# Thrust Augmentation of Micro-Resistojets by Steady Micro-Jet Blowing into Planar Micro-Nozzle
## 1 Introduction  
The micro-satellite is a device of great interest in space research, because it is cost-effective. In this field, the micro-thrusters present optimal characteristics for attitude control, orbit maintenance, and station maintenance, thanks a precise control of operating conditions [1]. In this regard, an onboard secondary propulsion system should be installed, which is able to provide small thrust forces from a few micro-newtons up to some milli-newton and high specific impulse, still satisfying the mass, volume, and power consumption constraints [2]. Different concepts of MEMS-based micro-propulsion systems, or micro-thrusters, have been developed to be implemented in small satellites [3, 4]: thanks to the reduced system complexity related to the simplicity of the working principle, micro-resistojets represent a very interesting choice. Among them, cold gas micro-thrusters (CGs) operate with gaseous propellants; when operating with liquid propellants, micro-resistojets are usually referred to as vaporizing liquid micro-thrusters (VLMs) [5, 6]. Increasing interest of the industrial and scientific communities on VLMs is due to their simplicity and compactness and the possibility to use water as green propellant. The use of water matches all requirements in terms of propulsive performance (associated mainly to thrust and specific impulse), system density (associated to mass and size), and safety (associated to flammability, instability, and health hazard), as discussed in [7]. However, by considering the global performance of micro-resistojets, the micro-nozzle overall efficiency is strongly affected by the entity of the viscous effects [8; 9]. These last ones are influenced by the degree of gas rarefaction, since it determines the mechanisms of interaction between gas-gas molecules and solid wall-gas molecules. In general, in micro-nozzles, the continuum assumption and the no slip condition at walls are violated, and two different regimes could establish based on the entity of the Knudsen number \(Kn\)[10], which is defined as the ratio between the mean free molecular path and the characteristic length as follows:  
\[Kn\ =\ \frac{\Lambda}{L}\ =\ \sqrt{\frac{k\pi}{2}}\frac{M}{Re}, \tag{1}\]  
where \(\Lambda\) is the mean free path of the gas molecule, \(L\) is the characteristic length, and \(M\) and \(Re\) are the Mach and the Reynolds numbers, respectively. Consequently, when \(0.01<Kn<0.1\) the slip flow regime occurs, while the transitional flow regime occurs if \(0.1<Kn<10\). In cases involving the first regime, the Navier-Stokes (NS) equations are still valid for numerical modeling, even in combination with partial slip models at walls [11]. In this regard, in [12], the optimization of the divergence half-angle of conical micro-nozzle was performed in order to increase the thrust; it was shown that the optimum angle little depends on the given nozzle length, and it is almost independent of gas species and heat loss. In [13], heating the convergent-divergent sidewall was studied as a method to improve the thrust level and specific impulse. However, a degradation of Mach number and velocity and an increase in the thickness of the sub-sonic boundary layer were found. The performance of the micro-nozzle is also influenced by surface roughness on outlet flow velocity [14]. In [15], it was shown that pressure disturbances in under-expanded micro-nozzle flow can propagate upstream due to the development of the boundary layer causing the flow blockage. In [16], surface discontinuities on the divergent contour of the micro-nozzle were investigated in order to assess the sensitivity of the pressure, skin friction, and heat transfer coefficients to these discontinuities as well as their impact on the specific impulse. The growth of the sub-sonic boundary layer reduced the thrust efficiency due to the viscous losses and the reduction of the actual cross section at the nozzle exit [17].