Momentum and mass balance in fluid mechanics

[Chuck88]

When I am reading the paper in fluid mechanics, I found a paragraph and two equations:

"Since the fluid is considered Newtonian and incompress-
ible of density ρ and constant viscosity μ, the dimensionless
mass and momentum balance equations are:"

$$\nabla \cdot \mathbf{v} = 0$$

$$Re[\frac{d\mathbf{v}}{dt}+(\mathbf{v} - \mathbf{x}^t) \cdot \nabla \mathbf{v}] = \nabla \cdot \mathbf{T}$$

"Where $\mathbf{T} = -p/Ca \mathbf{I} + (\nabla \mathbf{v} + \nabla \mathbf{v}^T)$ is the Cauchy stress tensor"

Can someone provide me with the information about the "mass and momentum balance?" Also, what does $Re$ mean in the formulae?

 

[boneh3ad]

Mass and momentum balance simply refer to two of the three conservation laws used to describe continuum mechanics: the conservation of mass and the conservation of momentum.

In this context, $\mathrm{Re} = \frac{\rho u L}{\mu}$ is the Reynolds number - a nondimensional number representing the ratio of inertial forces to viscous forces in the fluid.

 

[schliere]

If you're new to continuum mechanics, look up the Material Derivative. Applied to density, you can derive that first equation. You might use the Material Derivative to derive Reynolds Transport Theorem. Applying that to momentum, you can find the second equation, which is basically Newton's second law for a continuum flow.