# Presentation of Navier-Stokes eqns

This may be a trivial point, perhaps it's a lack of understanding on my part? When I was first introduced to fluid mechanics I was instructed to write the continuity and (generalized) Navier-Stokes equations as such

$$\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0,$$

$$\rho\left(\frac{\partial}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{u}=-\nabla p +\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\tau},$$

where the nabla operator in front of the pressure term is not a 'boldsymbol' operator since the fluid pressure is a scalar. However, in the literature this doesn't seem to be the general consensus. More often I find that these equations are expressed as

$$\nabla\cdot\boldsymbol{u}=0,$$

$$\rho\left(\frac{\partial}{\partial t}+\boldsymbol{u}\cdot\nabla\right)\boldsymbol{u}=-\nabla p +\nabla\cdot\boldsymbol{\tau},$$

where the neither the nabla or dot operators have the 'boldsymbol' typeface.

Just looking for some clarification really.

Thanks!

• fluid pressure is a scalar, but $\nabla$ is always a "vector" in the sense, that it has separate coordinates: $\nabla := (\frac{\partial}{\partial_x}, \frac{\partial}{\partial_y}, \frac{\partial}{\partial_z})$. $\nabla p$ is thus always a vector. – Lurco Jul 20 '15 at 7:14
• I agree with @Lurco. Either use bold face $\nabla$ throughout, or don't use bold face at all for $\nabla$. – Stephen Montgomery-Smith Jul 20 '15 at 7:16