# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 1 Introduction  
The convection of two-phase flows driven by the inertial forces in a pipe is one of the most fundamental hydrodynamic problems widely found in geophysical environments in nature, such as the circulation of magma in a continuously degassed volcanic conduit [1] and the countercurrent of the currents of different densities in oceanic straits [2]. They are also found in many industrial processes and have many applications in industrial processes, such as the exchange of air and combustion products in ventilation ducts during fires [3] and the transportation of viscous oil in lubrication pipelines through pipes by injecting water into the pipeline [4]. There are many laboratory studies of two-phase flow exchange flows in vertical pipelines [5; 6; 7; 8; 9], which are categorized according to the experimental setups in two ways: lock-exchange flows and long bubbles; lock-exchange flows are mainly exchange flows within a single pipe [5; 6; 7] between a heavy and a light fluid, each occupying one half of the pipe. The flow patterns can be categorized into three types, depending on the viscosity ratio (\(\beta\), the viscosity of the more viscous fluid divided by the viscosity of the less viscous fluid) of the two fluids [1]. While the long bubble flow is mainly the flow within a pipe connection between two reservoirs [8; 9], the flow regimes are mainly categorized into two types according to the viscosity ratios: core-annular flow (CAF) and side-by-side flow (SBS) [9]. There is no good analytical solution for fluid dynamics in inclined pipes. Here, we exemplify some major fluid dynamics studies carried out in vertical pipes. A theoretical model for laminar two-way core flow in a vertical pipe was presented by [PERSON] et al. [8]. They placed constraints on the pressure gradient within the pipe. They then provideda useful prediction of the magnitude of the fluid flux compared with the experimental. However, they did not compare the velocity of the fluid used in the experiment with the theoretical velocity. [PERSON] et al. [10] proposed a velocity model for side-by-side flow within a vertical pipe using the principle of maximum flux. However, the error obtained in the results of [PERSON] et al. 's [9] experimental velocities is large compared to their modelled velocities. We investigated the bi-stability of the exchange flow in the pipeline from our previous work [11]. By comparing these results with the experimental results of [PERSON] et al. [1], we obtain two mathematically valid solutions in the steady state using the core-loop solution in the vertical pipe. The increasing dynamics of a Taylor droplet in a vertical pipe were studied by [PERSON] et al. [12], who provided an analytical solution for the velocity of a Taylor droplet in a vertical pipe using a lubrication approximation, while also providing an analytical solution for the velocity of a continuously rising fluid under specific conditions (the radius of the rising fluid as a fraction of the radius of the pipe is 0.6). In addition to that, there are many studies on exchange flow in pipelines in terms of simulation [13; 14; 15; 16]. In actual nature, however, most of the channels involved in our study are inclined. The effect of an inclined pipe on the flow of fluid in the pipe is complex because the inclination of the pipe changes the shape of the fluid flow [17]. [PERSON] et al. [18] investigated the change in bubble rise velocity inside the pipe for different inclination angles (\(\theta\)), the change in bubble-rising velocity in the pipe, and found that with an increase, the velocity of the bubbles increased and then decreased. [PERSON] et al. [19] studied the exchange flow of light and heavy fluids in a pipe with different pipe diameters. The relationship between the forward velocity of a light fluid and viscosity was given. They found that the rising velocity of the light fluid increases and then decreases with an increase in the angle of inclination. This agrees with the velocity law found by [PERSON] et al. Further, [PERSON] et al.