# Analysis of Longitudinal Braking Stability of Lightweight Liquid Storage Mini-Track Vehicles
## 2 Analysis of LBS
### Mathematical Modeling
* Momentum conservation equation: \[ \begin{array}{l}\frac{\partial u_{i}}{\partial t}+u_{i}\frac{\partial u_{i} }{\partial x_{i}}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}+u_{k}\frac{ \partial u_{i}}{\partial x_{k}}=F_{i}-\frac{1}{\rho}\frac{\partial p}{ \partial x}+\frac{\mu}{\rho}\bigg{[}\frac{\partial^{2}u_{i}}{\partial x_{i} ^{2}}+\frac{\partial^{2}u_{i}}{\partial x_{i}^{2}}+\frac{\partial^{2}u_{i}}{ \partial x_{k}^{2}}\bigg{]}\\ \frac{\partial u_{i}}{\partial t}+u_{i}\frac{\partial u_{i}}{\partial x_{i}}+ u_{j}\frac{\partial u_{j}}{\partial x_{j}}+u_{k}\frac{\partial u_{i}}{\partial x _{k}}=F_{j}-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{\mu}{\rho} \bigg{[}\frac{\partial^{2}u_{j}}{\partial x_{i}^{2}}+\frac{\partial^{2}u_{j}} {\partial x_{j}^{2}}+\frac{\partial^{2}u_{j}}{\partial x_{k}^{2}}\bigg{]}\\ \frac{\partial u_{k}}{\partial t}+u_{i}\frac{\partial u_{k}}{\partial x_{i}}+ u_{j}\frac{\partial u_{k}}{\partial x_{j}}+u_{k}\frac{\partial u_{k}}{\partial x _{k}}=F_{k}-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{\mu}{\rho} \bigg{[}\frac{\partial^{2}u_{k}}{\partial x_{i}^{2}}+\frac{\partial^{2}u_{k}} {\partial x_{j}^{2}}+\frac{\partial^{2}u_{k}}{\partial x_{k}^{2}}\bigg{]}\end{array} \] (8) where \(F_{i},F_{j},\) and \(F_{k}\) are components of force in the \(x,y,\) and \(z\) directions, respectively; \(\rho\) is the density; \(\mu\) is the dynamic viscosity; and \(p\) is the liquid pressure. * Parametric equation of turbulence model \(K\) (turbulent kinetic energy) equation: \[\rho\frac{\partial K}{\partial t}+\rho u_{j}\frac{\partial K}{x_{j}}=\frac{ \partial}{\partial x_{i}}\bigg{[}\bigg{[}\mu+\frac{\mu_{t}}{\sigma_{K}}\bigg{]} \frac{\partial K}{\partial x_{j}}\bigg{]}+\mu_{t}\frac{\partial u_{i}}{ \partial x_{j}}\bigg{[}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\bigg{]}-\rho\varepsilon\] (9) \(\varepsilon\) (dissipation rate) equation: \[\rho\frac{\partial\varepsilon}{\partial t}+\rho u_{k}\frac{\partial \varepsilon}{x_{k}}=\frac{\partial}{\partial x_{k}}\bigg{[}\bigg{[}\mu+\frac{ \mu_{t}}{\sigma_{\varepsilon}}\bigg{]}\frac{\partial\varepsilon}{\partial x _{k}}\bigg{]}+\frac{c_{1}\varepsilon}{K}\mu_{t}\frac{\partial u_{i}}{\partial x _{j}}\bigg{[}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}\bigg{]}-c_{2}\rho\frac{\varepsilon^{2}}{K}\] (10) where \(K\) is the Reynolds mean velocity component; \(\mu_{t}\) is the turbulence viscosity coefficient; \(-\rho\varepsilon\) is the dissipation term; and \(\sigma_{K},\)\(\sigma_{\varepsilon},\)\(c_{1},\) and \(c_{2}\) are the empirical constants of the standard \(K\)-\(\varepsilon\) turbulence model. The longitudinal impact force \(F_{\text{c}}\) of the liquid can be obtained through the fluid control equation; however, it is difficult to solve the nonlinear and partial differential problems. Therefore, since the fluid simulation software FLUENT 2023 R1 integrates all the equations outlined above, it can simulate the fluid flow, and thus accurate CFD analysis results can be obtained. Therefore, solving the longitudinal impact force of the liquid in the tank via FLUENT 2023 R1 can improve the solving efficiency.