Erratum to: The effect of Knudsen layers on rarefied cylindrical Couette gas flows

1 Erratum to: Microfluid Nanofluid (2013) 14:31–43 DOI 10.1007/s10404-012-1019-2

The original publication of the article contains errors which need to be amended as mentioned below.

The text immediately after Eq. (8) should read as:

“where p(r) describes the probability a molecule will experience a collision while travelling a distance r.”

The corrected version of Fig. 1b is given here.

Fig. 1

a A gas molecule outside a solid cylinder and situated at a radial distance r from the centre of the cylinder of radius R ₁. R ⁻ is the travelling distance limit for a molecule moving towards the cylinder surface, for a given zenith angle θ ⁻. The largest travelling distance R ⁻_u is achieved for the zenith angle direction θ ⁻_u , above which the molecule by-passes the cylinder surface and travels into the bulk. b A gas molecule inside a cylindrical cavity of radius R ₂, at a wall normal distance of R ₂ − r, where r is the radial distance of the molecule from the centre of the cylinder. The molecule has a traveling distance of R ⁺ to the wall for a traveling direction of θ ⁺, where θ ⁺ is varied from 0 to π

The corrected version of Eq. (19) is given below:

$$ R_{2}^{2} = r^{2} + (R^{ + } )^{2} + 2rR^{ + } \cos (\theta^{ + } ), $$

After Eq. (20), in the second line of the paragraph text should read as:

“Using half symmetry, it is sufficient to integrate $\theta^{ + }$ from 0 to π.”

The corrected versions of Eqs. (21) to (24) can be written as follows:

$$ \lambda_{\text{eff(conc)}} = \lambda \left[ {1 - \frac{1}{\pi }\int\limits_{0}^{\pi } {\left( 1 + \frac{{R^{ + } (r,\theta^{ + } )}}{a} \right)^{(1 - n)} {\rm d}\theta^{ + } } } \right], $$

(21)

$$ \beta_{({\rm i})} = \frac{\lambda_{\text{eff(conc)}}}{\lambda } = 1 - \frac{1}{\pi }\int\limits_{0}^{\pi } {\left( {1 + \frac{{R^{ + } (r,\theta^{ + } )}}{a}} \right)^{(1 - n)} {\rm d}\theta^{ + } } , $$

(22)

$$ \lambda_{\text{eff}} = \lambda_{\text{eff(conv)}} \left( {\frac{{\theta_{u}^{ - } }}{\pi }} \right) + \lambda_{\text{eff(conc)}} \left[ {1 - \left( {\frac{{\theta_{u}^{ - } }}{\pi }} \right)} \right], $$

(23)

$$ \begin{gathered} \beta = \left( {\frac{{\theta_{u}^{ - } }}{\pi }} \right)\left[ {1 - \frac{1}{{\theta_{u}^{ - } }}\int\limits_{0}^{{\theta_{u}^{ - } }} {\left( {1 + \frac{{R^{ - } (r,\theta^{ - } )}}{a}} \right)}^{(1 - n)} {\rm d}\theta^{ - } } \right] + \hfill \\ \left[ {1 - \left( {\frac{{\theta_{u}^{ - } }}{\pi }} \right)} \right]\left[ {1 - \frac{1}{{\theta_{u}^{ + } }}\int\limits_{0}^{{\theta_{u}^{ + } }} {\left( {1 + \frac{{R^{ + } (r,\theta^{ + } )}}{a}} \right)}^{(1 - n)} {\rm d}\theta^{ + } } \right], \hfill \\ \end{gathered} $$

(24)

In Sect. 2.3, Line 10, the inline equation should be replaced by

$$ [1 - (\theta_{u}^{ - } /\pi )] $$

In Sect. 3.1, line 4 in the first paragraph, the text Fig. 1 is replaced by Fig. 5.