Inertial migration of a non-neutrally buoyant particle in a linear shear flow with thermal convection

We perform numerical simulations on inertial migration of a non-neutrally buoyant particle with a density ratio of 0.98–1.02 in a linear shear flow dominated channel with a Reynolds number up to 500 in the presence of thermal convection using a double-population lattice Boltzmann method. It is found that under the isothermal condition, the particle with a larger density difference from the fluid will either settle to the bottom of the channel or float to the top of the channel, while the particle with a smaller particle-fluid density difference remains suspended in the channel due to the inertial lift force. The presence of thermal convection (characterized by the Grashof number $\mathrm{Gr}$) induces an additional downward lift force, which results in distinctive migration behaviors that depend on whether the particle density is larger or smaller than that of the fluid. For a particle heavier than the fluid, the settling is enhanced by thermal convection due to the synergistic effect of the thermal lift force and the gravitational force. The critical Reynolds number for lifting the particle increases compared with the isothermal case and is linearly correlated with the dimensionless density ratio ($\sigma$). On the other hand, for a particle lighter than the fluid, an empirical dimensionless number ${\mathrm{Gr}}^{*}$, defined as $\mathrm{Gr}/\left[\sigma \left(1.59\mathrm{Re}+9.31\right)\right]$, is introduced to characterize the particle migration. It is discovered that the particle's equilibrium position depends on whether it migrates to the top wall or remains suspended in the shear flow under the isothermal condition. For the former case, when thermal convection is introduced, the particle stays at the top wall when ${\mathrm{Gr}}^{*}<1$, and becomes suspended in the channel when ${\mathrm{Gr}}^{*}>1$.

Figure 1

Schematic of the model setup.

Figure 2

Particle's lateral trajectory in a simple shear flow under isothermal condition. The open square denotes the results from Feng et al. [16].

Figure 3

Time evolution of the particle's horizontal position for different particle Grashof numbers. The normalized values are defined as $x^{*}=x/W$ and $t^{*}=t\nu /d^2$. The solid lines denote the results of the present work, while the dotted lines are results from [33].

Figure 4

Particle lateral position as a function of time for different lattice grids with the same Reynolds number $\mathrm{Re}=100$ and Grashof number $\mathrm{Gr}=1$.

Figure 5

Time evolution of the lateral position of particles with (a) different initial positions with the same ${\rho }_p=1.002$, and (b) different particle densities. The Reynolds number is fixed at $\mathrm{Re}=100$ and the Grashof number is $\mathrm{Gr}=0$. The dashed lines represent the bottom of the channel.

Figure 6

Equilibrium positions of the particle as a function of the Reynolds number for different densities. The dashed line represents the bottom of the channel.

Figure 7

The regime map showing whether a particle is settled or suspended in terms of the dimensionless density ratio and Reynolds number. The dashed line represents the results from Feng and Michaelides [18].

Figure 8

Equilibrium positions as a function of ${\lambda }_H$.

Figure 9

Equilibrium position as a function of Grashof number for different particle densities and Reynolds numbers: (a) $\mathrm{Re}=100$, (b) $\mathrm{Re}=200$, and (c) $\mathrm{Re}=300$. The dashed line represents the bottom of the channel.

Figure 10

A regime map showing whether a particle is settled or suspended in terms of $\mathrm{Re}$ and $\sigma$ at $\mathrm{Gr}=30$.

Figure 11

Lateral forces on a particle that is (a) heavier than the fluid and (b) lighter than the fluid in the simple shear flow.

Figure 12

(a) The effect of particle rotation on the lateral position for different particle densities. Time evolution of the particle's (b) slip velocity and (c) lateral positions for different Grashof numbers.

Figure 13

The pressure distribution of the flow field for (a) isothermal, (b) $\mathrm{Gr}=4$, and (c) $\mathrm{Gr}=20$ at the equilibrium state. The particle density is ${\rho }_p=1.005$ and the Reynolds number is fixed, $\mathrm{Re}=300$.

Figure 14

The vertical velocity contour of the flow field for (a) isothermal, (b) $\mathrm{Gr}=4$, and (c) $\mathrm{Gr}=20$ at $\gamma t=2.5$. The particle density is ${\rho }_p=1.005$ and the Reynolds number is fixed, $\mathrm{Re}=300$.

Figure 15

A regime map showing the two cases studied to explore the effect of thermal convection. The solid line refers to Eq. (34).

Figure 16

Equilibrium position as a function of Grashof number for different particle densities and Reynolds numbers: (a) $\mathrm{Re}=50$, (b) $\mathrm{Re}=100$, and (c) $\mathrm{Re}=150$. The dashed line represents the top of the channel.

Figure 17

(a) Critical Grashof number as a function of dimensionless density ratio for different Reynolds numbers. (b) Normalized critical Grashof number as a function of the Reynolds number.

Figure 18

Equilibrium position as a function of (a) $\mathrm{Gr}/\sigma$ and (b) $\frac{\mathrm{Gr}/\sigma }{{\lambda }_{L,\mathrm{crit}}}$ for different particle densities and Reynolds numbers in case A.

Figure 19

Equilibrium position as a function of Grashof number for different particle densities and Reynolds numbers: (a) $\mathrm{Re}=400$ and (b) $\mathrm{Re}=500$. The dashed line represents the top of the channel.

Figure 20

Equilibrium position as a function of (a) $\mathrm{Gr}/\sigma$ and (b) $\frac{\mathrm{Gr}/\sigma }{{\lambda }_{L,\mathrm{crit}}}$ for different particle densities and Reynolds numbers for case B.