# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 1 Introduction
[20] studied the displacement flow of two Newtonian fluids in an inclined pipe in terms of the Froude number (Fr), Reynolds number (Re), and angle of inclination (\(\theta\)), three parameters that characterize the flow regimes generated during the inclination process; the two fluids in the studied pipe have the same viscosity but different densities. They categorized the observed flow regimes into complete diffusion, transient diffusion, inertial flow, and viscous flow. For each flow regime, they give the frontal velocity as well as the axial diffusion coefficient. [PERSON] et al. [21] studied the exchange flow driven by inertial forces in a rectangular inclined pipe between two reservoirs, investigating the longitudinal aspect ratio (A) and the length-to-diameter ratio (B) of the pipe, as well as the angle of inclination (\(\theta\)), Reynolds number (Re) (based on the initial difference in inertial forces driving the flow), and Prandtl number (Pr) (considering both salt and temperature layers) of five parameters on the qualitative flow state (laminar, wavy, intermittent turbulent, or fully turbulent), the mass flux (the net transport of inertial forces between the reservoirs), and the interfacial thickness (the thickness of the intermediate density layer between the two countercurrent layers). The viscosity of the two fluids in the inclined pipes studied in the above experiments are all the same, and both of them are used to study the change in their parameters by making the fluids undergo exchange flow through the difference in density. In nature and industrial applications, the viscosity difference between the two fluids in which the exchange flow occurs is relatively large in many channels [1; 4]. [PERSON] et al. [1] found that the lock-exchange flow of two fluids in a pipe is only related to their \(\beta\). Three different transient flow patterns occur for two pure fluids in a vertical pipe in the range of three \(\beta\); when \(\beta<10\), the heavy fluid separates from the wall and assumes approximately the same morphology as the light fluid; when \(10<\beta<300\), the descending fluid eventually separates from the wall and splits into individual droplets that descend along the centre of the pipe; when \(\beta>300\), the falling fluid sticks to the wall. We build on this by tilting the pipe to a certain angle to observe the changes in the two fluid flow regimes inside the pipe. The main advantage of this study compared to the current experiments is that the combination of the change in viscosity ratio with the change in tilt angle is more relevant to the physical phenomena observed in real nature, as well as industrial applications. In this study, in order to understand the effect of viscous forces on the flow regimes of exchange flow in an inclined pipe, we adopted a combination of experimental and theoretical modes to investigate the effect of viscosity ratio and angle of inclination on the changes in the flow regimes of the two types of fluids in the pipe, and we recorded the changes in the velocity of the ascending light fluid; then, we compared them with our previously obtained model of the velocity of the light fluid. We found that for different viscosity ratios, the flow regimes of the descending heavy fluid, as well as the ascending light fluid, changed in the inclined pipe. We observe four different flow regimes for the rising light fluid, which are related to the angle of inclination. However, as the viscosity ratio increases, the tilt angle for the change in flow regime is also different. We also investigated the effect of the viscosity ratio and inclination angle on the ratio of the width of the upward-flowing light fluid to the width of the pipe.