1 Introduction
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Continuum regime (\(\text {Kn} < 0.01\)), where the continuum assumption holds and the Navier–Stokes (N–S) equations with no-slip boundary conditions are applied in numerical solutions of gas particle flows.
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Slip flow regime (\(0.01< \text {Kn} < 0.1\)), where the velocity at the particle boundary no longer satisfies the no-slip conditions; hence, the slip flow regime is observed. The fluid flow can still be resolved by solving the Navier–Stokes equations by applying the slip velocity boundary conditions.
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Transitional regime (\(0.1< \text {Kn} < 10\)), where the continuum theory together with slip conditions begin to break down.
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Free molecular regime (\(\text {Kn} > 10\)), where the continuum theory is no longer valid.
2 Drag force on a spherical microparticle
3 Numerical solution of 3D flow past microparticle
3.1 Slip boundary condition
Authors | \({\left. u_{\tau } \right| _w}\) | \({\beta }\) |
---|---|---|
Maxwell [35] (orig.) | \(\left. u_{\tau } \right| _w \approx \alpha \tau _{\tau }\) | |
\(\rightarrow \) conventional: | \( \left. u_{\tau } \right| _w = \left. \text {Kn}\,d_p \frac{du_\tau }{dn}\right| _w\) | \(\beta _{MWC} = \text {Kn}\) |
\(\rightarrow \) generalised: | \(\left. u_{\tau } \right| _w =\left. \text {Kn}\,d_p \;\left[ \frac{du_\tau }{dn} + \frac{du_n}{dt} \right] \right| _{w}\) | \(\beta _{MWG} = \text {Kn}\) |
Schaaf and Chambre [9] | \(\left. u_{\tau } \right| _w = \left[ \frac{2-\sigma }{\sigma } \right] \left. \text {Kn}\,d_p \frac{du_\tau }{dn}\right| _w\) | \(\beta _{SC} = \frac{2-\sigma }{\sigma }\;\text {Kn}\) |
Pan et al. [20] | \(\left. u_{\tau } \right| _w = 1.1254\;\text {Kn}\,d_p \left. \frac{du_\tau }{dn}\right| _w\) | \(\beta _{P}= 1.1254\,\text {Kn}\) |
Barber et al. [19] | \(\left. u_{\tau } \right| _w = \left[ \frac{2-\sigma }{\sigma } \right] \left. \text {Kn}\,d_p\;\left[ \frac{du_\tau }{dn} + \frac{du_n}{dt} \right] \right| _{ w}\) | \(\beta _{B} = \frac{2-\sigma }{\sigma }\;\text {Kn}\) |
Sharipov [25] | \(\left. u_{\tau } \right| _w = \sigma _{p_{S}}\left. \text {Kn}\,d_p \frac{du_\tau }{dn}\right| _w\,,\) | \(\beta _{S} = \sigma _{p_{S}}\;\text {Kn}\) |
\(\sigma _{p_{S}} = {\left\{ \begin{array}{ll} 1.0,\text {diffuse scattering}\\ \frac{1.772-0.754 \; \sigma }{\sigma },\text {diffuse specular scattering} \end{array}\right. } \) |
3.2 Geometric model and computational domain
Mesh ID | Total cell count | Cube el.\(^\mathrm{a}\) | \({C_d}\cdot \text {Re}_{p,1}\) | \({C_d}\cdot \text{ Re}_{p,2}\) |
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M0 | 13,176 | 9 | 24.498 | 24.344 |
M1 | 24,136 | 16 | 24.532 | 24.386 |
M2 | 62,664 | 36 | 24.605 | 24.466 |
M3 | 226,890 | 100 | 24.654 | 24.520 |
M4 | 553,658 | 196 | 24.676 | 24.542 |
M5 | 1,096,828 | 324 | 24.680 | 24.548 |
M6 | 1,914,578 | 484 | 24.684 | 24.552 |
3.3 Verification of the numerical model: the continuum flow over a sphere
3.4 Discussion on conventional and generalised Maxwell model
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Conventional Maxwell model: \(C_{c,{conv}}=\left[ 1+ 2\sigma _p\lambda /d\right] /\left[ 1+ 4\sigma _p\lambda /d\right] \,,\)
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Generalised Maxwell model: \(C_{c,{gen}}=\left[ 1+ 4\sigma _p\lambda /d\right] /\left[ 1+ 6 \sigma _p\lambda /d\right] \,.\)
No. | Particles | Fluid | Parameter A | Author |
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1 | Oil drops | Air | \(1.155+0.471\,\exp (-0.596/[2\mathrm {Kn}])\) | [11] |
2 | Oil drops | Air | \(1.209+0.441\,\exp (-0.781/[2\mathrm {Kn}])\) | [16] |
3 | Oil drops | Air | \(1.099+0.518\,\exp (-0.425/[2\mathrm {Kn}])\) | [13] |
4 | PSL | Air | \(1.142+0.558\,\exp (-0.999/[2\mathrm {Kn}])\) | [12] |
5 | PSL | Air | \(1.231+0.469\,\exp (-1.178/[2\mathrm {Kn}])\) | [14] |
6 | PSL | Air | \(1.165+0.480\,\exp (-1.001/[2\mathrm {Kn}])\) | [15] |
4 Computational procedure to determine the \(\beta =\beta (\mathrm {Kn})\) relationship
5 Results and analysis
5.1 The single slip correction case
Author | Polynomial model (\(b_0 = 0\)) | |||
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Fitted param. | \({{b_4}}\) | \({{b_3}}\) | \({{b_2}}\) | \({{b_1}}\) |
Rader [16] | 1137 | − 94.62 | 10.04 | 1.244 |
Allen and Raabe [11] | 1030 | − 77.88 | 8.310 | 1.209 |
Buckley and Loyalka [13] | 1052 | − 71.14 | 7.292 | 1.156 |
Hutchins et al. [14] | 1048 | − 79.39 | 9.669 | 1.274 |
Allen and Raabe [12] | 679.0 | − 36.88 | 6.413 | 1.219 |
Jung et al. [15] | 754.6 | − 44.93 | 7.097 | 1.236 |
5.2 The \(\beta \) analysis
Model nr. | Particle group | Polynomial model (\(b_0 = 0\)) | |||
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\({{b_4}}\) | \({{b_3}}\) | \({{b_2}}\) | \({{b_1}}\) | ||
1 | Oil droplets in air\(^\mathrm{a}\) | 1073 | − 81.21 | 8.548 | 1.203 |
2 | PSL spheres in air\(^\mathrm{b}\) | 827.1 | − 53.73 | 7.727 | 1.243 |
3 | Ensemble data\(^\mathrm{a, b}\) | 950.0 | − 67.47 | 8.137 | 1.223 |