1 Introduction

During the last decades, the rapid development of the micro-electro-mechanical systems has led to the increasing number of microfluidic devices. The gas-phase microflow in microtubes has a broad range of application, such as extracting biological samples, cooling integrated circuits, detecting air-borne pollutants and generating dangerous or expensive chemicals (Barber and Emerson 2006, Tang et al. 2008). Hence, it is necessary to get a good understanding of the flow in microtubes.

For microscale flows, the typical Reynolds numbers are of the order of 100 to 101 (Gloss and Herwig 2009). In other words, the microscale flow is usually laminar. According to the famous Moody chart (1944), surface roughness plays a significant role in the turbulent flows in tubes while has no effect on the laminar flows. However, Gloss and Herwig (2009) analyzed the influence of wall roughness based on the entropy production considerations and pointed out that the surface roughness effects always existed in laminar flows. The results also showed that when the scales were changed from macro- to microscales, there were no scaling effects. In addition, Gloss and Herwig (2010) experimentally studied the gas flow through channels with heights from 20 to 400 μm. The results demonstrated that the wall roughness that has significant effect on turbulent flows also played a non-negligible role in laminar flows. Many researchers also found that surface roughness influenced the microscale laminar flows by experiments (Cui et al. 2004; Mala and Li 1999; Kandlikar et al. 2005; Tang et al. 2007; Gloss and Herwig 2010; Lin et al. 2014). Tang et al. (2007) summarized the experimental results on friction factors and experimentally analyzed the effect of rarefication, compressibility and surface roughness on the gas flow in microtubes. He pointed out that the rarefication effect resulted in the reduction of the friction factor while surface roughness led to the increase.

It is difficult to control the surface topography of microtubes and quantify its effect on flow characteristics in experiments, then a number of researchers numerically analyzed the flow field taking into account the surface roughness (Rawool et al. 2006; Natrajan and Christensen 2010; Celata et al. 2007). In general, the rough surfaces are described as periodic and random types (Zhang et al. 2012a), as listed in Table 1. To simply model the roughness effect on micro-flows, some researchers represented the surface roughness as periodic distributions, including rectangular, triangular, sinusoidal and circular surface roughness (Ji et al. 2006; Cao et al. 2006; Herwig et al. 2008, 2010; Liu et al. 2011; Sun et al. 2012; Rovenskaya and Croce 2013; Rovenskaya 2013; Noorian et al. 2014; Wagner and Kandlikar 2012; Dharaiya and Kandlikar 2013). Herwig et al. (2008) proposed a useful approach to understanding the surface roughness effect on friction factors by taking the entropy production into account. Their work provided a theoretical background to the famous Moody chart, which was mainly based on experimental results. Rovenskaya and Croce (2013) modeled the roughness by a series of triangular obstructions and investigated the flow filed using a coupling approach by decomposing the computational physical domain into kinetic and continuum subdomains. Wagner and Kandlikar (2012) systematically quantified the effect of structured roughness geometries modeled by sinusoidal waves on friction factors in the laminar and turbulent flows. However, most microtube fabrication strategies generate random surface topography (Jaeger et al. 2012). Then, several researchers paid attention to modeling random surface roughness. Croce and D’Agaro (2004) explicitly modeled the surface roughness through a set of random generated peaks along an ideal smooth surface in the microtubes. Chen et al. (2009) introduced the fractal geometry to characterize the multi-scale self-affine surface profile of microtubes and revealed the effect of roughness on flow characteristics.

Table 1 Overview of several descriptions of periodic and random rough surfaces
Full size table

It should be noted that a number of references focused on the surface roughness effect, including roughness height (Dharaiya and Kandlikar 2013; Heck and Papavassiliou 2013), roughness distribution (Dharaiya and Kandlikar 2013) and roughness shape (Liu et al. 2011), but without consideration of the rarefication effect. Rarefication effect is a key factor for the gas-phase microflow, and it can be characterized by the Knudsen number Kn, which is defined as the ratio of the mean free path of molecules λ to the characteristic length L s . It is accepted that the rarefied flow can be generally divided into four regimes according to the Knudsen number as follows: the continuum regime (Kn < 0.001), the slip flow regime (0.001 < Kn < 0.1), the transition regime (0.1 < Kn < 10) and the free molecular regime (Kn > 10). In the slip flow regime, the rarefied gas can be modeled by continuum methods, such as the Navier–Stokes equation with the velocity slip boundary condition (Shams et al. 2009; Konh and Shams 2014), or particle-based methods, such as the lattice Boltzmann method (Zhang et al. 2014) and the molecular dynamics (Cao et al. 2006). For the rarefied gas flow in microtubes, the effect of surface roughness is more complex and difficult to measure (Ji et al. 2006). Ji et al. (2006) studied the two-dimensional (2D) compressible gas flow in the slip regime using the computational fluids dynamics to analyze the effect of surface roughness, which was simulated by rectangular elements. The results showed that roughness effect led to an increase in the Poiseuille number with increasing roughness height and decreasing element spacing. Cao et al. (2006) investigated the effect of surface roughness on slip flows using the molecular dynamics simulation method. The surface roughness was modeled by triangular, rectangular, sinusoidal and randomly triangular waves. Zhang et al. (2014) conducted a lattice Boltzmann simulation of gas flow in microtubes incorporating the effect of surface roughness as characterized by the fractal Cantor structure to study the temperature jump at rough gas–solid interfaces in the slip flow regime. However, these investigations on the rarefied gas flow usually used the 2D surface profile to describe the 3D surface topography (Cao et al. 2006; Ji et al. 2006; Shams et al. 2009; Zhang et al. 2012a; Chen et al. 2012; Zhang et al. 2014). This simplification is correct when the width of rectangular microchannels is much larger than the height but unsuitable for microtubes whose cross-section is circular. Moreover, the 3D surface topography can mimic the actual surface more exactly than the 2D surface profile. Therefore, recently some researchers have carried much effort to model the 3D rough surface topography although there were many difficulties. Xiong and Chung (2010a) proposed a new bottom-up approach to generate a 3D microtube surface. The surface roughness height obeyed a Gaussian distribution and a bi-cubic Coons patch was used to form the curved surface. Jaeger et al. (2012) obtained the microtube’s surface topography through a 3D Optical Surface Profiler and directly used the data to model the surface roughness at the same spatial resolution. They investigated the effect of 3D surface topography without considering the rarefication effect.

It can be found from the literature that the researchers either focused on the surface roughness effect without taking the effect of rarefication and compressibility into account or studied the coupled effect of rarefication and surface roughness by simplifying the real 3D surface topography. There may be two difficulties in modeling the rarefied gas flow in microtubes incorporating the effect of 3D rough surface topography: (1) for the continuum method, the slip velocity is hard to obtain because of the complex surface textures; (2) for the particle-based method, it costs too much computational resource to model 3D gas flow. In this paper, two key issues are focused on that have not been well addressed by previous investigations. Firstly, an extended slip model suitable for surfaces with an arbitrary normal vector is proposed to characterize the slip velocities at the wall with 3D rough surface topography. In addition, the 3D surface topography is represented by the modified two-variable fractal Weierstrass-Mandelbrot (W-M) function (Ausloos and Berman 1985) to characterize the multi-scale self-affine features. The effect of rarefication, compressibility, relative roughness height and fractal dimension are investigated and discussed.

2 3D random surface topography

It has been well documented that rough surface profiles demonstrate a multi-scale and self-affine property, and the roughness at all magnifications appears qualitatively similar (Majumdar and Bhushan 1990; Majumdar and Tien 1990). In addition, the surface topography is best represented by a non-stationary random structure (Sayles and Thomas 1978). The continuity, self-affine and non-differentiability of the surface profile can be preserved by the one-variable fractal W-M function,

$$R\left( x \right) = G^{d - 1} \sum\limits_{{n = n_{1} }}^{\infty } {\frac{{{ \cos }\left( {2\pi \gamma^{n} x} \right)}}{{\gamma^{(2 - d)n} }}}$$
(1)

where G is the scaling constant, d(1 < d < 2) is the fractal dimension of the surface, γ is a scaling parameter. The surface profile described by Eq. (1) is a simplification of the real 3D fractal surface topography, which can be characterized by a two-variable W-M function given by Yan and Komvopoulos (1998). The function is suitable for the Cartesian coordinate system. And for a cylindrical coordinate system (r, θ, z), it can be written as

$$\begin{aligned} R_{r} (\theta ,z) & = R_{0} + L_{0} \left( {\frac{G}{{L_{0} }}} \right)^{D - 2} \cdot \left( {\frac{\ln \gamma }{M}} \right)^{\frac{1}{2}} \\ & \quad \times \sum\limits_{m = 1}^{m} {\sum\limits_{n = 0}^{{n_{{\rm max} } }} {\gamma^{(D - 3)n} \left( {{ \cos }\varphi_{m,n} - { \cos }\left\{ {\frac{{2\pi \gamma^{n} \left( {(\theta R_{0} - x_{0} )^{2} + z^{2} } \right)^{\frac{1}{2}} }}{{L_{0} }} \cdot { \cos }\left[ {\tan^{ - 1} \left( {\frac{z}{{\theta R_{0} - x_{0} }}} \right) - \frac{\pi m}{M}} \right] + \varphi_{m,n} } \right\}} \right)} } \\ \end{aligned}$$
(2)

where R r is the real radius of the microtube considering the surface roughness, R 0 is the nominal radius, L 0 is the sample length, G is the scaling constant independent of frequency, D (2 < D < 3) is the fractal dimension of the surface, γ is a scaling parameter equaling 1.5 in general (Majumdar and Tien 1990), M is the number of ridges, n is a frequency index, \(n_{{\text{max}}} = \text{int} [\log (L_{0} /L_{s} )/\log (\gamma )]\) represents the upper limit of n, L s is a cutoff length and φ m,n is a random phase. This function provides deterministic means of creating 3D random rough surfaces.

Figure 1 shows the 3D rough surface topographies constructed by the modified two-variable W-M function. The parameter σ is the root-mean-square (rms) roughness, which is defined as below,

$$\sigma = \sqrt {\frac{1}{{\text{Num}}}\sum\limits_{i = 1}^{{\text{Num}}} {(R_{r\_i} - R_{0} )^{2} } }$$
(3)

where Num is the total number of points used to construct the rough surface utilizing a bottom-up approach proposed by Xiong and Chung (2010a). As illustrated in Fig. 1, three features of 3D surface topographies can be found: (1) the rms roughness σ describes the averaged roughness height of the topography. (2) For topographies with the same statistical roughness height σ, the topographies can be very different as the self-affine fractal dimension D varies. That is, D and σ are two independent parameters. (3) The larger the self-affine fractal dimension is, the more frequent the variation of surface roughness over the surface will be. The fractal dimension demonstrates the fractal nature of the real surface topography (Chen et al. 2009) and directly exhibits the irregularity of surface. The fractal surface topography makes it more difficult to obtain the slip velocities, and this problem is solved in the next section.

Fig. 1
figure 1

Different 3D surface topographies constructed by the modified two-variable W-M function with a comparison to the real surface reported by Tang et al. (2007)

Full size image

3 Mathematic model

3.1 The extended slip model

The well-known Maxwell first-order slip model can be written as (Maxwell 1879),

$$U_{s} = \frac{{2 -\upsigma_{t} }}{{\upsigma_{t} }}\lambda \frac{{\text{d}U}}{{{\text{d}}z}}\left| {_{w} } \right.$$
(4)

where U s is the velocity slip at the surface, i.e., the difference between gas velocity adjacent to the wall and the wall velocity, σ t is the tangential momentum accommodation coefficient (TMAC) at the wall, λ denotes the mean free path of gas molecules, and \(\frac{{{\text{d}}U}}{{{\text{d}}z}}\left| {_{w} } \right.\) is the normal gradient of gas velocity at the wall. It is the earliest and most widely used linear slip model (Cao et al. 2009) and is suitable for smooth surfaces because the incoming flow is assumed to be parallel to the wall. For rough surfaces, especially for surfaces with 3D rough topography considered in this paper, the incoming flow near the wall is disturbed by the roughness and the gas slip behavior is more complex.

Figure 2 shows the gas molecules interact with the rough surface. Define the unit normal vector of the surface as \(\overrightarrow {n} = (l,\;m,\;n)\) which points from the surface to the gas. Due to the randomness of the surface topography, the direction of the unit normal vector at any point of the surface is also random. To simplify this problem, the rough surface is regarded as a combination of many smooth surfaces,

$$S = \cup s_{k}$$
(5)

where S denotes the whole rough surface and s k denotes one tiny smooth surface with the unit normal vector \(\overrightarrow {n}_{k} = (l_{k} ,\;m_{k} ,\;n_{k} )\). On each tiny smooth surface, \(\overrightarrow {n}_{k}\) remains the same. As illustrated in Fig. 2, when the incident molecules strike tiny smooth surfaces, a portion (σ t ) is adsorbed and then re-emitted to the space above the wall in all angles equiprobably. This kind of molecules is called the re-emitted molecules and the tangential momentums of these are lost. Another portion (1 − σ t ) experiences direct elastic collisions and is specularly reflected into the space. This kind is called the reflected molecules and the tangential momentums of these are held. Hence, the gas near the wall is made up of three kinds of molecules: the incident molecules, the re-emitted molecules and the reflected molecules. These three kinds of molecules transport momentum to the surface and the slip velocity can be derived based on the conservation of the transported momentum.

Fig. 2
figure 2

The gas–solid interactions at the surface with 3D rough topography

Full size image

Assume that the global Cartesian coordinate system is (x, y, z), and the local Cartesian coordinate system (x k , y k , z k ) is established according to \(\overrightarrow {n}_{k}\): The y k axis is chosen as the\(\overrightarrow {n}_{k}\), and the directions of the other two axis are chosen arbitrarily. The gas velocity in the global system is denoted as (u, v, w) and the one in the local system is denoted as (u k , v k , w k ). In the local system, the total flux of momentum in the normal direction (y k ) is

$$P = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {V_{k} \varphi (U_{k} ,\;V_{k} ,\;W_{k} )f(U_{k} ,V_{k} ,W_{k} ){\text{d}}U_{k} {\text{d}}V_{k} {\text{d}}W_{k} } } }$$
(6)

where (U k , V k , W k ) is the molecular thermal motion velocity, φ(U k , V k , W k ) is the momentum transported by one molecule and f(U k , V k , W k ) is the velocity distribution function. The function φ can be represented by φ = m(u sk  + U k ), φ = m(v sk  + V k ) or φ = m(w sk  + W k ), where m is the mass of a molecule, and (u sk , v sk , w sk ) is the slip velocity in the local system. The total flux P consists of three parts,

$$P = P_{i} + (1 - \sigma_{t} )P_{r} + \sigma_{t} P_{e}$$
(7)

where P i is the momentum transported by the incident molecules, P r is the one transported by the reflected molecules and P e denotes the one transported by the re-emitted molecules. As analyzed above, the reflected molecules hold the momentum, while the re-emitted molecules lose. As a result, P r  = −P i and P e  = 0. Hence, Eq. (7) can be simplified to

$$P = \sigma_{t} P_{i}$$
(8)

where P i can be represented as,

$$P_{i} = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{0} {\int\limits_{ - \infty }^{ + \infty } {V_{k} \varphi (U_{k} ,\;V_{k} ,\;W_{k} )f(U_{k} ,V_{k} ,W_{k} ){\text{d}}U_{k} {\text{d}}V_{k} {\text{d}}W_{k} } } }$$
(9)

The velocity distribution function f in Eqs. (6) and (9) satisfies the Boltzmann equation, and the first-order approximation of f given by Chapman and Cowling (1970) is used here,

$$f = n\left( {\frac{m}{2\pi kT}} \right)^{\frac{3}{2}} e^{{ - \frac{m}{2kT}\left( {U_{k}^{2} + V_{k}^{2} + W_{k}^{2} } \right)}} \left[ {1 - \frac{{4k_{c} \beta^{2} }}{5nk}\left( {\beta^{2} C^{2} - \frac{5}{2}} \right)C_{i} \frac{{\partial }}{{{\partial }x_{i} }}\ln T - \frac{{4\mu \beta^{4} }}{\rho }C_{i}^{o} C_{j} \frac{{{\partial }c_{i} }}{{{\partial }x_{j} }}} \right]$$
(10)

where n is the number density of molecules, k is the Boltzmann constant, \(\beta = {1 \mathord{\left/ {\vphantom {1 {\sqrt {2RT} }}} \right. \kern-0pt} {\sqrt {2RT} }}\) and R is the gas constant, (C 1, C 2, C 3) = (U k , V k , W k ), (x 1, x 2, x 3) = (x k , y k , z k ), (c 1, c 2, c 3) = (u k , v k , w k ) and \(C_{i}^{0} C_{j} = C_{i} C_{j} - C^{2} \delta_{ij} /3\). Due to the isothermal flow, the temperature gradient is neglected. If the gas flow is along the x direction and parallel to the smooth surface, only ∂u k /∂y k is considered and the Maxwell’s model can be derived. However, the random surface topography disturbs the velocity field near the wall. Therefore, all the velocity gradients are taken into account here. By substituting Eq. (10) into Eqs. (6) and (9) and integrating with respect U k , V k and W k , the result can be given as below,

$$P = \mu \left( {\frac{{{\partial }u_{k} }}{{{\partial }y_{k} }}\text{ + }\frac{{{\partial }v_{k} }}{{{\partial }x_{k} }}} \right)$$
(11)
$$P_{i} = \left[ {1 + \frac{{2\mu \beta^{2} }}{3\rho }\left( {\frac{{{\partial }u_{k} }}{{{\partial }x_{k} }} - 2\frac{{{\partial }v_{k} }}{{{\partial }y_{k} }} + \frac{{{\partial }w_{k} }}{{{\partial }z_{k} }}} \right)} \right]\frac{{n\overline{c} }}{4}mu_{s} + \frac{1}{2}\mu \left( {\frac{{{\partial }u_{k} }}{{{\partial }y_{k} }} + \frac{{{\partial }v_{k} }}{{{\partial }x_{k} }}} \right)$$
(12)

Substitute Eqs. (11) and (12) into Eq. (8),

$$\mu \left( {\frac{{{\partial }u_{k} }}{{{\partial }y_{k} }} + \frac{{{\partial }v_{k} }}{{{\partial }x_{k} }}} \right) = \sigma_{t} \left\{ {\frac{1}{2}\mu \left( {\frac{{{\partial }u_{k} }}{{{\partial }y_{k} }} + \frac{{{\partial }v_{k} }}{{{\partial }x_{k} }}} \right) + \left[ {1 + \frac{{2\mu \beta^{2} }}{3\rho }\left( {\frac{{{\partial }u_{k} }}{{{\partial }x_{k} }} - 2\frac{{{\partial }v_{k} }}{{{\partial }y_{k} }} + \frac{{{\partial }w_{k} }}{{{\partial }z_{k} }}} \right)} \right]n\sqrt {\frac{RT}{2\pi }} mu_{sk} } \right\}$$
(13)

In this equation, the term of the left hand means the viscous shear stress at the wall, the first term at the right denotes the viscous shear stress induced by the incident molecules (one half of the total molecules), and the second means the flux of tangential momentum generated by the slip velocity u sk . From Eq. (13), it is easy to obtain u sk ,

$$u_{sk} = \frac{{2 - \sigma_{t} }}{{\sigma_{t} }}\lambda \frac{{\frac{{{\partial }u_{k} }}{{{\partial }y_{k} }}\text{ + }\frac{{{\partial }v_{k} }}{{{\partial }x_{k} }}}}{{1\text{ + }\frac{\mu }{3p}\left( {\frac{{{\partial }u_{k} }}{{{\partial }x_{k} }}\text{ - }2\frac{{{\partial }v_{k} }}{{{\partial }y_{k} }}\text{ + }\frac{{{\partial }w_{k} }}{{{\partial }z_{k} }}} \right)}}$$
(14)

The additive derivative ∂v k /∂x k in the numerator can also be derived from the tangential shear stress at the surface (Lockerby et al. 2004). The additive term in the denominator denotes the ratio of the viscous stress in the y k direction to the static pressure and it is small enough to be neglected. Then, Eq. (14) has a similar form with the 2D slip model proposed by Lockerby et al. (2004). The slip velocity u sk given by Eq. (14) is in the local system and the slip velocity (u s , v s , w s ) can be obtained based on the principle of coordinate transformation. The extended slip model suitable for random rough surfaces is represented as,

$$\begin{gathered} u_{s} = \frac{{2 - \sigma_{t} }}{{\sigma_{t} }}\lambda \left\{ {\frac{\partial }{{\partial y_{k} }}\left[ {(1 - l_{k}^{2} )u - l_{k} m_{k} v - l_{k} n_{k} w} \right] + (1 - l_{k}^{2} )\frac{{\partial v_{k} }}{\partial x} - l_{k} m_{k} \frac{{\partial v_{k} }}{\partial y} - l_{k} n_{k} \frac{{\partial v_{k} }}{\partial z}} \right\} \hfill \\ v_{s} = \frac{{2 - \sigma_{t} }}{{\sigma_{t} }}\lambda \left\{ {\frac{\partial }{{\partial y_{k} }}[(1 - m_{k}^{2} )v - l_{k} m_{k} u - m_{k} n_{k} w] + (1 - m_{k}^{2} )\frac{{\partial v_{k} }}{\partial y} - l_{k} m_{k} \frac{{\partial v_{k} }}{\partial x} - m_{k} n_{k} \frac{{\partial v_{k} }}{\partial z}} \right\} \hfill \\ w_{s} = \frac{{2 - \sigma_{t} }}{{\sigma_{t} }}\lambda \left\{ {\frac{\partial }{{\partial y_{k} }}[(1 - n_{k}^{2} )w - l_{k} n_{k} u - m_{k} n_{k} v] + (1 - n_{k}^{2} )\frac{{\partial v_{k} }}{\partial z} - l_{k} n_{k} \frac{{\partial v_{k} }}{\partial x} - m_{k} n_{k} \frac{{\partial v_{k} }}{\partial y}} \right\} \hfill \\ \end{gathered}$$
(15)

where ∂/∂ ∂y k . ∂y k  = l k  ∂/∂ ∂ x. ∂x + m k  ∂/∂ ∂y. ∂y + n k  ∂/∂ ∂z. ∂z and v k  = l k u + m k v + n k w.

3.2 Governing equations

In order to investigate the effect of 3D surface topography on the gas slip flow in microtubes, three-dimensional pressure-driven compressible gas flow is analyzed. As depicted in Fig. 3, the pressure at the inlet and the outlet are imposed to drive the gas flow through the rough microtubes with a radius of R 0. R 0 is selected as 25.2 μm (Ewart et al. 2006). The Knudsen number of the gas in the microtube is defined as,

$$Kn = \frac{\lambda }{{2R_{0} }}$$
(16)

where λ is the mean free path of gas molecules and can be given by (Ewart et al. 2006)

$$\lambda = k_{\lambda } \frac{\mu }{p}\sqrt {2{\text{RT}}}$$
(17)

where k λ is the coefficient depending on the molecular interaction model, μ is the dynamic viscosity of the gas, p is the static pressure, and T is the temperature. In general, \(k_{\lambda } = {{\sqrt \pi } \mathord{\left/ {\vphantom {{\sqrt \pi } 2}} \right. \kern-0pt} 2}\) (a value close to that obtained from the hard sphere model (Chapman and Cowlin g 1970)) is retained. In the slip regime (0.001 < Kn < 0.1), gas flow is modeled by the N–S equation with the slip boundary condition (Zhang et al. 2012a; Yan and Wang 2009). Thus, the continuity, momentum and energy equations for gas flow in rough microtubes still can be written as:continuity equation:

$$\nabla \cdot (\rho {\mathbf{V}}) = 0$$
(18)

momentum equation (N–S equation):

Fig. 3
figure 3

Schematic of the microtube with an enlarged view of the rough surface topographies

Full size image
$$\frac{\partial }{\partial t}(\rho {\mathbf{V}}) + \nabla \cdot (\rho {\mathbf{VV}}) = - \nabla p + \nabla \cdot (\varvec{\tau})$$
(19)
$$\frac{\partial }{\partial t}(\rho E) + \nabla \cdot ({\mathbf{V}}(\rho E + p)) = \nabla \cdot [k_{c} \nabla T + (\varvec{\tau}\cdot {\mathbf{V}})]$$
(20)

energy equation:

$$E = H - \frac{p}{\rho } + \frac{{{\mathbf{V}}^{2} }}{2}$$
(21)
$$\tau = \mu \left[ {(\nabla {\mathbf{V}} + \nabla {\mathbf{V}}^{T} ) - \frac{2}{3}\nabla \cdot {\mathbf{VI}}} \right]$$
(22)

where ρ is the fluid density, \({\mathbf{V}} = (u,\,\,v,\,\,w)\) is the velocity vector, τ is the viscous stress tensor, E is the total energy, k c is the thermal conductivity and H is the enthalpy. Make the system of equations closed by invoking the ideal gas law equation of state

$$p = \rho RT$$
(23)

The boundary conditions of the gas at the inlet, the outlet and the wall are as below:

$${\text{inlet}}:\;p = p_{\text{in}} ,\;T = T_{\text{in}}$$
(24)
$$\text{outlet}:\;p = p_{{\text{out}}} ,\;T = T_{{\text{out}}}$$
(25)
$$\text{wall}:\;\;u = u_{s} ,\;v = v_{s} ,\;w = w_{s} ,\;T = T_{w}$$
(26)

where p in and p out are the pressure at the inlet and the outlet, T in, T out and T w are the temperature of the inlet, the outlet and the wall, u s , v s and w s are the slip velocities in the x, y and z directions given by Eq. (15). The pressure at the inlet p in is higher than p out to drive the gas and the temperature T in, T out and T w are imposed as the same value to ensure isothermal flows.

The gas flow can be characterized by the Poiseuille number (friction constant), which is defined as

$$Po = F \cdot Re = \frac{{ - 2({\text{d}}\bar{p}/{\text{d}}z) \cdot D_{h}^{2} }}{{\mu \bar{u}}}$$
(27)

where F is the friction factor, Re is the Reynolds number, \({\text{d}}\bar{p}/{\text{d}}z\) is the average pressure gradient in the flow direction, D h is the hydraulic diameter, and \(\overline{u}\) is the average velocity in the cross-section. From Eq. (27), it can be found that the hydraulic diameter has a significant effect on the Poiseuille number. In general, the hydraulic diameter can be written as,

$$D_{h} = \frac{4A}{C}$$
(28)

where A is the area of the cross-section and C is the perimeter. For the smooth microtube, the hydraulic diameter can be expressed as D h  = 2R 0. For the rough microtube, D h should be determined through an appropriate approach. Herwig et al. (2008) summarized several appropriate approaches to describe D h for rough walls and pointed out that the hydraulic diameter D h can be determined using a certain geometrical mean value. Therefore, the mean diameter of the rough microtube can be used as the hydraulic diameter as follows,

$$D_{h} = \begin{array}{*{20}c} {\text{mean}} & {(2R_{r} )} \\ \end{array}$$
(29)

where mean() represents the mean value and R r is the real radius of the rough microtube given by Eq. (2). The same selection to determine the hydraulic diameter was also employed by Xiong and Chung (2010b).

The numerical solution of Eqs. (18)–(23) for the flow field in rough microtubes is obtained by means of the control volume finite-difference technique. The slip velocity at the wall expressed by Eq. (15) is solved through a user-defined function. The work gas is selected as nitrogen. The entire microtube is taken as the computational domain to investigate the effect of 3D surface topography on the flow characteristics. The roughness elements have been integrated with the flow region as a part of the computational domain. To evaluate the effect of mesh grids on the accuracy of numerical solutions, grid-independence check is performed as the mesh grid size is refined until acceptable differences between the last two mesh grid sizes are found. This check made sure that the results presented in this paper are independent of the mesh. The solution is regarded as convergent not only when the residual levels of velocity are below 10−8 but also the slip velocities become constant. The numerical results are verified by comparing to the experimental and theoretical results in different ways to ensure the validity of the numerical analysis. And the comparison is shown in the next section.

4 Results and discussion

4.1 Rarefication effect

Figure 4 depicts the Poiseuille number as a function of the Knudsen number in smooth and rough microtubes. The Knudsen number denotes the mean Knudsen number that is based on the mean pressure p m  = 0.5(p in + p out). The relative roughness height ɛ is defined as ɛ = σ/D h . It is evident that the numerical results of the smooth microtube have a good agreement with the experimental results (Ewart et al. 2006) and the theoretical results (Yan and Wang 2009) in the slip regime (0.003 < Kn < 0.1), which can verify the correctness of this present model. The Poiseuille numbers in both smooth and rough microtubes decrease with the increasing Knudsen number. It indicates that the rarefication effect decreases the frictional drag on the wall. This phenomenon can be explained by the fact that as the Knudsen number increases, the collisions between the gas molecules and the surface becomes weaker (Zhang et al. 2012b). In addition, when the walls are no longer smooth, the Poiseuille numbers are obviously larger. The underlying physical mechanism is the increased dissipation rate near the roughness elements (Gloss and Herwig 2010).

Fig. 4
figure 4

Comparisons of Poiseuille numbers among the present results with the experimental results (Ewart et al. 2006), and theoretical results (Yan and Wang 2009)

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As an important component of the microscale gas flow, the velocity distribution can present the fundamental information for understanding the gas flow in microtubes. Figure 5 shows the velocity distribution along the radial direction in the smooth microtube for different Knudsen numbers. The velocities in the figure are normalized by the maximum velocity in the cross-section, in other words, v* = V/V max. The velocity profiles show parabolic shapes with different degrees of slip. As the Knudsen number increases, the slip velocities increase remarkably. For example, for Kn = 0.003, the gas slip velocity at the wall is too small to be observed. And for Kn = 0.1, the slip velocity approaches to 0.4. Moreover, the whole velocity level for a larger Knudsen number is higher than that for a smaller Knudsen number. This trend indicates that rarefication effect increases the gas slip and the whole velocity level.

Fig. 5
figure 5

The normalized velocity distribution across the smooth microtube under different Knudsen numbers with a partial enlarged view

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4.2 Surface topography effect

The streamlines and the local velocity field in rough microtubes with different roughness height and fractal dimensions are illustrated in Fig. 6. As seen from this figure, the 3D surface topography observably perturbs the local gas flow. As a result, the streamlines adjacent to the walls are twisted into the shape of skew curves. In the valley, the streamlines are expanded, which implies the obstruction effect of the roughness. In addition, in microtubes with higher roughness height or fractal dimension, the surface topography effect is more evident.

Fig. 6
figure 6

The streamlines in smooth and rough microtubes with different roughness heights σ and fractal dimensions D (Kn = 0.1)

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Figure 7 shows the slip velocities normalized by the maximum velocity at the inlet to further analyze the effect of surface topography on the boundary interaction. As depicted in Fig. 7, the slip behavior is significantly affected by the topography. There exists a good corresponding relationship between the peak (or valley) values of surface topography and the valley (or peak) values of slip velocities. The slip velocities in the valley are so small that the gas at these locations can be regarded as no-slip (Zhang et al. 2012b). However, the gas slips evidently on the peak of the rough wall. This phenomenon indicates that the obstruction effect in the valley of surface topography is larger than that on the peak.

Fig. 7
figure 7

The 3D surface topography and the slip velocity distribution on the surface (D = 2.8, ɛ = 3 %) with an enlarged drawing to demonstrate the corresponding relationship between the surface topography and the slip velocities

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Figure 8 depicts the distribution of the normalized velocity across smooth and rough microtubes for different Knudsen numbers. This figure reveals the coupled effect of rarefication and surface topography on the gas flow. By comparing the four plots in Fig. 8 in turn, it can be found that as the rarefication effect increases, the surface roughness effect becomes more significant. When the Knudsen number is so small that the gas flow approaches to the continuum regime, the surface topography has no obvious effect on the velocity distribution, as illustrated in Fig. 8a. However, for Kn = 0.04 and Kn = 0.1, the effect of surface topography is so evident: on one hand, the presence of surface roughness reduces the gas slip near the surface and the whole velocity level. On the other hand, the velocity profiles affected by the surface topography cannot hold symmetry. For Kn = 0.1, The normalized slip velocity in the smooth microtube is 0.38 both at y/R = −1 and y/R = 1. But for the rough microtube with ɛ = 3 %, the normalized slip velocity at y/R = −1 is 0.16 while at y/R = 1 is 0.25. This phenomenon is due to the randomness of the 3D surface topography. In the rough microtube, surface roughness induces velocity components normal to the wall, and as a result, the term ∂v/ ∂x should be considered (Colin 2012). Figure 9 shows the effect of ∂v/ ∂x n the velocity distribution. For the smooth microtube, ∂v/ ∂x has no effect. However, for the rough microtube with ɛ = 3 %, ∂v/∂x reduces the slip velocity at y/R = −1 about 20 %.

Fig. 8
figure 8

The coupled effect of rarefication and surface topography on the velocity distribution in the radial direction of microtubes. a Kn = 0.003. b Kn = 0.02. c Kn = 0.04. d Kn = 0.1

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Fig. 9
figure 9

The effect of ∂v/ ∂x on the velocity distribution in the smooth and rough microtubes (Kn = 0.1)

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The variation of velocity induced by surface topography may affect the pressure drop along the microtubes. Figure 10 shows the gas pressure drop along the smooth and rough microtubes and the pressure contours in the longitudinal sections. As illustrated in the figure, surface roughness increases the pressure drop in rough microtubes (Duan and Muzychka 2010). And the increase both in roughness height and fractal dimension leads to the increase in pressure drop. In addition, the pressure drop in rough microtubes is not in a linear pattern as that in the smooth microtube but rather fluctuates. Moreover, the magnitude of fluctuation is larger for a rough surface with a higher roughness height or fractal dimension. This phenomenon is induced by the surface topography that can be observed intuitively from the pressure contours. The pressure distribution near the wall is significantly affected by the surface topography. In the region away from the rough surface, the pressure distribution across the microtube comes to be nearly in a linear pattern. In other words, surface topography mainly affects the pressure distribution in the near-wall region but has no evident impact on the flow away from the surface.

Fig. 10
figure 10

The pressure drop along the flow direction and the pressure contours in the longitudinal sections of the smooth and rough microtubes with different a surface roughness heights and b fractal dimensions

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Figure 11 depicts the surface topography effect on the Poiseuille number. For lower Knudsen numbers (Kn = 0.003 and Kn = 0.01), the Poiseuille number increases slightly with the roughness height and the fractal dimension. And for higher Knudsen numbers (Kn = 0.04 and Kn = 0.1), the Poiseuille number increases rapidly. That is, surface topography effect becomes more significant on rarefied gas flows when the Knudsen number is increased (Karniadakis and Beskok 2002). This can be explained according to the molecule collisions, which include the collisions between molecules and the rough surface (c w ) and the inter-collisions between molecules (c m ). For a low Knudsen number, the frequency of c m is more often than that of c w and in the continuum regime c w is not considered. As the Knudsen number increases, c w comes to be dominant. Therefore, the effect of surface topography increases with the increasing rarefication effect. Figure 11 also reveals the effect of ∂v/ ∂x on the Poiseuille number. The term of ∂v/∂x is of ϑ(Kn 2) (Colin 2012) and the velocity components normal to the wall are induced by the surface topography. Hence, as seen from Fig. 11, the effect of ∂v/∂x increases not only with the increasing Knudsen number but also with the roughness height and the fractal dimension. For example, for Kn = 0.1, the increase in the Poiseuille number resulted from ∂v/∂x is 4, 5.2, 6.2 and 7.8 % for D = 2.2, D = 2.4, D = 2.6 and D = 2.8, respectively. For the rough microtube with D = 2.8, the increase is 0.25, 1, 2.8 and 7.8 % for Kn = 0.003, Kn = 0.01, Kn = 0.04 and Kn = 0.1.

Fig. 11
figure 11

The effect of the a roughness heights and b fractal dimensions on the Poiseuille numbers in rough microtubes with a comparison between results considering ∂v/∂x and those neglecting ∂v/∂x

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4.3 Compressibility effect

Figure 12 demonstrates the variation of Poiseuille numbers with the Mach numbers in the smooth microtube. The numerical results without consideration of the rarefication effect have a good agreement with the experimental results (Tang et al. 2007) and the theoretical results (Guo and Li 2003). It can be found that the Poiseuille number increases with the increasing Mach number and the decreasing Knudsen number. Furthermore, the Poiseuille number is less sensitive to the Mach number for a higher Knudsen number. For example, the rate of change of the Poiseuille number to the Mach number at Ma = 0.25 is about 31.6 for no-slip, 18.3 for Kn = 0.006 and 10.8 for Kn = 0.01, respectively. This is due to the coupled effect of compressibility and rarefication. On one hand, the compressibility effect increases the dimensionless velocity gradient at the wall and consequently leads to the increment of the Poiseuille number (Guo and Wu 1997). On the other hand, the increase in the velocity gradient can induce slip velocities becoming larger and decrease the Poiseuille number. Figure 13 illustrates the Poiseuille numbers in rough microtubes. The increase in the fractal dimension leads to the increment of Poiseuille numbers. In addition, a larger fractal dimension makes the Poiseuille number increase with the Mach number more rapidly.

Fig. 12
figure 12

The coupled effect of rarefication and compressibility on the Poiseuille number in the smooth microtube with a comparison to the experimental results (Tang et al. 2007) and theoretical results (Guo and Li 2003)

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Fig. 13
figure 13

The variation of Poiseuille numbers with the Mach numbers in smooth and rough microtubes for different Knudsen numbers

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4.4 Tangential momentum accommodation coefficient (TMAC)

Tangential momentum accommodation coefficient is a key factor for the slip model, and many researchers investigated TMAC experimentally by measuring the mass flow rate (Yamaguchi et al. 2011; Arkilic et al. 2001; Maurer et al. 2003; Colin et al. 2004; Ewart et al. 2007). Define the dimensionless mass flow rate S as follows,

$$S = {{Q_{m} } \mathord{\left/ {\vphantom {{Q_{m} } {\frac{{\pi \varDelta pp_{m} }}{8\mu RTL}}}} \right. \kern-0pt} {\frac{{\pi \varDelta pp_{m} }}{8\mu RTL}}}(R_{0} )^{4}$$
(30)

where Q m is the mass flow rate, Δp = p in − p out. If the Maxwell’s slip model is taken into account, the relationship can be given by,

$$S = 1 + 8\frac{{2 - \sigma_{t} }}{{\sigma_{t} }}Kn$$
(31)

From Eq. (28) and (29), it demonstrates that TMAC can be obtained by measuring the mass flow rate. Figure 14 shows the relationship between the dimensionless mass flow rate S and the Knudsen number Kn in the smooth and rough microtubes. The TMAC is chosen as the value measured from the experiment (Ewart et al. 2006), and then, the numerical results have a good agreement with the experimental ones. For a lower Knudsen number (about Kn < 0.02), surface roughness has no obvious effect on the dimensionless mass flow rate. However, as the Knudsen number increases, the decrease in the mass flow rate induced by surface roughness becomes more notable. Furthermore, the relationships between Kn and S influenced by different surface topographies are still nearly linear and they can be well fitted by straight lines as shown in Fig. 14.

Fig. 14
figure 14

The variation of the dimensionless mass flow rate with the Knudsen number in the smooth and rough microtubes with a comparison to the experimental results (Ewart et al. 2006)

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From the fitted lines, the TMACs affected by surface topography are obtained and they are compared with the results measured in experiments, as listed in Table 2. It can be found that the surface topography has a significant effect on the TMAC. In general, it is suggested that if the relative surface roughness is less than 1 %, the effect of surface roughness on gas flow in microtubes can be neglected and the conventional laminar prediction can be applied (Tang et al. 2007; Dai et al. 2014). However, the surface with a relative roughness height of 1 % makes the TMAC increase about by 10 %. In addition, there are obvious differences in the experimentally measured TMACs listed in Table 2, and this may be due to the different surface topographies of the tested microtubes.

Table 2 Comparison of TMACs between the present work and the experimental measurements for nitrogen reported in literature
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Figure 15 shows the relationship between the surface roughness and the TMACs. The increase in the roughness height and the fractal dimension can both result in the increment of TMACs. Chew (2009) summarized the measured TMAC in the literature and pointed out that most of the TMACs were below unity, but the TMAC increased above unity for rough surfaces. And in Fig. 15, it is notable that the TMAC for the microtube with ɛ = 3 % and D = 2.6 is slightly larger than unity. Because TMAC is very sensitive to surface roughness, the tested microtubes in experiments should be fabricated as smooth as possible to get more actual TMACs.

Fig. 15
figure 15

The effect of surface topography on TMAC

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5 Conclusions

A 3D model of gaseous slip flow in microtubes with random surface topographies is developed and analyzed numerically. The modified two-variable W-M function is employed to represent the 3D rough topography. In order to characterize the slip velocities at the rough wall exactly, an extended slip model is proposed and applied to the numerical analysis. The flow field is obtained by solving the Navier–Stokes equation with the extended slip model. The effect of surface topography, rarefication and compressibility are investigated. The influence of surface roughness on TMAC is also analyzed. The results show that the rarefication effect can increase the slip velocity at the wall and consequently decrease the Poiseuille number. The rough surface topography induces the Poiseuille number increasing and the increase in the roughness height and fractal dimension can both lead to higher Poiseuille numbers. With respect to the compressibility effect, as the Mach number increases, the Poiseuille number becomes larger. Moreover, the effect of surface topography, rarefication and compressibility are coupled: (1) the effect of surface roughness increases with the increasing rarefication effect; (2) the rarefication effect reduces the effect of compressibility and (3) a more rough surface makes the Poiseuille number more sensitive to the Mach number. The additive derivative ∂v/∂x in the proposed slip model is discussed. The effect of ∂v/∂x increases with the increasing Knudsen number and the irregularity of the rough surfaces. Last but not least, TMAC is influenced significantly by surface topography, and with increasing roughness height and fractal dimension, TMAC increases rapidly. Hence, the surface roughness effect on TMAC should be considered in experiments.