An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip

Abstract

The tremendous advances in micro-fabrication technology have brought numerous applications to the field of micro-scale science and engineering in recent decades. Microchannels are inseparable part of microfluidic technology which necessitate knowledge of flow behavior inside microchannels. For gaseous flows, the mean free path of a gas is comparable with characteristic length of a microchannel due to the micro-scale dimension of the channel. So, no-slip velocity assumption on the boundaries of channel is no longer valid, and a slip velocity needs to be defined. Although rigorous modeling of rarefied flows requires molecular solutions, researchers proposed use of slip models for applicability of the continuum equations. In slip-flow regime (i.e. Knudsen numbers up to 0.1), well-known Maxwell’s first-order slip model is applicable. For higher Knudsen numbers, higher-order slip models can be implemented to extend the applicability limit of the continuum equations. In the present study, Langhaar’s assumptions for entrance region of two-dimensional microchannels (microtube, slit-channel and concentric annular microchannel) have been implemented using high-order slip models. Different slip models proposed in the literature have been used and velocity profile, entrance length and apparent friction factor have been obtained in integral forms.

Appendix: Velocity Profile for Concentric Annular Microchannel

Velocity profile is expressed as

$$\begin{aligned} \lambda (\gamma ,\text {q})=\mathcal {A}(\gamma )I_0(\gamma \text {q})+\mathcal {B}(\gamma )K_0(\gamma \text {q}) + \mathcal {C}(\gamma ) \end{aligned}$$

(57)

Using the first-order slip model coefficients $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$ can be defined as:

$$\begin{aligned} \mathcal {A}=\frac{\mathcal {A}_1^{(f)}}{\mathcal {A}_2^{(f)}}, \quad \mathcal {B}=\frac{\mathcal {B}_1^{(f)}}{\mathcal {B}_2^{(f)}},\quad \mathcal {C}=\frac{\mathcal {C}_1^{(f)}}{\mathcal {C}_2^{(f)}} \end{aligned}$$

(58)

Coefficient $\mathcal {A}$ can be expressed as:

$$ \begin{aligned} \begin{aligned} \mathcal {A}_1^{(f)}&= \gamma (m^2-1) \Pi _1\\ \mathcal {A}_2^{(f)}&= \left[ \Delta _1I_1(\gamma m)-\Delta _2I_0(\gamma m)-2I_1(\gamma )\right] \Pi _1 \& \left[ \Delta _1K_1(\gamma m)-\Delta _2K_0(\gamma m)-2K_1(\gamma )\right] \Pi _2 \end{aligned} \end{aligned}$$

(59)

where

(60)

Coefficient $\mathcal {B}$ can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {B}_1^{(f)}&= \gamma (m^2-1) \Pi _2\\ \mathcal {B}_2^{(f)}&= \Pi _3I_1(\gamma m) + \Pi _4I_0(\gamma m) + \Pi _5I_1(\gamma ) + \Pi _6I_0(\gamma )-4/\gamma \\ \end{aligned} \end{aligned}$$

(61)

where

(62)

Coefficient $\mathcal {C}$ can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {C}_1^{(f)}&= \left[ \Delta _5I_1(\gamma m)-\Delta _2I_0(\gamma m)\right] \Pi _7 - \left[ \Delta _5I_1(\gamma )-\Delta _2I_0(\gamma )\right] \Pi _8 \\ \mathcal {C}_2^{(f)}&= -\mathcal {B}_2^{(f)} \end{aligned} \end{aligned}$$

(63)

where

(64)

Using general slip model, coefficients $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$ can also be expressed as:

$$\begin{aligned} \mathcal {A} =\frac{\mathcal {A}_1^{(g)}}{\mathcal {A}_2^{(g)}}, \quad \mathcal {B}=\frac{\mathcal {B}_1^{(g)}}{B_2^{(g)}}, \quad \mathcal {C}=\frac{\mathcal {C}_1^{(g)}}{\mathcal {C}_2^{(g)}} \end{aligned}$$

(65)

Coefficient $\mathcal {A}$ can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {A}_1^{(g)}&= (1-m^2) \Pi _9\\ \mathcal {A}_2^{(g)}&= \left\{ \Delta _6 I_1(\gamma m) + (m^2-1) I_0(\gamma m)-2/\gamma \left[ m I_1(\gamma m) + I_1(\gamma )\right] \right\} \Pi _9\\&\quad +\left\{ \Delta _6 K_1(\gamma m) - (m^2-1) K_0(\gamma m)-2/\gamma \left[ m K_1(\gamma m) + K_1(\gamma )\right] \right\} \Pi _{10} \end{aligned} \end{aligned}$$

(66)

where

(67)

Coefficient $\mathcal {B}$ can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {B}_1^{(g)}&= \Delta _5\left[ \tilde{b}_2I_1(\gamma m) + \tilde{b}_1I_1(\gamma )\right] + \Delta _2\tilde{b}_1\tilde{b}_2\left[ I_0(\gamma m) + I_0(\gamma )\right] \\ \mathcal {B}_2^{(g)}&= \Pi _{11}I_1(\gamma m) + \Pi _{12}I_0(\gamma m) + \Pi _{13}I_1(\gamma ) + \Pi _{14}I_0(\gamma )-4\tilde{b}_1\tilde{b}_2/\gamma \\ \end{aligned} \end{aligned}$$

(68)

where

(69)

The coefficients $b_1$ and $b_2$ are defined in Eq. (38). Coefficient $\mathcal {C}$ can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {C}_1^{(g)}&= \left[ \Delta _5I_1(\gamma m)+\tilde{b}_1\Delta _2I_0(\gamma m)\right] \Pi _{15} + \left[ \Delta _5K_1(\gamma m)+\tilde{b}_1\Delta _2K_0(\gamma m)\right] \Pi _{16} \\ \mathcal {C}_2^{(g)}&= -\mathcal {B}_2^{(g)}\\ \end{aligned} \end{aligned}$$

(70)

where

(71)

Fully-developed velocity profile using first-order slip model slip model can be written as:

$$\begin{aligned} \lambda _{fd}^{(f)}=\frac{\mathcal {Q}_1}{\mathcal {Q}_2} \end{aligned}$$

(72)

where

(73)

and for general slip model, fully-developed velocity can be written as:

$$\begin{aligned} \lambda _{fd}^{(g)}=\frac{\mathcal {S}_1}{\mathcal {S}_2} \end{aligned}$$

(74)

where

(75)

(76)

The coefficients $b_1$ and $b_2$ are defined in Eq. (39), and and .