# Convection current

I am a master student (geology), I seek to ask how one can simulate convection current movement. What the difference between Boussinesq approximation and non Boussinesq approximation, and what are the physical parameters control decreasing the density of fluid?

How can handle this with smoothed particles hydrodynamics (SPH)?

• While I think the question is on topic here, you might find more people familiar with the subject on either Earth Science or Computational Science. – dmckee Feb 14 '18 at 20:55
• it is related to thermodynamic and fluid dynamics, and both of them consider physics. so this topic is not about earth science. – Rostom Recrucified Feb 14 '18 at 21:50
• I would say your first question is on topic here, your second question is more suitable for CompSci. It usually is frowned upon to ask multiple question in the same post, this is especially the case here as the answers to both questions are both very extensive and only somewhat related. – nluigi Feb 15 '18 at 8:05

## 1 Answer

Consider a fluid between two plates, the bottom plate at $T\left(0\right)=T_b$ and the top plate at $T\left(H\right)=T_t$ with $T_b>T_t$ (i.e. the bottom plate is hotter than the top plate). The bottom plate will heat the fluid causing the fluid to expand (because density is dependent on temperature) and hot fluid will rise towards the top plate. Simultaneously, the top plate is cooling the fluid causing the fluid to become denser and fall towards the bottom plate. In a stationary state, the hot fluid rising is replaced by the cool fluid falling and this results in a convection current.

The steady-state movement of convection currents is determined by hydrodynamic and thermal equations simultaneously:

$$\vec{\nabla}\cdot\rho\vec{v} = 0$$ $$\vec{\nabla}\cdot\rho\vec{v}\vec{v} = -\vec{\nabla}\cdot p + \mu\vec{\nabla}\cdot\left[\vec{\nabla}\vec{v}+\left(\vec{\nabla}\vec{v}\right)^T\right] + \rho\vec{g}$$ $$\vec{\nabla}\cdot\rho c_p \vec{v}T = \kappa\nabla^2T$$

This is the most general description of the compressible fluid movement due to a differences in density (bouyancy) caused by a temperature gradient. This is what you call non-Boussinesque approximation, it does not make any assumptions about fluid properties in relation to bouyancy and the thermal and hydrodynamic equations are coupled through the temperature dependence of the density.

The problem with the above system of equations is that it doesn't lend itself well for analysis because of it's compressible nature. Generally, scientist/engineers prefer to make the assumption of incompressibility which is a good assumption for liquids:

$$\vec{\nabla}\cdot\vec{v} = 0$$ $$\vec{v}\cdot\vec{\nabla}\vec{v} = -\frac{1}{\rho}\vec{\nabla}\cdot p + \mu\nabla^2\vec{v} + \vec{g}$$ $$\vec{v}\cdot\vec{\nabla}T = \alpha\nabla^2T$$

This assumption simplifies the equations considerably but we lose coupling between the thermal and hydrodynamic equations and with it the capability to simulate bouyancy because density differences are not possible (due to the incompressibility assumption).

To circumvent this we make the Bousinesque assumption: We consider the density changes $\Delta\rho$ only small compared to some reference density $\rho_0$ (i.e. $\Delta\rho/\rho_0\ll1$) so that we may linearize the density:

$$\rho\approx\rho_0\left(1+\frac{\Delta\rho}{\rho_0}\right)\approx\rho_0\left(1+\beta\Delta T\right)$$

where $\beta$ is the thermal expansion coefficient and $\Delta T = T_b - T$ the local temperature gradient. Substituting this into the system of equations yields:

$$\vec{\nabla}\cdot\vec{v} = 0$$ $$\vec{v}\cdot\vec{\nabla}\vec{v} = -\frac{1}{\rho_0}\vec{\nabla}\cdot p + \mu\nabla^2\vec{v} + \beta\Delta T\vec{g}$$ $$\vec{v}\cdot\vec{\nabla}T = \alpha\nabla^2T$$

As you can see these equations are similar to the incompressible equations but the acceleration term now has a dependence on the temperature, i.e. there is again a coupling between the thermal and hydrodynamic equations. If the local temperature gradient is positive, the fluid will fall, conversely it will rise.

This is now the starting point for further analysis of this fluid system.