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Next: The Boussinesq approximation Up: Rayleigh-Benard Convection Previous: Intrduction

The exact equations of motion

We consider a layer of fluid of depth $ d$ have infinite horizontal dimensions. The temperature at the bottom and the top are maintained at $ T_b$ and $ T_u$ respectively. We use tensor notation in writing down the equations of motion. Thus we use cartesian tensor with $ \boldsymbol {x}=x_j$ and velocity $ \boldsymbol {u}=u_j$ for j=1,2,3. [Note: We frequently mix vector and tensor notation, e.g. the velocity vector can be written as $ \boldsymbol {u} = (u,v,w) = u_j,j=1,2,3$]

$ \bullet$
Equation of continuity

$\displaystyle \frac{\ensuremath{\partial}\rho}{\ensuremath{\partial}t} + \frac{\ensuremath{\partial}}{\ensuremath{\partial}x_j}\left(\rho u_j\right)=0,
$

or in equaivalent form

$\displaystyle \frac{\ensuremath{\partial}\rho}{\ensuremath{\partial}t} + u_j \f...
...partial}x_j} = -\rho \frac{\ensuremath{\partial}u_j}{\ensuremath{\partial}x_j},$ (1)

where $ \rho$ is the density of the fluid.

$ \bullet$
Equations of momentum

$\displaystyle \rho\left(\frac{\ensuremath{\partial}u_i}{\ensuremath{\partial}t}...
...rho\delta_{i3}+\frac{\ensuremath{\partial}\tau_{ij}}{\ensuremath{\partial}x_j},$ (2)

where

$\displaystyle \tau_{ij}= -p \delta_{ij}+\mu\left[\frac{\ensuremath{\partial}u_i...
...{3}\frac{\ensuremath{\partial}u_k}{\ensuremath{\partial}x_k}\delta_{ij}\right]
$

is the stress tensor where $ \mu$ is the shear viscosity. Strictly speaking, the pressure term $ p$ in the stress tensor is the mechanical tensor which is different than the thermodynamic pressure term. But their difference is so small that we assume they are same for all the practical case.
$ \bullet$
Equation of energy

$\displaystyle \rho\left(\frac{\ensuremath{\partial}E}{\ensuremath{\partial}t} +...
...}x_j}\right)-p\frac{\ensuremath{\partial}u_k}{\ensuremath{\partial}x_k} + \Phi.$ (3)

$ E=$ internal energy per unit mass of fluid [$ =c_v T$ ($ c_v=$ specific heat at constant volume) for gases and $ =cT$ ($ c=$ specific heat) for liquid], $ k=$ thermal diffusivity, $ T=$ temperature and rate of viscous dissipation is

$\displaystyle \Phi = \frac{\mu}{2}\left[\frac{\ensuremath{\partial}u_i}{\ensure...
...{3}\mu\left[\frac{\ensuremath{\partial}u_k}{\ensuremath{\partial}x_k}\right]^2
$


next up previous
Next: The Boussinesq approximation Up: Rayleigh-Benard Convection Previous: Intrduction
A/C for Homepage of Dr. S Ghorai 2003-01-16