{{p_{pf} }} }}{\partial y}} \right) = \frac{{\partial \lambda_{pf} }}{\partial y}\frac{\partial T}{\partial y} + \lambda_{pf} \frac{{\partial^{2} T}}{{\partial y^{2} }}$$

or

$$\begin{aligned} & \left( {\rho_{pf} c_{{p_{pf} }} \frac{\partial T}{\partial x} + \rho_{pf} T\frac{{\partial c_{{p_{pf} }} }}{\partial x}} \right)w_{x} + \left( {\rho_{pf} c_{{p_{pf} }} \frac{\partial T}{\partial y} + \rho_{pf} T\frac{{\partial c_{{p_{pf} }} }}{\partial y}} \right)w_{y} \\ & = \frac{{\partial \lambda_{pf} }}{\partial y}\frac{\partial T}{\partial y} + \lambda_{pf} \frac{{\partial^{2} T}}{{\partial y^{2} }} \\ \end{aligned}$$

(k)

Where

$$\frac{\partial T}{\partial x} = - \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\frac{1}{4}\eta_{pf} x^{ - 1}$$

$$\frac{\partial T}{\partial y} = \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1}$$

$$\frac{{\partial^{2} T}}{{\partial y^{2} }} = (T_{w} - T_{\infty } )\frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}\left({\frac{1}{4}Gr_{x,\infty ,pf}}\right )^{1/2} x^{ - 2}$$

Consulting (e) and (f), we have

$$\frac{{\partial \lambda_{pf} }}{\partial y} = \left({\frac{1}{4}Gr_{x,\infty ,pf}} \right)^{1/4} x^{ - 1} \frac{{d\lambda_{pf} }}{{d\eta_{pf} }}$$

$$\frac{{\partial c_{p_{pf}} }}{\partial x} = - \frac{1}{4}x^{ - 1} \eta_{pf} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}$$

$$\frac{{\partial c_{p_{pf}} }}{\partial y} = (\frac{1}{4}Gr_{x,\infty ,pf} )^{1/4} x^{ - 1} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}$$

Then, (k) is changed to

$$\begin{aligned} & \left[ {\rho_{pf} c_{{p_{pf} }} \left( { - \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\frac{1}{4}\eta_{pf} x^{ - 1} } \right) + \rho_{pf} T\left( { - \frac{1}{4}x^{ - 1} \eta_{pf} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right]2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} W_{x} \\ & \quad \quad + \left[ {\rho_{pf} c_{{p_{pf} }} \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1} } \right. \\ & \left. {\quad \quad + \rho_{pf} T\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{{ - \frac{1}{4}}} W_{y} \\ & \quad = \left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1} \frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/4} x^{ - 1} \\ & \quad \quad + \lambda_{pf} \left( {T_{w} - T_{\infty } } \right)\frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}\left( {\frac{1}{4}Gr_{x,\infty ,pf} } \right)^{1/2} x^{ - 2} \\ \end{aligned}$$

i.e.

$$\begin{aligned} & \left[ {\rho_{pf} c_{{p_{pf} }} \left( { - \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}\frac{1}{4}\eta_{pf} x^{ - 1} } \right) + \rho_{pf} T\left( { - \frac{1}{4}x^{ - 1} \eta_{pf} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right] \\ & \times 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} W_{x} + \left[ {\rho_{pf} c_{{p_{pf} }} \left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }}x^{ - 1} + \rho_{pf} Tx^{ - 1} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right] \\ & \times 2\sqrt {gx\cos \alpha } \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} W_{y} = \left( {\frac{1}{4}\frac{{g\cos \alpha \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|x^{3} }}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} x^{ - 2} \frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\left( {T_{w} - T_{\infty } } \right)\frac{d\theta }{{d\eta_{pf} }} \\ & \quad \quad + \lambda_{pf} \left( {T_{w} - T_{\infty } } \right)\frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}\left( {\frac{1}{4}\frac{{g\cos \alpha \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|x^{3} }}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} x^{ - 2} \\ \end{aligned}$$

The above equation is divided by $\sqrt {\frac{g\cos \alpha }{x}} \left| {\frac{{\rho_{\infty ,pf} }}{{\rho_{w,pf} }} - 1} \right|^{1/2} \left( {T_{w} - T_{\infty } } \right)$ and this leads to

$$\begin{aligned} & \left[ {\rho_{pf} c_{{p_{pf} }} \left( { - \frac{d\theta }{{d\eta_{pf} }}\frac{1}{4}\eta_{pf} } \right) + \rho_{pf} \frac{T}{{T_{w} - T_{\infty } }}\left( { - \frac{1}{4}\eta_{pf} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right]2W_{x} \\ & \quad \quad + \left[ {\rho_{pf} c_{{p_{pf} }} \frac{d\theta }{{d\eta_{pf} }} + \rho_{pf} \frac{T}{{T_{w} - T_{\infty } }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]2W_{y} \\ & \quad = \left( {\frac{1}{4}\frac{1}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} \frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \lambda_{pf} \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}\left( {\frac{1}{4}\frac{1}{{\nu_{\infty ,pf}^{2} }}} \right)^{1/2} \\ \end{aligned}$$

The above equation is multiplied by $\frac{{2\nu_{\infty ,pf} }}{{\lambda_{pf} }}$ and is simplified, then

$$\begin{aligned} & \frac{{\rho_{pf} \nu_{\infty ,pf} }}{{\lambda_{pf} }}\left[ {c_{p_{pf}} \left( { - \frac{d\theta }{{d\eta_{pf} }}\eta_{pf} } \right) + \frac{T}{{T_{w} - T_{\infty } }}\left( { - \eta_{pf} \frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right]W_{x} \\ & \quad \quad + \frac{{\rho_{pf} \nu_{\infty ,pf} }}{{\lambda_{pf} }}\left[ {c_{{p_{pf} }} \frac{d\theta }{{d\eta_{pf} }} + \frac{T}{{T_{w} - T_{\infty } }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]4W_{y} \\ & \quad = \frac{1}{{\lambda_{pf} }}\frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }} \\ \end{aligned}$$

i.e.

$$\begin{aligned} & \frac{{\rho_{pf} \nu_{\infty ,pf} c_{p_{pf}} }}{{\lambda_{pf} }}\left[ {\left( { - \frac{d\theta }{{d\eta_{pf} }}\eta_{pf} } \right) + \frac{T}{{T_{w} - T_{\infty } }}\left( { - \eta_{pf} \frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right]W_{x} \\ & \quad \quad + \frac{{\rho_{pf} \nu_{\infty ,pf} c_{p_{pf}} }}{{\lambda_{pf} }}\left[ {\frac{d\theta }{{d\eta_{pf} }} + \frac{T}{{T_{w} - T_{\infty } }}\frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]4W_{y} \\ & \quad = \frac{1}{{\lambda_{pf} }}\frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }} \\ \end{aligned}$$

Since $\frac{{\rho_{pf} \nu_{\infty ,pf} c_{p_{pf}} }}{{\lambda_{pf} }} = Pr_{pf} \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}$, the above equation can be simplified to

$$\begin{aligned} & Pr_{pf} \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\left[ {\left( { - \frac{d\theta }{{d\eta_{pf} }}\eta_{pf} } \right) + \frac{T}{{T_{w} - T_{\infty } }}\left( { - \eta_{pf} \frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right)} \right]W_{x} \\ & + Pr_{pf} \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\left[ {\frac{d\theta }{{d\eta_{pf} }} + \frac{T}{{T_{w} - T_{\infty } }}\frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]4W_{y} \\ & = \frac{1}{{\lambda_{pf} }}\frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }} \\ \end{aligned}$$

Or

$$Pr_{pf} \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\left[ {\frac{d\theta }{{d\eta_{pf} }} + \frac{T}{{T_{w} - T_{\infty } }}\frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right]\left( { - \eta_{pf} W_{x} + 4W_{y} } \right) = \frac{1}{{\lambda_{pf} }}\frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}$$

Since $\frac{T}{{T_{w} - T_{\infty } }} = \frac{{\left( {T_{w} - T_{\infty } } \right)\theta - T_{\infty } }}{{T_{w} - T_{\infty } }} = \theta + \frac{{T_{\infty } }}{{T_{w} - T_{\infty } }}$, the above equation becomes

$$Pr_{pf} \frac{{\nu_{\infty ,pf} }}{{\nu_{pf} }}\left( { - \eta_{pf} W_{x} + 4W_{y} } \right)\left[ {\frac{d\theta }{{d\eta_{pf} }} + \left( {\theta + \frac{{T_{\infty } }}{{T_{w} - T_{\infty } }}} \right)\frac{1}{{c_{p_{pf}} }}\frac{{dc_{p_{pf}} }}{{d\eta_{pf} }}} \right] = \frac{1}{{\lambda_{pf} }}\frac{{d\lambda_{pf} }}{{d\eta_{pf} }}\frac{d\theta }{{d\eta_{pf} }} + \frac{{d^{2} \theta }}{{d\eta_{pf}^{2} }}$$

(4.25)