# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Particle Motion  
#### 4.2.1 Flow Following Behavior  
Seeding-based measurement techniques rely on the assumption that the seeding particles follow the flow with no slip. Since this assumption is only approximately valid, the particle-fluid interaction leads to a fundamental measurement limit. By neglecting the interaction between the particles and assuming a homogeneous, laminar flow field around the particle, [PERSON] derived the equation of motion for a spherical particle with a homogeneous material (uniform composition) and a smooth surface [147]:  
\[\frac{\pi}{6}d_{\mathrm{P}}^{3}\rho_{\mathrm{P}}\frac{\mathrm{d} \vec{v}_{\mathrm{p}}}{\mathrm{d}t} = -3\pi\eta d_{\mathrm{P}}(\vec{v}_{\mathrm{p}}-\vec{v})+\frac{\pi} {6}d_{\mathrm{P}}^{3}\rho\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}-\frac{1}{2} \frac{\pi}{6}d_{\mathrm{P}}^{3}\rho\frac{\mathrm{d}(\vec{v}_{\mathrm{p}}-\vec{ v})}{\mathrm{d}t} \tag{6}\] \[-\frac{3}{2}d_{\mathrm{P}}^{2}\sqrt{\pi\rho\eta}\int\limits_{t_{0 }}^{t}\frac{\mathrm{d}(\vec{v}_{\mathrm{p}}-\vec{v})}{\mathrm{d}t^{\prime}} \frac{1}{\sqrt{t-t^{\prime}}}\mathrm{d}t^{\prime}+\vec{F}.\]  
The symbols \(\vec{v}_{\mathrm{p}}\), \(\vec{v}\) are the particle and the flow velocity; \(\rho_{\mathrm{P}}\), \(\rho\) are the particle and the fluid density; \(\eta\) is the dynamic fluid viscosity; \(d_{\mathrm{P}}\) the is particle diameter; \(t_{0}\) is the initial time; and \(\vec{F}\) represents the external forces acting on the particle, e.g., the centrifugal force in a vortex, the buoyancy force in a shear flow or the gravitational force [148; 149; 150]. External forces are not considered here. A detailed derivation of the equation of motion is contained for instance in [151]. The condition of no particle interaction is considered to be applicable, if the mean distance between the particles is larger than 1000 times the particle diameter [26]. For \(d_{\mathrm{P}}=1\,\mathrm{\SIUnitSymbolMicro m}\), this means for instance a maximal particle concentration of about \(10^{9}\) /m\({}^{3}\)[152]. For this reason, the condition of no particle interaction is usually applicable. In particular for DGV measurements, however, higher particle concentrations can occur [139], and particle interactions have to be taken into account. For the approximation of the relation between the mean free particle distance and the particle concentration, the space for each particle is considered as a cube with identical dimension. The edge length of the cube can be interpreted as the mean free particle distance. The resulting relation is shown in Figure 7 for two different particle diameter \(d_{\mathrm{P}}\). Figure 7: Mean free distance of the seeding particles divided by the particle diameter \(d_{\mathrm{P}}\) over the seeding particle concentration for two different particle diameters. For gaseous flows, it is usually \(\rho_{\mathrm{P}}\gg\rho\), so that Equation (6) can be simplified according to [PERSON] and [PERSON] [152; 153]. One component of the velocity then follows the relation:  
\[\frac{\mathrm{d}v_{\mathrm{P}}}{\mathrm{d}t}=-\underbrace{\frac{18\eta}{\rho_{ \mathrm{P}}d_{\mathrm{P}}^{2}}}_{=1/\tau_{\mathrm{P}}}(v_{\mathrm{P}}-v) \tag{7}\]  
with \(v_{\mathrm{P}}\), \(v\) as the components of the particle and flow velocity, respectively, and \(\tau_{\mathrm{P}}\) as the characteristic time constant of the particle. The deviation \(\Delta v\) between the particle and the flow velocity for a spherical particle thus reads:  
\[\Delta v=v_{\mathrm{P}}-v=-\tau_{\mathrm{P}}\frac{\mathrm{d}v_{\mathrm{P}}}{ \mathrm{d}t}. \tag{8}\]  
As an example, the time constant amounts to \(3.4\,\mathrm{\SIUnitSymbolMicro s}\) and \(2.3\,\mathrm{\SIUnitSymbolMicro s}\) for the particles listed in Table 1 in air flows at room temperature (\(\eta=15\times 10^{-6}\,\mathrm{Pas}\), \(\rho=1.2\,\mathrm{kg/m^{3}}\)).