Effect of inertial lift on a spherical particle suspended in flow through a curved duct

Abstract

We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrates. The Navier-Stokes equations determine the hydrodynamic forces acting on a particle. A leading order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales, specifically the particle radius, duct height, and duct bend radius. Additionally, a trapezoidal shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.

Figures (14)

• Figure 1: Curved duct with rectangular cross-section containing a spherical particle located at $\mathbf{x}_{p}=\mathbf{x}(\theta_{p},r_{p},z_{p})$. The enlarged view of the cross-section around the particle illustrates the origin of the local $r,z$ coordinates at the centre of the duct. The bend radius $R$ is with respect to the centre-line of the duct and is of modest size for illustration.
• Figure 2: The force $\boldsymbol{F}_{p}^{\prime}$ on neutrally buoyant particles at different locations in a curved duct with square cross-section ($W=H=2$) and bend radius $R=160$. The colour background shows the magnitude of $\boldsymbol{F}_{p}^{\prime}$. Black and white contours are the zero level set curves of the horizontal and vertical components respectively whereas the arrows indicate the sign of each component in the area bounded by the respective contour. The left wall is on the inside of the bend. The dashed red line shows where the centre of the particle lies when its surface touches the wall.
• Figure 3: Approximate trajectories of neutrally buoyant particles within a curved duct with square cross-section ($W=H=2$) and bend radius $R=160$. Trajectories from several starting positions have been super-imposed. Green, orange and red markers show the location of stable, (unstable) saddle and unstable equilibria respectively (with marker size indicative of particle size). The left wall is on the inside of the bend. The dashed red line shows where the centre of the particle lies when its surface touches the wall.
• Figure 4: Approximate trajectories of neutrally buoyant particles within a curved duct with square cross-section ($W=H=2$). Trajectories from several starting positions have been super-imposed. Green, orange and red markers show the location of stable, (unstable) saddle and unstable equilibria respectively (with marker size indicative of particle size). The left wall is on the inside of the bend. The dashed red line shows where the centre of the particle lies when its surface touches the wall.
• Figure 5: The force $\boldsymbol{F}_{p}^{\prime}$ on neutrally buoyant particles at different locations in a curved duct with rectangular cross-section ($W/2=H=2$) and bend radius $R=160$. The colour background shows the magnitude of $\boldsymbol{F}_{p}^{\prime}$. Black and white contours are the zero level set curves of the horizontal and vertical components respectively whereas the arrows indicate the sign of each component in the area bounded by the respective contour. The left wall is on the inside of the bend. The dashed red line shows where the centre of the particle lies when its surface touches the wall.