# MEMS Vaporizing Liquid Microthruster: A Comprehensive Review
## 4 Instability Control into VLMs
### Flow Boiling Instabilities in Microchannels  
Because of the more rapid growth and collapse of the vapor bubbles in a confined space, boiling in microchannels is more prone to flow instabilities than macroscale flow boiling. This causes significant pressure fluctuations, which alter the local state of the flow affecting the heat transfer process. Consequently, additional undesired effects are triggered, such as a premature critical heat flux, thermal stress, and mechanical vibration in the microchannels. A first assessment of the confinement effect was provided in [27], suggesting the following relation:  
\[D_{bubble}=0.0208\ \theta\ \sqrt{\frac{\sigma_{t}}{g_{0}(\rho_{v}-\rho_{l})}} \tag{1}\]  
where \(\theta\) is the wetting angle expressed in degrees, \(g_{0}\) = 9.81 m/s\({}^{2}\) is the gravitational acceleration2, \(\rho_{l}\) and \(\rho_{v}\) are the liquid and vapor densities in kg/m\({}^{3}\), and \(\sigma_{t}\) is the surface tension in N/m. The equation above can be used to verify whether the theoretical diameter of the bubbles is larger than the diameter of the duct and, therefore, quantify the entity of the confinement effect. Dimensions numbers have also been proposed to indicate the relative strength of the different forces playing a key role in multiphase flows in microchannels:  
* Eotvos number \(Eo=\frac{g_{0}(\rho_{v}-\rho_{l})D_{h}^{2}}{\sigma_{t}}\), which measures the importance of gravity compared to the surface tension force;
* Reynolds numbers \(Re=\frac{GD_{h}}{\mu}\), which compares inertial force to the viscous force;  
Figure 7: Explosive boiling phenomena [20]. * Weber number \(We=\frac{G^{2}D_{h}}{\alpha_{l}\rho_{l}}\), which estimates the balance between the inertial force and surface tension force;
* Capillary number \(Ca=\frac{\mu G}{\alpha_{l}}\), which quantifies the viscous force in relation to the surface tension force. where \(G\) is the mass flux in kg/m\({}^{2}\)/s, \(D_{h}\) is the hydraulic diameter of the microchannel in m, and \(\mu\) is the dynamic viscosity in Pa-s. When considering the Eotvos number, the possibility of the occurrence of instability increases as this one approaches zero. A new parameter was then proposed in [28], i.e., the instability parameter \(K_{inst}\) is expressed as:  
\[K_{inst}=\sqrt{\frac{F_{kk}}{F_{rad}}} \tag{2}\]  
where \(F_{kk}\) is the backward evaporation momentum and \(F_{fund}\) is the forward liquid inertia force:  
\[F_{kk}=\frac{1}{4\rho_{l}A_{cs}}\Bigg{(}\frac{\dot{Q}}{h_{L}}\Bigg{)}^{2},F_{ rad}=\frac{G^{2}A_{cs}}{4\rho_{l}} \tag{3}\]  
In the above expressions, \(A_{cs}\) is the cross-section area in m\({}^{2}\) and \(\dot{Q}\) is the total supplied heat in W. Therefore, if \(K_{inst}\) is less than the unity, the flow would be stable, while a higher value of \(K_{inst}\) indicates that the evaporation momentum forces are dominant; thus, altering the interface movement and yielding more unstable behavior. It is worth observing that the expression of \(F_{kk}\) and \(F_{fund}\) is explicitly valid for a single channel, and they must be modified for channels with inlet restrictors or a variable cross-section. It is also worth observing that the influence of the evaporation momentum force is better evaluated by considering the density ratio in combination with the boiling number; consequently, the instability parameter \(K_{inst}\) includes the boiling number \(Bo\) and the liquid to vapor density ratio, since \(K_{inst}=Bo(\rho_{l}/\rho_{v})\). Flow boiling instabilities can be classified into two main groups, namely, static and dynamic instabilities. The former occurs when a disturbance to an unstable steady state drives the system to shift towards a stable, steady state; the latter generally exhibits oscillatory behavior of flow parameters such as pressure, mass flux, and temperature. If reducing the number of parallel channels and the length-to-diameter ratio enhances the static stability, dynamic instabilities involve a higher complexity.