Complete Similarity Mathematical Models on Laminar Free Convection Film Condensation from Vapor–Gas Mixture

Abstract

By means of the new similarity analysis method, the governing partial differential equations of laminar free convection film condensation of vapor–gas mixture are transformed into the complete dimensionless mathematical models. The transformed complete governing mathematical models are equivalent to the system of dimensionless governing equations, which involve (1) the continuity, momentum, and energy equations for both liquid and vapor–gas mixture films, as well as species conservation equation with mass diffusion in the vapor–gas mixture film, (2) a set of interfacial physical matching conditions, such as those for two-dimensional velocity component balances, shear force balance, mass flow rate balance, temperature balance, heat transfer balance, concentration condition, as well as the balance between the condensate mass flow and vapor mass diffusion. On the other hand, the transformed complete similarity mathematical models of the film condensation of vapor–gas mixture are very well coupled with a series of physical property factors, such as the density factor, absolute viscosity factor, thermal conductivity factor, of the medium of liquid film and the vapor–gas mixture film. Thus, the transformed complete similarity mathematical models are advanced ones for consideration of variable physical properties.

Appendix A: Similarity Transformation of the Governing Partial Differential Equations (18.1)–(18.7) and Their Boundary Condition Equations (18.8)–(18.16)

1.1 For Liquid Film

With the assumed transformation variables for the liquid film shown in Eqs. (18.17)–(18.21), the governing partial differential equations (18.1)–(18.3) of the liquid film are transformed into dimensionless ordinary ones as follows, respectively:

1.1.1 Similarity Transformation of Eq. (18.1)

Equation (18.1) is changed to

$$\begin{aligned} W_{x\mathrm{l}}\frac{\partial \rho _\mathrm{l}}{\partial x}+w_{y\mathrm{l}}\frac{\partial \rho _\mathrm{l}}{\partial y}+\rho _\mathrm{l}\left( \frac{\partial w_{x\mathrm{l}}}{\partial x}+\frac{\partial w_{y\mathrm{l}}}{\partial y}\right) =0 \end{aligned}$$

(A1)

With Eq. (18.20) we have

$$\begin{aligned} \frac{\partial w_{x\mathrm{l}}}{\partial x}=\sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}+2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2} \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\frac{\partial \eta _\mathrm{l}}{\partial x} \end{aligned}$$

With Eqs. (18.17) and (18.18) we have

$$\begin{aligned} \frac{\partial \eta _\mathrm{l}}{\partial x}&=\frac{\partial }{\partial x}\left[ \left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty })}{{{\nu _{\text{ l},\mathrm{s}}}^{2}}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}\frac{y}{x^{1/4}}\right] \nonumber \\&=-\frac{1}{4}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}\frac{y}{x^{2}} \end{aligned}$$

$$\begin{aligned} =-\frac{1}{4}x^{-1}\eta _\mathrm{l} \end{aligned}$$

(A2)

Then,

$$\begin{aligned} \frac{\partial w_{x\mathrm{l}}}{\partial x}&=\sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2} W_{x\mathrm{l}}+2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta }\frac{1}{4}x^{-1}\eta _\mathrm{l} \end{aligned}$$

$$\begin{aligned} =\sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho }_{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( W_{x\mathrm{l}}-\frac{1}{2}\eta _\mathrm{l}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) \end{aligned}$$

(A3)

With Eqs. (18.17) and (18.21) we have

$$\begin{aligned} \frac{\partial w_{y\mathrm{l}}}{\partial y}=2\sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2} \frac{\mathrm{d}W_{y\mathrm{l}} }{\mathrm{d}\eta _\mathrm{l}} \end{aligned}$$

(A4)

With Eq. (18.17) we have

$$\begin{aligned} \frac{\partial \rho _\mathrm{l}}{\partial x}=\frac{\mathrm{d}\rho }{\mathrm{d}\eta _\mathrm{l}}\frac{\mathrm{d}\eta _\mathrm{l}}{\mathrm{d}x} \end{aligned}$$

With Eq. (A2), we have

$$\begin{aligned} \frac{\partial \rho _\mathrm{l}}{\partial x}=-\frac{1}{4}\eta _\mathrm{l}x^{-1}\frac{\mathrm{d}\rho _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}} \end{aligned}$$

(A5)

In addition, with Eq. (18.17), we have

$$\begin{aligned} \frac{\partial \rho _\mathrm{l} }{\partial y}=\frac{\mathrm{d}\rho }{\mathrm{d}\eta _\mathrm{l} }\frac{\mathrm{d}\eta _\mathrm{l}}{\mathrm{d}y}=\frac{\mathrm{d}\rho _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{Gr}}_{{xl}, \text{ms}}\right) ^{1/4}x^{-1} \end{aligned}$$

(A6)

With Eqs. (A3)–(A6), Eq. (A1) is changed to

$$\begin{aligned} \,&2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}\left( -\frac{1}{4}\eta _\mathrm{l}{x}^{-1}\frac{\mathrm{d}\rho _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) \\&+ 2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{l},\infty }\right) ^{-1/4}W_{y\mathrm{v}}\frac{\mathrm{d}\rho _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{l},s}\right) ^{1/4}x^{-1}\\&+ \rho _\mathrm{l}\left[ \sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( W_{x\mathrm{l}}-\frac{1}{2}\eta _\mathrm{l}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) \right. \nonumber \\&\left. +2\sqrt{\frac{g}{x}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\frac{\mathrm{d}W_{y\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right] = 0 \end{aligned}$$

The above equation is simplified to

$$\begin{aligned}&2\sqrt{gx} W_{x\mathrm{l}} \left( -\frac{1}{4}\eta _\mathrm{l} x^{-1}\frac{\mathrm{d}\rho _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\right) +2\sqrt{gx} W_{y\mathrm{l}} \frac{\mathrm{d}\rho _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }x^{-1}\nonumber \\&+\rho _\mathrm{l} \left[ \sqrt{\frac{g}{x}} \left( W_{x\mathrm{l}} -\frac{1}{2}\eta _\mathrm{l} \frac{\mathrm{d}W_{x\mathrm{l}} }{\mathrm{d}\eta _\mathrm{l} }\right) +2\sqrt{\frac{g}{x}} \frac{\mathrm{d}W_{y\mathrm{l}} }{\mathrm{d}\eta _\mathrm{l} }\right] = 0 \end{aligned}$$

The above equation is divided by $\sqrt{\frac{g}{x}}$, and simplified to

$$\begin{aligned} 2W_{x\mathrm{l}} -\eta _\mathrm{l} \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}+4\frac{\mathrm{d}W_{y\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}-\frac{1}{\rho _\mathrm{l}}\frac{\mathrm{d}\rho _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}(\eta _\mathrm{l} W_{x\mathrm{l}} -4W_{y\mathrm{l}})=0 \end{aligned}$$

(18.28)

1.1.2 Similarity Transformation of Eq. (18.2)

Equation (18.2) can be rewritten as

$$\begin{aligned} \rho _\mathrm{l}\left( w_{x\mathrm{l}} \frac{\partial w_{x\mathrm{l}}}{\partial x}+w_{y\mathrm{l}} \frac{\partial w_{x\mathrm{l}}}{\partial y}\right) =\mu _\mathrm{l}\frac{\partial ^{2}w_{x\mathrm{l}} }{\partial y^{2}}+\frac{\partial \mu _\mathrm{l}}{\partial y} \frac{\partial w_{x\mathrm{l}} }{\partial y}+g(\rho _\mathrm{l} -\rho _{\mathrm{m},\infty }) \end{aligned}$$

(A7)

With Eqs. (18.17), (18.18), and (18.21) we have

$$\begin{aligned} \frac{\partial w_{x\mathrm{l}} }{\partial y}&=2\sqrt{gx} \left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}\frac{\mathrm{d}W_{x\mathrm{l}} }{\mathrm{d}\eta }\frac{\partial \eta }{\partial y} \nonumber \\&=2\sqrt{gx} \left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}\frac{\mathrm{d}W_{y\mathrm{l}} }{\mathrm{d}\eta }x^{-1}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}} \right) ^{1/4} \end{aligned}$$

(A8)

Then,

$$\begin{aligned} \frac{\partial ^{2}w_{x\mathrm{l}} }{\partial y^{2}}=2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2} \frac{\mathrm{d}^{2}W_{x\mathrm{l}} }{\mathrm{d}\eta ^{2}}x^{-2} \left( {1 \over 4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/2} \end{aligned}$$

(A9)

In addition,

$$\begin{aligned} \frac{\partial \mu _\mathrm{l}}{\partial y}=\frac{\mathrm{d}\mu _\mathrm{l}}{\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1} \end{aligned}$$

(A10)

With (A3), (A8), (A9), (18.36), and (A10), Eq. (A7) is changed to

$$\begin{aligned}&\rho _\mathrm{l}\left[ 2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}\sqrt{\frac{g}{x}}\left( \frac{\rho _{\text{ l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( W_{x\mathrm{l}}-\frac{1}{2}\eta _\mathrm{l}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) \right. \\&\quad +2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{-1/4}W_{y\mathrm{l}}2\sqrt{gx}\left( \frac{\rho _{\text{ l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\\&\quad \times \left. \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}x^{-1}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\right) ^{-1/4}\right] \\&=\mu _\mathrm{l}2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{v},\mathrm{m}\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\frac{\mathrm{d}^{2}W_{x\mathrm{l}}}{\mathrm{d}\eta _{l^{2}}}\left( \!\frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\!\right) ^{1/2}x^{-2}\\&\quad +\frac{\mathrm{d}\mu _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \!\frac{1}{4}{\text{ Gr }}_{x\mathrm{v}},\text{ ms }\!\right) ^{1/4}x^{-1}2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\!\right) ^{1/2}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}x^{-1}\left( \!\frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\!\right) ^{-1/4}\\&\quad +g\left( \rho _\mathrm{l}-\rho _{\mathrm{m},\infty }\right) \end{aligned}$$

The above equation is divided by $g\frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }$, and simplified to

$$\begin{aligned}&\rho _\mathrm{l}\left[ 2W_{x\mathrm{l}}\left( W_{x\mathrm{l}} -\frac{1}{2}\eta _\mathrm{l} \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) +2W_{y\mathrm{l}} 2\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right] \\&=\mu _\mathrm{l} 2\frac{\mathrm{d}^{2} W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}^2}\left( {\frac{1}{4}}\frac{1}{\nu _{\text{ l},\mathrm{s}}^2}\right) ^{1/2}+\frac{\mathrm{d}\mu _{\text{ v},\mathrm{w}}}{\mathrm{d}\eta _\mathrm{m}}2\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l} }\left( {\frac{1}{4}}\frac{1}{{\nu _{\text{ l},\mathrm{s}}}^2}\right) ^{1/2}+\frac{\left( \rho _\mathrm{l} -\rho _{\mathrm{m},\infty } \right) }{\frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }} \end{aligned}$$

The above equation is divided by $\frac{\nu _{\text{ l},\mathrm{s}} }{\mu _\mathrm{l} }$, and further simplified to

$$\begin{aligned}&\frac{\nu _{\text{ l},\mathrm{s}}}{\nu _\mathrm{l}}\left[ W_{x\mathrm{l}}\left( 2W_{x\mathrm{l}}-\eta _\mathrm{l}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) +4W_{y\mathrm{l}}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right] \end{aligned}$$

$$\begin{aligned} \quad =\frac{\mathrm{d}^{2} W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}^2 }+\frac{1}{\mu _\mathrm{l} }\frac{\mathrm{d}\mu _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l} }+\frac{\mu _{\text{ l},\mathrm{s}}}{\mu _\mathrm{l}}\frac{\left( \rho _\mathrm{l}-\rho _{\mathrm{m},\infty }\right) }{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }} \end{aligned}$$

(18.29)

1.1.3 Similarity Transformation of Eq. (18.3)

Equation (18.3) can be rewritten as

$$\begin{aligned} \rho _\mathrm{l}c_{p_\mathrm{l}}\left( w_{x\mathrm{l}}\frac{\partial t_\mathrm{l}}{\partial x}+w_{y\mathrm{l}}\frac{\partial t_\mathrm{l}}{\partial x}\right) =\lambda _\mathrm{l}\frac{\partial ^{2}t_\mathrm{l}}{\partial y^{2}}+\frac{\partial \lambda _\mathrm{l}}{\partial _{y}}\frac{\partial t_\mathrm{l}}{\partial y} \end{aligned}$$

(A11)

where

$$\begin{aligned} \frac{\partial t_\mathrm{l} }{\partial x}=(t_\mathrm{w} -t_{\text{ s},\text{ int }} )\frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\frac{\partial \eta _\mathrm{l} }{\partial x} \end{aligned}$$

With Eq. (A2) the above equation becomes

$$\begin{aligned} \frac{\partial t_\mathrm{l} }{\partial x}=-\frac{1}{4}\eta _\mathrm{l}x^{-1}(t_\mathrm{w} -t_{\text{ s},\text{ int }} )\frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} } \end{aligned}$$

(A12)

In addition

$$\begin{aligned} \frac{\partial t_\mathrm{l} }{\partial y}=-(t_\mathrm{w} -t_{\text{ s},\text{ int }} )\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l} }\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\right) ^{1/4}x^{-2} \end{aligned}$$

(A13)

$$\begin{aligned} \frac{\partial ^{2}t_\mathrm{l}}{\partial y^{2}}=(t_\mathrm{w} -t_{\text{ s},\text{ int }} )\frac{\mathrm{d}^{2}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\right) ^{1/2}x^{-2} \end{aligned}$$

(A14)

$$\begin{aligned} \frac{\partial \lambda _\mathrm{l}}{\partial y}=-\frac{\mathrm{d}\lambda _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l} }\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}^{\prime }\right) ^{1/4}x^{-1} \end{aligned}$$

(A15)

With (A12)–(A15), and (18.20) and (18.21), Eq. (A11) is changed to

$$\begin{aligned}&\rho _\mathrm{l}c_{p_\mathrm{l}}\left[ -2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{v},\mathrm{m}\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}\right) \eta _\mathrm{l}x^{-1}\right. \\&\quad \left. +2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{v},\mathrm{m}\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{l},\mathrm{m}\infty }\right) ^{-1/4}W_{y,\mathrm{l}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right. \\&\quad \left. \left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}\right] \\&=\lambda _\mathrm{l}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\frac{\mathrm{d}^{2}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/2}x^{-2}+\frac{\mathrm{d}\lambda _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\\&\quad \times \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1} \end{aligned}$$

The above equation is divided by $(T_\mathrm{w} -T_\mathrm{s})$, and simplified to

$$\begin{aligned}&\rho _\mathrm{l}c_{p_\mathrm{l}}\left[ -2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}\right) \eta _\mathrm{l}x^{-1}\right. \\&\left. +2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{y,\mathrm{l}}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}x^{-1}\right] \\&\quad =\lambda _\mathrm{l}\frac{\mathrm{d}^{2}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/2}x^{-2}+\frac{\mathrm{d}\lambda _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1} \end{aligned}$$

With definition of ${\text{ Gr }}_{x{\text{ l},\mathrm{s}}}$, the above equation is further simplified to

$$\begin{aligned}&\rho _\mathrm{l}c_{p_\mathrm{l}}\left[ -2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x\mathrm{l}}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\left( \frac{1}{4}\right) \eta _\mathrm{l}x^{-1}\right. \\&\qquad \qquad \left. +2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{y,\mathrm{l}}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}x^{-1}\right] \\&\quad {} =\lambda _\mathrm{l}\frac{\mathrm{d}^{2}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}^{2}}\left( \frac{1}{4} \frac{g\left( \rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}\right) x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}x^{-2}\\&\qquad \quad \; +\frac{\mathrm{d}\lambda _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}x^{-1}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}} \left( \frac{1}{4}\frac{g\left( \rho _{\mathrm{l},\mathrm{w}}-{\rho _{\mathrm{m},\infty }}\right) x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}x^{-1} \end{aligned}$$

The above equation is divided by $[\frac{g(\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty } )}{x\rho _{\text{ l},\mathrm{s}} }]^{1/2}$, and is further simplified to

$$\begin{aligned} {}\rho _\mathrm{l} c_{p_\mathrm{l} } \left[ \!-2W_{x\mathrm{l}} \frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\!\left( \!\frac{1}{4}\!\right) \eta _\mathrm{l}+2W_{y,\mathrm{l}} \frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\right] = \lambda _\mathrm{l} \frac{\mathrm{d}^{2}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} ^2}\!\left( \!\frac{1}{4}\frac{1}{{\nu _{\text{ l},\mathrm{s}}}^{2}}\!\right) ^{\!1/2}+ \frac{\mathrm{d}\lambda _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\!\left( \!\frac{1}{4}\frac{1}{{\nu _{\text{ l},\mathrm{s}}}^{2}}\right) ^{\!1/2} \end{aligned}$$

The above equation is divided by $2\frac{\nu _{\text{ l},\mathrm{s}} }{\lambda _\mathrm{l} }$, and is further simplified to

$$\begin{aligned} \frac{\rho _\mathrm{l} c_{p_\mathrm{l}} \nu _{\text{ l},\mathrm{s}} }{\lambda _\mathrm{l} }\left[ -W_{x\mathrm{l}} \frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\eta _\mathrm{l}+4W_{y\mathrm{l}} \frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\right] =\frac{\mathrm{d}^{2}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} ^2}+\frac{1}{\lambda _\mathrm{l} }\frac{\mathrm{d}\lambda _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} }\frac{\mathrm{d}\theta _\mathrm{l} }{\mathrm{d}\eta _\mathrm{l} } \end{aligned}$$

i.e.,

$$\begin{aligned} \Pr \nolimits _\mathrm{l} \frac{\nu _{\text{ l},\mathrm{s}}}{\nu _\mathrm{l}}(-\eta _\mathrm{l}{W}_{x\mathrm{l}} +4W_{y\mathrm{l}} )\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}=\frac{\mathrm{d}^{2}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}^{2}}+\frac{1}{\lambda _\mathrm{l}}\frac{\mathrm{d}\lambda _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}} \end{aligned}$$

(18.30)

1.2 For Vapor–Gas Mixture Film

With the assumed similarity transformation variables for vapor–gas mixture film shown in Eqs. (18.22)–(18.27), the governing partial differential equations (18.4)–(18.7) for vapor–gas mixture film are transformed equivalently into the following ones, respectively:  

1.2.1 Similarity Transformation of Eq. (18.4)

Equation (18.4) is rewritten as

$$\begin{aligned} W_{x\mathrm{m}}\frac{\partial \rho _\mathrm{m}}{\partial x}+W_{y\mathrm{m}}\frac{\partial \rho _\mathrm{m}}{\partial y}+\rho _\mathrm{m}\left[ \frac{\partial W_{x\mathrm{m}}}{\partial x}+\frac{\partial W_{y\mathrm{m}}}{\partial y}\right] =0 \end{aligned}$$

(A16)

With Eq. (18.25) we have

$$\begin{aligned} \frac{\partial W_{x\mathrm{m}}}{\partial x}=\sqrt{\frac{g}{x}}(\rho _{\text{ m}, \mathrm{s}}/{\rho _{\mathrm{m},\infty }}-1)^{1/2}W_{x\mathrm{m}}+2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/{\rho _{\mathrm{m},\infty }}-1)^{1/2}\frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial x} \end{aligned}$$

Similar to the derivation for Eq. (A2), we have

$$\begin{aligned} \frac{\partial \eta _\mathrm{m}}{\partial x}==-\frac{1}{4}x^{-1}\eta _\mathrm{m} \end{aligned}$$

(A17)

Then,

$$\begin{aligned} \frac{\partial w_{x\mathrm{m}}}{\partial x}=\sqrt{\frac{g}{x}} \left( \rho _{\text{ m}, \mathrm{s}} /{\rho _{\mathrm{m},\infty }}-1\right) ^{1/2}\left( W_{x\mathrm{m}} -\frac{1}{2}\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\right) \end{aligned}$$

(A18)

With Eq. (18.26) we have

$$\begin{aligned} \frac{\partial w_{y\mathrm{m}}}{\partial y}&=2\sqrt{g x} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} \frac{\mathrm{d}W_{y\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial y} \nonumber \\&=2\sqrt{g x} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} \frac{\mathrm{d}W_{y\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m}}x^{-1}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4} \end{aligned}$$

$$\begin{aligned} =2\sqrt{\frac{g}{x}} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}\frac{\mathrm{d}W_{y\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}} \end{aligned}$$

(A19)

In addition

$$\begin{aligned} \frac{\partial \rho _\mathrm{m}}{\partial x}=\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial {x}} \end{aligned}$$

With Eq. (A17), the above equation becomes

$$\begin{aligned} \frac{\partial \rho _\mathrm{m}}{\partial x}=-\frac{1}{4}x^{-1}\eta _\mathrm{m}\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}} \end{aligned}$$

(A20)

While,

$$\begin{aligned} \frac{\partial \rho _\mathrm{m}}{\partial y}&=\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial y} \nonumber \\&=\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A21)

With Eqs. (A18)–(A21), and (18.25) and (18.26), Eq. (A16) is changed to

$$\begin{aligned}&{}-2\sqrt{g x}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}W_{x\mathrm{m}} \frac{1}{4}x^{-1}\eta _\mathrm{m}\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}} \\&{}+2\sqrt{g x}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4}W_{y\mathrm{m}} \frac{\mathrm{d}\rho _{\mathrm{v},\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\mathrm{s}}\right) ^{1/4}x^{-1} \\&{}+\rho _\mathrm{m}\left[ \!\sqrt{\frac{g}{x}}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }{-}1)^{1/2}\left( W_{x\mathrm{m}}{-} \frac{1}{2}\eta _\mathrm{m}\frac{\mathrm{d}W_{x,\mathrm m}}{\mathrm{d}\eta _\mathrm{m}}\right) \right. \\&{}+2\left. \sqrt{\frac{g}{x}}(\rho _{\mathrm{m},\mathrm{s}}/\rho _{\mathrm{m},\infty }{-}1)^{1/2} \frac{\mathrm{d}W_{y\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\!\right] =0 \end{aligned}$$

The above equation is divided by $\rho _\mathrm{m}\sqrt{\frac{g}{x}}(\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}$, and simplified to

$$\begin{aligned} W_{x\mathrm{m}}-\frac{1}{2}\eta _\mathrm{m}\frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}+\frac{\mathrm{d}W_{y\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}-\frac{1}{\rho _\mathrm{m}}\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{2}\eta _\mathrm{m}W_{x\mathrm{m}}-2W_{y\mathrm{m}}\right) =0 \end{aligned}$$

i.e.,

$$\begin{aligned} 2W_{x\mathrm{m}} -\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }+4\frac{\mathrm{d}W_{y\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }-\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }(\eta _\mathrm{m} W_{x\mathrm{m}} -4W_{y\mathrm{m}} )=0 \end{aligned}$$

(18.31)

1.2.2 Similarity Transformation of Eq. (18.5)

Equation (18.5) is changed to

$$\begin{aligned} \rho _\mathrm{m} \left( w_{x\mathrm{m}} \frac{\partial w_{x\mathrm{m}} }{\partial x}+w_{y\mathrm{m}} \frac{\partial w_{x\mathrm{m}} }{\partial y}\right) =\mu _\mathrm{m} \frac{\partial ^{2}w_{x\mathrm{m}} }{\partial y^2}+\frac{\partial \mu _\mathrm{m} }{\partial y}\frac{\partial w_{x\mathrm{m}} }{\partial y}+g(\rho _\mathrm{m} -\rho _{\mathrm{m},\infty } ) \end{aligned}$$

(A22)

With Eqs. (18.22) and (18.25), we have

$$\begin{aligned} \frac{\partial w_{x\mathrm{m}}}{\partial y}=2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2} \frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A23)

$$\begin{aligned} \frac{\partial ^{2} w_{x\mathrm{m}} }{\partial y^{2} }=2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2} \frac{\mathrm{d}^{2} W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m}^{2} }\left( {\frac{1}{4}}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2} \end{aligned}$$

(A24)

Additionally,

$$\begin{aligned} \frac{\partial \mu _\mathrm{m} }{\partial y}=\frac{\mathrm{d}\mu _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A25)

With Eqs. (A18), (A23)–(A25), and (18.25) and (18.26), Eq. (A22) is changed to

$$\begin{aligned}&\rho _\mathrm{m} \left[ 2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}W_{x\mathrm{m}} \sqrt{\frac{g}{x}} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) ^{1/2}\right. \\&\qquad \;\times \left( W_{x\mathrm{m}}-\frac{1}{2}\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta }\right) +2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} \\&\qquad \;\times \left. W_{y\mathrm{m}} 2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }x^{-1} \left( \frac{1}{4} {\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4} \right] \\&=\mu _\mathrm{m} 2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2} \frac{\mathrm{d}^{2} W_{x\mathrm{m}} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }^{\prime }\right) ^{1/2}x^{-2} \\&\quad \;+\frac{\mathrm{d}\mu _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty } \right) ^{1/4}x^{-1}2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}\\&\quad \;\times \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }x^{-1} \left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty } \right) ^{1/4} +g(\rho _\mathrm{m} -\rho _{\mathrm{m},\infty }) \end{aligned}$$

With the definition of ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation is changed to

$$\begin{aligned}&\rho _\mathrm{m} \left[ 2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)W_{x\mathrm{m}} \sqrt{\frac{g}{x}} \left( W_{x\mathrm{m}} -\frac{1}{2}\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta }\right) \right. \\&\qquad \left. \,+2\sqrt{gx} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) W_{y\mathrm{m}} 2\sqrt{gx} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }x^{-1} \right] \\&=\mu _\mathrm{m} 2\sqrt{gx} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) ^{1/2} \frac{\mathrm{d}^{2} W_{x\mathrm{m}} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)x^3 }{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2} x^{-2} \\&\;+\frac{\mathrm{d}\mu _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) x^3 }{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-1}2\sqrt{gx} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) ^{1/2}\\&\;\times \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }x^{-1} +g\left( \rho _\mathrm{m} -\rho _{\mathrm{m},\infty } \right) \end{aligned}$$

The above equation is divided by $\frac{\mu _\mathrm{m}}{\nu _{\mathrm{m},\infty }}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)$, and simplified to

$$\begin{aligned}&\frac{\nu _{\mathrm{m},\infty } \rho _\mathrm{m} }{\mu _\mathrm{m} }\left[ W_{x\mathrm{m}} \left( 2W_{x\mathrm{m}} -\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta }\right) +4W_{y\mathrm{m}} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{v} }\right] \\&=\frac{\mathrm{d}^{2} W_{x\mathrm{m}} }{{\mathrm{d}\eta _\mathrm{m}}^{2} }+\frac{1}{\mu _\mathrm{m} }\frac{\mathrm{d}\mu _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }+\frac{\nu _{\mathrm{m},\infty } \rho _{\mathrm{m},\infty } }{\mu _\mathrm{m} }\frac{\rho _\mathrm{m} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ m}, \mathrm{s}} -\rho _{\mathrm{m},\infty } } \end{aligned}$$

i.e.,

$$\begin{aligned}&\frac{\nu _{\mathrm{m},\infty }}{\nu _\mathrm{m}}\left[ W_{x\mathrm{m}}\left( 2W_{x\mathrm{m}} -\eta _\mathrm{m} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m}}\right) +4W_{y\mathrm{m}}\frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\right] \nonumber \\&\quad {}= \frac{\mathrm{d}^{2}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}^{2}}+ \frac{1}{\mu _\mathrm{m}} \frac{\mathrm{d}\mu _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}} \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }+ \frac{\mu _{\mathrm{m},\infty }}{\mu _\mathrm{m}}\cdot \frac{\rho _\mathrm{m} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ m}, \mathrm{s}} -\rho _{\mathrm{m},\infty }} \end{aligned}$$

 

1.2.3 Similarity Transformation of Eq. (18.6)

Equation (18.6) is changed to

$$\begin{aligned} \rho _\mathrm{m} Cp_\mathrm{m} \left( w_{x\mathrm{m}} \frac{\partial t}{\partial x}+w_{y\mathrm{m}} \frac{\partial t}{\partial y}\right) =\,&\lambda _\mathrm{m} \frac{\partial ^{2}t}{\partial y^2}+\frac{\partial \lambda _\mathrm{m} }{\partial y}\cdot \frac{\partial t}{\partial y}-D_\mathrm{v} (c_{p_\mathrm{v}}-c_{p_\mathrm{g}} )\nonumber \\&\;\times \left[ \rho _\mathrm{m} \frac{\partial C_{\text{ mv }} }{\partial y}\frac{\partial t}{\partial y}+t\rho _\mathrm{m} \frac{\partial ^{2}C_{\text{ mv }} }{\partial y^{2}}+t\frac{\partial C_{\text{ mv }} }{\partial y}\frac{\partial \rho _\mathrm{m} }{\partial y}\right] \end{aligned}$$

(A26)

With Eq. (18.24) we have

$$\begin{aligned} \frac{\partial t}{\partial x}=(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\frac{\partial \eta _\mathrm{m} }{\partial x} \end{aligned}$$

With Eq. (A17), the above equation becomes

$$\begin{aligned} \frac{\partial t}{\partial x}==-\frac{1}{4}\eta _\mathrm{m}x^{-1}(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{l} } \end{aligned}$$

(A27)

With Eqs. (18.22) and (18.24) we have

$$\begin{aligned} \frac{\partial t}{\partial y}=(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A28)

$$\begin{aligned} \frac{\partial ^{2}t}{\partial y^{2}}=(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}^{2}\theta _\mathrm{m} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2} \end{aligned}$$

(A29)

With Eqs. (18.22) and (18.24) we have

$$\begin{aligned} \frac{\partial \lambda _\mathrm{m} }{\partial y}=\frac{\mathrm{d}\lambda _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A30)

$$\begin{aligned} \frac{\partial \rho _\mathrm{m} }{\partial y}=\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A31)

$$\begin{aligned} \frac{\partial C_{\text{ mv }} }{\partial y}=\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A32)

$$\begin{aligned} \frac{\partial ^{2}C_{\text{ mv }} }{\partial y^{2}}=\frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2} \end{aligned}$$

(A33)

With Eqs. (A27)–(A31), Eq. (A26) is changed to

$$\begin{aligned} \rho _\mathrm{m} c_{p_\mathrm{m} }&\left[ -2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}W_{x\mathrm{m}} (t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\right) \eta _\mathrm{m} x^{-1}\right. \\&\quad \;+2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }^{\prime }\right) ^{-1/4}\\&\quad \left. \;\times W_{y\mathrm{m}} (t_{\text{ s},\text{ int }}-t_\infty )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\right] \\&=\lambda _\mathrm{m} (t_{\text{ s},\text{ int }} -t_\infty )\frac{\mathrm{d}^{2}\theta _\mathrm{m}}{{\mathrm{d}\eta _\mathrm{m}} ^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2}+\frac{\mathrm{d}\lambda _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\\&\cdot (t_{\text{ s},\text{ int }} -t_\infty )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} -D_\mathrm{v} (c_{p_\mathrm{v}} -c_{p_\mathrm{g}} ) \\&\;\times \left[ \rho _\mathrm{m} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\right. \\&\qquad \;+t\rho _\mathrm{m} \frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2}+t\frac{\mathrm{d}C_{\mathrm{m},\mathrm{v}} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\\&\qquad \left. \;\times \frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\right] \end{aligned}$$

With definition of ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation is changed to

$$\begin{aligned}&\rho _\mathrm{m} c_{p_\mathrm{m} } \left[ -2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}W_{x\mathrm{m}} (t_{\text{ s},\text{ int }} -T_{\infty })\frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\right) \eta _\mathrm{m}x^{-1} \right. \\&\qquad \qquad \left. +\,2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}W_{y\mathrm{m}}(t_{\text{ s},\text{ int }} -t_{\infty })\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }x^{-1}\right] \\&=\lambda _\mathrm{m} (t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}^{2}\theta _\mathrm{m} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1\right) x^{3} }{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-2}\\&\qquad \;+\frac{\mathrm{d}\lambda _\mathrm{m} }{\mathrm{d}\eta _\mathrm{v} }\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1\right) x^{3} }{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-1} \nonumber \\&\qquad \;\cdot (t_{\text{ s},\text{ int }}-t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}x^{-1}-D_\mathrm{v}(c_{{p}_\mathrm{v}}-c_{{p}_\mathrm{g}} )\nonumber \\&\qquad \;\times \left[ \rho _{\mathrm{v},\mathrm{m}} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{v}}\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty } -1\right) x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-1}(t_{\text{ s},\text{ int }}-t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}x^{-1}\right. \\&\qquad \qquad \;+\,t\rho _\mathrm{m}\frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-2}\\&\qquad \qquad \left. \;+t\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{v}}\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1\right) x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-1}\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }x^{-1}\right] \end{aligned}$$

The above equation is divided by $\sqrt{\frac{g}{x}} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}$, and simplified to

$$\begin{aligned}&\rho _\mathrm{m}c_{{p}_\mathrm{m}}\left[ -2W_{x\mathrm{v}} (t_{\text{ s},\text{ int }} -t_{\infty })\frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}\right) \eta _\mathrm{m}+2W_{y\mathrm{m}}(t_{\text{ s},\text{ int }}-t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\right] \\&=\lambda _\mathrm{m}(t_{\text{ s},\text{ int }}-t_{\infty })\frac{\mathrm{d}^{2}\theta _\mathrm{m}}{{\mathrm{d}\eta _\mathrm{m}} ^{2}}\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}+\frac{\mathrm{d}\lambda _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2} \\&\qquad \cdot (t_{\text{ s},\text{ int }} -t_{\infty })\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}}-D_\mathrm{v}(c_{{p}_\mathrm{v}}-c_{{p}_\mathrm{g}}) \\&\qquad \;\times \left[ \rho _\mathrm{m} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}(t_{\text{ s},\text{ int }} -t_{\infty } )\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }+t\rho _\mathrm{m} \frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m} }^{2}}\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}\right. \\&\qquad \qquad \left. \;+t\frac{\mathrm{d}C_{\text{ v},\mathrm{w}} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\right] \end{aligned}$$

The above equation is multiplied by $2\frac{\nu _{\mathrm{m},\infty }}{\lambda _\mathrm{m}(t_{\text{ s},\text{ int }} -t_{\infty })}$, and simplified to

$$\begin{aligned}&\frac{\nu _{\mathrm{m},\infty } \rho _{\text{ mv }} \mu _\mathrm{m} Cp_\mathrm{m} }{\lambda _\mathrm{m} \mu _\mathrm{m}}\left[ -\eta _\mathrm{m} W_{x\mathrm{m}} +4W_{y\mathrm{m}} \right] \frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}} \\&=\frac{\mathrm{d}^{2}\theta _\mathrm{m} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\lambda _\mathrm{m} }\frac{\mathrm{d}\lambda _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }-\frac{D_\mathrm{v} \rho _\mathrm{m} }{\lambda _\mathrm{m} }(c_{{p}_\mathrm{v}}-c_{{p}_\mathrm{g}}) \\&\quad \;\times \left[ \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}}+\frac{t}{t_{\text{ s},\text{ int }} -t_{\infty } }\frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}}\frac{t}{t_{\text{ s},\text{ int }} -t_{\infty } }\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\right] \end{aligned}$$

The above equation is changed to

$$\begin{aligned}&\frac{\nu _{\mathrm{m},\infty } \rho _\mathrm{m} \mu _\mathrm{m}c_{{p}_\mathrm{m}}}{\lambda _\mathrm{m} \mu _\mathrm{m} }[-\eta _\mathrm{v} W_{x\mathrm{m}} +4W_{y\mathrm{m}} ]\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} } \nonumber \\&=\frac{\mathrm{d}^{2}\theta _\mathrm{m} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\lambda _\mathrm{m} }\frac{\mathrm{d}\lambda _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }-\frac{D_\mathrm{v} \rho _\mathrm{m} }{\lambda _\mathrm{m} }(c_{{p}_\mathrm{v}-c_{{p}_\mathrm{g}}})\nonumber \\&\quad \;\times \left[ \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }+\left( \theta _\mathrm{m} +\frac{t_{\infty } }{t_{\text{ s},\text{ int }}-t_{\infty } }\right) \frac{\mathrm{d}^{2}C_{\text{ mv }} }{\mathrm{d}\eta _{\infty } ^{2}}\right. \nonumber \\&\quad \qquad \left. \;+\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \theta _\mathrm{v} +\frac{t_{\infty } }{t_{\text{ s},\text{ int }}-t_{\infty }}\right) \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\right] \end{aligned}$$

(A34)

With (18.27) we have

$$\begin{aligned} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{v} }=(C_{\text{ mv},\mathrm{s}} -C_{\mathrm{m}\mathrm{v},\infty } )\frac{\mathrm{d}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{v}} \end{aligned}$$

(A35)

$$\begin{aligned} \frac{\mathrm{d}^{2}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{v}^{2}}=(C_{\text{ mv},\mathrm{s}} -C_{\mathrm{m}\mathrm{v},\infty } )\frac{\mathrm{d}^{2}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{v}^{2}} \end{aligned}$$

(A36)

Then, Eq. (A34) becomes

$$\begin{aligned}&\frac{\nu _{\mathrm{m},\infty }\Pr _\mathrm{m} }{\nu _\mathrm{m} }[-\eta _\mathrm{m}W_{x\mathrm{m}} +4W_{y\mathrm{m}} ]\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} } \\&=\frac{\mathrm{d}^{2}\theta _\mathrm{m} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\lambda _\mathrm{m} }\frac{\mathrm{d}\lambda _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} } -\frac{D_\mathrm{v}\rho _\mathrm{m}}{\lambda _\mathrm{m}}(C_{\text{ mv},\mathrm{s}} -C_{\mathrm{m}\mathrm{v},\infty } )(c_{{p}_\mathrm{v}}-c_{{p}_\mathrm{g}})\\&\quad \;\times \left[ \frac{\mathrm{d}\Gamma _{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m}}\frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}}+\left( \theta _\mathrm{m} +\frac{t_{\infty }}{t_{\text{ s},\text{ int }}-t_{\infty }}\right) \frac{\mathrm{d}^{2}\Gamma _{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\right. \end{aligned}$$

$$\begin{aligned} \qquad \qquad \;\left. +\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\left( \theta _\mathrm{m} +\frac{t_{\infty }}{t_{\text{ s},\text{ int }}-t_{\infty }}\right) \frac{\mathrm{d}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right] \end{aligned}$$

(18.33)

 

1.2.4 Similarity Transformation of Eq. (18.7)

The left-hand side of Eq. (18.7) is

$$\begin{aligned}&\frac{\partial (w_{x\mathrm{m}}\rho _\mathrm{m}C_{\text{ mv }})}{\partial x}+\frac{\partial (w_{y\mathrm{m}} \rho _\mathrm{m} C_{\text{ mv }} )}{\partial y} \\&=w_{x\mathrm{m}}\left[ \frac{\partial (\rho _\mathrm{m}C_{\text{ mv }} )}{\partial x}\right] +\rho _\mathrm{m}C_{\text{ mv }} \frac{\partial w_{x\mathrm{m}} }{\partial x}+w_{y\mathrm{m}}\left[ \frac{\partial (\rho _\mathrm{m}C_{\text{ mv }} )}{\partial y}\right] +\rho _\mathrm{m} C_{\text{ mv }} \frac{\partial w_{y\mathrm{m}} }{\partial y} \\&=w_{x\mathrm{m}} \rho _\mathrm{m} \frac{\partial (C_{\text{ mv }})}{\partial x}+w_{x\mathrm{m}}C_{\text{ mv }} \frac{\partial \rho _\mathrm{m}}{\partial x}+\rho _\mathrm{m} C_{\text{ mv }} \frac{\partial w_{x\mathrm{m}} }{\partial x}\\&\quad \;+w_{y\mathrm{m}} \rho _\mathrm{m} \frac{\partial (C_{\text{ mv }} )}{\partial y}+w_{y\mathrm{m}} C_{\text{ mv }} \frac{\partial \rho _\mathrm{m}}{\partial y}+\rho _\mathrm{m} C_{\text{ mv }} \frac{\partial w_{y\mathrm{m}}}{\partial y} \\&=w_{x\mathrm{m}}\rho _\mathrm{m}\frac{\partial (C_{\text{ mv }})}{\partial x}+w_{y\mathrm{m}} \rho _\mathrm{m} \frac{\partial (C_{\text{ mv }})}{\partial y}\\&\quad \;+C_{\mathrm{m},\mathrm{v}}\left( w_{x\mathrm{m}} \frac{\partial \rho _\mathrm{m} }{\partial x}+\rho _\mathrm{m}\frac{\partial w_{x\mathrm{m}} }{\partial x}+w_{y\mathrm{m}} \frac{\partial \rho _\mathrm{m}}{\partial y}+\rho _\mathrm{m}\frac{\partial w_{y\mathrm{m}} }{\partial y}\right) \end{aligned}$$

With Eq. (18.4) (the continuity equation of vapor–gas mixture film), the above equation is changed to

$$\begin{aligned} \frac{\partial (w_{x\mathrm{m}}\rho _\mathrm{m}C_{\text{ mv }})}{\partial x} + \frac{\partial (w_{y\mathrm{m}} \rho _\mathrm{m}C_{\text{ mv }})}{\partial y}=w_{x\mathrm{m}}\rho _\mathrm{m}\frac{\partial (C_{\text{ mv }} )}{\partial x}+w_{y\mathrm{m}}\rho _\mathrm{m}\frac{\partial (C_{\text{ mv }})}{\partial y} \end{aligned}$$

(A37)

Then, Eq. (18.7) becomes

$$\begin{aligned} w_{x\mathrm{m}} \frac{\partial C_{\text{ mv }} }{\partial x}+w_{y\mathrm{m}} \frac{\partial C_{\text{ mv }} }{\partial y}=D_\mathrm{v} \left( \frac{\partial ^2C_{\text{ mv }} }{\partial y^2}+\frac{1}{\rho _\mathrm{m} }\frac{\partial \rho _\mathrm{m} }{\partial y}\frac{\partial C_{\text{ mv }} }{\partial y}\right) \end{aligned}$$

(A38)

where $D_\mathrm{v}$ is regarded as constant variable.

With Eq. (18.22) we have

$$\begin{aligned} \frac{\partial C_{\text{ mv }}}{\partial x}=\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial x}=-\frac{1}{4}\eta _\mathrm{m} x^{-1}\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}} \end{aligned}$$

(A39)

$$\begin{aligned} \frac{\partial C_{\text{ mv }}}{\partial x}=\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\frac{\partial \eta _\mathrm{m}}{\partial x}=\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

(A40)

$$\begin{aligned} \frac{\partial ^{2}C_{\text{ mv }} }{\partial y^2}&=\frac{\partial }{\partial y}\left( \frac{\partial C_{\text{ mv }} }{\partial y}\right) \end{aligned}$$

$$\begin{aligned} =\frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}x^{-2} \end{aligned}$$

(A41)

With Eqs. (A31), (A39)–(A41), Eq. (A38) is changed to

$$\begin{aligned}&-2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}W_{x\mathrm{m}} \frac{1}{4}\eta _\mathrm{m} x^{-1}\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} } +2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}\\&\times \left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} W_{y\mathrm{m}} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \\&\qquad =D_{\text{ v},\mathrm{w}} \left[ \frac{\mathrm{d}^{2}C_{\text{ mv }}}{{\mathrm{d}\eta _\mathrm{m}}^2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty } \right) ^{1/2}x^{-2}+\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\right. \\&\qquad \quad \qquad \;\times \left. \frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}\right] \end{aligned}$$

With the definition of ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation is further changed to

$$\begin{aligned}&-2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}W_{x\mathrm{m}} \frac{1}{4}\eta _\mathrm{m}x^{-1}\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m}} \\&\quad \;+2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{1/2}W_{y\mathrm{m}} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m}}x^{-1} \\&=D_\mathrm{v}\left[ \frac{\mathrm{d}^{2}C_{\text{ mv }}}{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1\right) x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-2}\right. \\&\left. \qquad \qquad \;+\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}x^{-1}\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\frac{g\left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1\right) x^{3} }{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}x^{-1}\right] \end{aligned}$$

The above equation is divided by $\sqrt{\frac{g}{x}} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}$, and simplified to

$$\begin{aligned}&\;-2W_{x\mathrm{m}}\frac{1}{4}\eta _\mathrm{m} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }+2W_{y\mathrm{m}} \frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }\\&\quad {}=D_\mathrm{v}\left[ \frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}+\frac{1}{\rho _\mathrm{m}}\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m} }\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/2}\frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right] \end{aligned}$$

The above equation is multiplied by $\frac{2\nu _{\mathrm{m},\infty }}{D_\mathrm{v}}$, and further simplified to

$$\begin{aligned} \frac{\nu _{\mathrm{m},\infty } }{D_\mathrm{v} }(-W_{x\mathrm{m}} \eta _\mathrm{m} +4W_{y\mathrm{m}} )\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} }=\frac{\mathrm{d}^{2}C_{\text{ mv }} }{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\rho _\mathrm{m} }\frac{\mathrm{d}\rho _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m} }\frac{\mathrm{d}C_{\text{ mv }} }{\mathrm{d}\eta _\mathrm{m} } \end{aligned}$$

(A42)

With Eqs. (A35) and (A36), Eq. (A42) becomes

$$\begin{aligned} (-\eta _\mathrm{m}W_{x\mathrm{m}}+4W_{y\mathrm{m}})\frac{\mathrm{d}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}=\frac{1}{{\text{ Sc }}_{\mathrm{m},\infty }}\left( \frac{\mathrm{d}^{2}\Gamma _{\text{ mv }}}{{\mathrm{d}\eta _\mathrm{m}}^{2}}+\frac{1}{\rho _\mathrm{m}}\frac{\mathrm{d}\rho _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\frac{\mathrm{d}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right) \end{aligned}$$

(18.34)

where the local Schmidt number${\text{ Sc }}_{\mathrm{m},\infty }$ is defined as

$$\begin{aligned} {\text{ Sc }}_{\mathrm{m},\infty }=\frac{\nu _{\mathrm{m},\infty } }{D_\mathrm{v} } \end{aligned}$$

(A43)

1.3 For Boundary Condition Equations

1.3.1 Similarity Transformation of Eq. (18.8)

With Eqs. (18.20) and (18.21), Eq. (18.8) can be easily transformed into the following dimensionless form:

$$\begin{aligned} \eta _\mathrm{l} =0{:} \quad W_{x\mathrm{l}}=0, \quad W_{y\mathrm{l}}=0, \quad \theta _{1}=1 \end{aligned}$$

(18.35)

1.3.2 Similarity Transformation of Eq. (18.9)

With Eqs. (8.20) and (18.25), Eq. (18.9) is easily transformed as

$$\begin{aligned} W_{x\mathrm{m},\mathrm{s}} =\left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1)^{-1/2}W_{x{\text{ l},\mathrm{s}}} \end{aligned}$$

(18.36)

1.3.3 Similarity Transformation of Eq. (18.10)

With Eqs. (18.17) and (18.18) we have

$$\begin{aligned} \delta _\mathrm{l}=\eta _{\mathrm{l}\delta }\left( \frac{1}{4}\frac{g\left( \rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }\right) x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}x \end{aligned}$$

Then,

$$\begin{aligned} \left( \frac{\partial \delta _\mathrm{l}}{\partial x}\right) =\frac{1}{4}\eta _{\mathrm{l}\delta } \left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{-1/4} \end{aligned}$$

(A44)

Similarly,

$$\begin{aligned} \left( \frac{\partial \delta _\mathrm{m}}{\partial x}\right) =\frac{1}{4}\eta _{\mathrm{m}\delta } \left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} \end{aligned}$$

(A45)

With Eqs. (18.20), (18.21), (18.25), (18.26), (A44), and (A45), Eq. (18.10) is changed to

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}}\left[ 2\sqrt{gx}\left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x{\text{ l},\mathrm{s}}} \frac{1}{4}\eta _{\mathrm{l}\delta }\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}} \right) ^{-1/4}\right. \\&\qquad \qquad \left. \,-2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{-1/4}W_{ybad hbox}\right] \\&=\rho _{\text{ m}, \mathrm{s}} C_{\text{ mv},\mathrm{s}} \left[ 2\sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}W_{x\mathrm{m},\mathrm{s}} \frac{1}{4}\eta _{\mathrm{m}\delta } \left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty } \right) ^{-1/4} \right. \\*&\quad \qquad \qquad \qquad \left. \,-2\sqrt{gx} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} W_{y{\text{ m},\mathrm{s}}}\right] \end{aligned}$$

Since $\eta _{\mathrm{m}\delta } = 0$ at the liquid–vapor mixture interface, the above equation becomes

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}} \left[ 2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}W_{x{\text{ l},\mathrm{s}}} \frac{1}{4}\eta _{\mathrm{l}\delta } \left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}} \right) ^{-1/4} \right. \\&\qquad \qquad \left. \,-2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}} \right) ^{-1/4}W_{ybad hbox}\right] \\&=-2\rho _{\text{ m}, \mathrm{s}} C_{\mathrm{m},\text{ vs }} \sqrt{gx} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

With definitions of ${\text{ Gr }}_{x{\text{ l},\mathrm{s}}}$ and ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation is changed to

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}} \left[ 2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}W_{x{\text{ l},\mathrm{s}}} \frac{1}{4}\eta _{\mathrm{l}\delta }\left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty })x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}} }\right) ^{-1/4} \right. \\&\qquad \qquad \left. \,-2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/2}\left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty })x^{3} }{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{-1/4}W_{ybad hbox}\right] \\&=-2\rho _{\text{ m}, \mathrm{s}} C_{\mathrm{m},\text{ vs }} \sqrt{gx} (\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty } -1)^{1/2}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^2}\right) ^{-1/4} W_{y\mathrm{m},\mathrm{s}} \end{aligned}$$

The above equation is further simplified to

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}} \left[ 2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4}W_{x{\text{ l},\mathrm{s}}} \frac{1}{4}\eta _{\mathrm{l}\delta } \left( \frac{1}{4}\frac{gx^{3} }{{\nu _{\text{ l},\mathrm{s}}}^{2}}\right) ^{-1/4}\right. \\&\qquad \qquad \left. \,-2\sqrt{gx} \left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}\left( \frac{1}{4}\frac{gx^{3}}{{\nu _{\text{ l},\mathrm{s}}} ^{2}}\right) ^{-1/4}W_{{\rm yl}, {\rm s}}\right] \\&=-2\rho _{\text{ m}, \mathrm{s}} C_{\mathrm{m},\text{ vs }} \sqrt{gx} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1\right) ^{1/4}\left( \frac{1}{4}\frac{gx^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{-1/4} W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

The above equation is multiplied by $x^{1/4}g^{-1/4}$, and simplified to

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}}\left[ 2\left( \frac{\rho _{\text{ l},\mathrm{m}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4}W_{x{\text{ l},\mathrm{s}}} \frac{1}{4}\eta _{\mathrm{l}\delta }\left( \frac{1}{4}\frac{1}{{\nu _{\text{ l},\mathrm{s}}} ^{2}}\right) ^{-1/4} \right. \\&\qquad \qquad \left. \,-2\left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4}\left( \frac{1}{4}\frac{1}{{\nu _{\text{ l},\mathrm{s}}}^{2}}\right) ^{-1/4}W_{\rm yl, s} \right] \\&=-2\rho _{\text{ m}, \mathrm{s}} C_{\mathrm{mv},\text{ s }} (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{1/4}\left( \frac{1}{4}\frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{-1/4}W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

i.e.,

$$\begin{aligned}&\rho _{\text{ l},\mathrm{s}}\left[ \frac{1}{4}\left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4}W_{x{\text{ l},\mathrm{s}}} \eta _{\mathrm{l}\delta } {\nu _{\text{ l},\mathrm{s}}}^{1/2}-\left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}{\nu _{\text{ l},\mathrm{s}}}^{1/2}W_{\rm yl, s}\right] \\&=-\rho _{\text{ m}, \mathrm{s}}C_{\mathrm{mv},\text{ s }} \left( \rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty }-1\right) ^{1/4}{\nu _{\mathrm{m},\infty }}^{1/2}W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

Then,

$$\begin{aligned} W_{y{\text{ m},\mathrm{s}}} =&{}-\frac{1}{C_{\text{ mv},\mathrm{s}} }\frac{\rho _{\text{ l},\mathrm{s}} }{\rho _{\text{ m}, \mathrm{s}} }\left( \frac{\nu _{\text{ l},\mathrm{s}} }{\nu _{\mathrm{m},\infty } }\right) ^{1/2}\left( \frac{\rho _{\text{ l},\mathrm{m}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4} \end{aligned}$$

$$\begin{aligned} \times (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{-1/4}\left( \frac{1}{4}\eta _{\mathrm{l}\delta } W_{x{\text{ l},\mathrm{s}}} -W_{ybad hbox} \right) \end{aligned}$$

(18.37)

1.3.4 Similarity Transformation of Eq. (18.11)

With Eqs. (A8) and (A19), Eq. (18.11) is changed to

$$\begin{aligned}&\mu _{\text{ l},\mathrm{s}}2\sqrt{gx}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}x^{-1}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}\\&=\mu _{\text{ m}, \mathrm{s}}2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _{\nu }}\right) _\mathrm{s}x^{-1}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4} \end{aligned}$$

With the definitions of ${\text{ Gr }}_{x{\text{ l},\mathrm{s}}}$ and ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation becomes

$$\begin{aligned}&\mu _{\text{ l},\mathrm{s}}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{\rho _{\text{ l},\mathrm{s}}}\right) ^{1/2}\left( \frac{\mathrm{d}W_{x\mathrm{l}}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}x^{-1}\left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty })x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}\\&=\mu _{\text{ m}, \mathrm{s}}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{\mathrm{d}W_{x\mathrm{m}}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}x^{-1}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4} \end{aligned}$$

Therefore,

$$\begin{aligned} \left( \frac{\mathrm{d}W_{x\mathrm{m}} }{\mathrm{d}\eta _\mathrm{m} }\right) _\mathrm{s}=\,&\frac{\mu _{\text{ l},\mathrm{s}} }{\mu _{\text{ m}, \mathrm{s}} }\left( \frac{\nu _{\mathrm{m},\infty } }{\nu _{\text{ l},\mathrm{s}} }\right) ^{1/2} \left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{3/4} \end{aligned}$$

$$\begin{aligned} \times (\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{-3/4} \left( \frac{\mathrm{d}W_{x\mathrm{l}} }{\mathrm{d}\eta _\mathrm{l} }\right) _\mathrm{s} \end{aligned}$$

(18.38)

1.3.5 Similarity Transformation of Eq. (18.12)

With Eqs. (A13), (A25), (A28), (A45), and (18.26), Eq. (18.12) is changed to

$$\begin{aligned}&\lambda _{\text{ l},\mathrm{s}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}\left( \frac{1}{4}{\text{ Gr }}_{x{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}\\[6pt]&=\lambda _{\text{ m}, \mathrm{s}}(t_{\text{ s},\text{ int }}-t_{\infty })\left( \frac{\mathrm{d}\theta _{\infty }}{\mathrm{d}\eta _{\infty }}\right) _\mathrm{s}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1}+h_\text{ fg } \rho _{\text{ m}, \mathrm{s}}C_{\text{ mv},\mathrm{s}}\\[6pt]&\qquad \qquad \left[ 2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}W_{x\mathrm{m}}\frac{1}{4}\eta _{\mathrm{m}\delta }\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\mathrm{s}}\right) ^{-1/4}\right. \\[6pt]&\left. \qquad \qquad -2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4}W_{y{\text{ m},\mathrm{s}}}\right] _\mathrm{s} \end{aligned}$$

With the definitions of ${\text{ Gr }}_{x{\text{ l},\mathrm{s}}}$ and ${\text{ Gr }}_{x\mathrm{m},\infty }$, the above equation is changed to

$$\begin{aligned}&\lambda _{\text{ l},\mathrm{s}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}\left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty })x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}\\[6pt]&=\lambda _{{\rm v, \, ms}}(t_{\text{ s},\text{ int }}-t_{\infty })\left( \frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}x^{-1}+h_\text{ fg } \rho _{\text{ m}, \mathrm{s}}C_{\mathrm{mv},\text{ s }}\\[6pt]&\qquad \;\times \left[ 2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}W_{x\mathrm{m}}\frac{1}{4}\eta _{\mathrm{m}\delta }\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}\right. \\[6pt]&\left. \qquad \qquad -2\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{-1/4}W_{y\mathrm{m}}\right] _\mathrm{s} \end{aligned}$$

Since $\eta _{\mathrm{m}\delta }= 0$ at the liquid–vapor mixture interface, the above equations becomes

$$\begin{aligned}&\lambda _{\text{ l},\mathrm{s}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}\left( \frac{1}{4}\frac{g(\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty })x^{3}}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}x^{-1}\\[6pt]&=\lambda _{\text{ m}, \mathrm{s}}(t_{\text{ s},\text{ int }}-t_{\infty })\left( \frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}x^{-1}\\[6pt]&\;-2h_\text{ fg } \rho _{\text{ m}, \mathrm{s}}C_{\mathrm{m},\text{ vs }}\sqrt{gx}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/2}\left( \frac{1}{4}\frac{g(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)x^{3}}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

The above equation is divided by $\left( \frac{g}{4x}\right) ^{1/4}$, and simplified to

$$\begin{aligned}&\lambda _{\text{ l},\mathrm{s}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}\\[6pt]&=\lambda _{\text{ m}, \mathrm{s}}(t_{\text{ s},\text{ int }}-t_{\infty })\left( \frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\left( \frac{\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}\\[6pt]&\qquad -4h_\text{ fg } \rho _{\text{ m}, \mathrm{s}}C_{\text{ mv},\mathrm{s}}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/4}\left( \frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{-1/4} W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

i.e.,

$$\begin{aligned}&\lambda _{bad hbox}(t_{\text{ s},\text{ int }}-t_{\infty })\left( \frac{\mathrm{d}\theta _\mathrm{m}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\left( \frac{\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{1/4}\\[6pt]&=\lambda _{\text{ l},\mathrm{s}}(t_\mathrm{w}-t_{\text{ s},\text{ int }})\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s}\left( \frac{\rho _{\mathrm{l},\mathrm{w}}-\rho _{\mathrm{m},\infty }}{{\nu _{\text{ l},\mathrm{s}}}^{2}\rho _{\text{ l},\mathrm{s}}}\right) ^{1/4}\\[6pt]&\qquad +4h_\text{ fg } \rho _{\text{ m}, \mathrm{s}}C_{\text{ vw},\mathrm{s}}(\rho _{\text{ m}, \mathrm{s}}/\rho _{\mathrm{m},\infty }-1)^{1/4}\left( \frac{1}{{\nu _{\mathrm{m},\infty }}^{2}}\right) ^{-1/4} W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

Therefore,

$$\begin{aligned}&\left( \frac{\mathrm{d}\theta _\mathrm{m} }{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s} \nonumber \\&=\frac{\lambda _{\text{ l},\mathrm{s}} (t_\mathrm{w} -t_{\text{ s},\text{ int }})\left( \frac{\nu _{\mathrm{m},\infty } }{\nu _{\text{ l},\mathrm{s}} }\right) ^{1/2}\left( \frac{\rho _{\mathrm{l},\mathrm{w}} -\rho _{\mathrm{m},\infty } }{\rho _{\text{ l},\mathrm{s}} }\right) ^{1/4}(\rho _{\text{ m}, \mathrm{s}} /\rho _{\mathrm{m},\infty } -1)^{-1/4}\left( \frac{\mathrm{d}\theta _\mathrm{l}}{\mathrm{d}\eta _\mathrm{l}}\right) _\mathrm{s} +4h_\text{ fg } \rho _{\text{ m}, \mathrm{s}} C_{\text{ mv},\mathrm{s}} \nu _{\mathrm{m},\infty } W_{y{\text{ m},\mathrm{s}}} }{\lambda _{\text{ m}, \mathrm{s}} (t_{\text{ s},\text{ int }} -t_\infty )}\nonumber \\ \end{aligned}$$

(18.44)

1.3.6 Similarity Transformation of Eq. (18.13)

With Eqs. (18.19) and (18.24), Eq. (18.13) can easily be changed to

$$\begin{aligned} \theta _\mathrm{l}=0, \quad \theta _\mathrm{v} = 1 \end{aligned}$$

(18.40)

1.3.7 Similarity Transformation of Eq. (18.14)

With Eq. (18.27), Eq. (18.14) is easily transformed into

$$\begin{aligned} \Gamma _{\mathrm{mv}} =1 \end{aligned}$$

(18.41)

1.3.8 Similarity Transformation of Eq. (18.15)

With Eqs. (18.20), (18.21), (A40), and (A44), Eq. (18.15) is changed to

$$\begin{aligned}&\rho _{\text{ m}, \mathrm{s}}C_{\text{ mv},\mathrm{s}}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/2}\cdot 2\cdot x^{-1}\nu _{\mathrm{m},\infty }2\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{-1/4} \left( \!\frac{1}{4}\eta _{\mathrm{m}\delta }W_{x\mathrm{m},\mathrm{s}}-W_{y{\text{ m},\mathrm{s}}}\!\right) \\[6pt]&=D_\mathrm{v}\rho _{\text{ m}, \mathrm{s}} \left( \frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\left( \frac{1}{4}{\text{ Gr }}_{x\mathrm{m},\infty }\right) ^{1/4}x^{-1} \end{aligned}$$

i.e.,

$$\begin{aligned} \rho _{\text{ m}, \mathrm{s}}C_{\text{ mv},\mathrm{s}}\nu _{\mathrm{m},\infty }(\eta _{\mathrm{m}\delta }W_{x\mathrm{m},\mathrm{s}}-4W_{y{\text{ m},\mathrm{s}}} )=D_\mathrm{v}\rho _{\text{ m}, \mathrm{s}}\left( \frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}\end{aligned}$$

Since $\eta _{\mathrm{m}\delta } = 0$ at the liquid–vapor mixture interface, we have

$$\begin{aligned} \left( \frac{\mathrm{d}C_{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}=-4\frac{C_{\text{ mv},\mathrm{s}} \nu _{\mathrm{m},\infty } }{D_\mathrm{v} }W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

With Eq. (18.27), the above equation becomes

$$\begin{aligned} \left( \frac{\mathrm{d}\Gamma _{\text{ mv }}}{\mathrm{d}\eta _\mathrm{m}}\right) _\mathrm{s}=-4\frac{C_{\text{ mv},\mathrm{s}}}{(C_{\text{ mv},\mathrm{s}} -C_{\mathrm{m\nu},\infty })}\frac{\nu _{\mathrm{m},\infty }}{D_\mathrm{v}}W_{y{\text{ m},\mathrm{s}}} \end{aligned}$$

(18.42)

1.3.9 Similarity Transformation of Eq. (18.16)

At $y\rightarrow \infty$, with Eqs. (18.24), (18.25), and (18.27), Eq. (18.16) is easily transformed into

$$\begin{aligned} \eta _\mathrm{m} \rightarrow \infty : \quad W_{x\mathrm{m}} =0, \quad \theta _\mathrm{m} =0, \quad \Gamma _{\mathrm{m}\mathrm{v},\infty } =0 \end{aligned}$$

(18.43)