# What are the limitations of this form of the Navier-Stokes equation?

$$\frac{∂u}{∂t}+u\frac{∂u}{∂x}+v\frac{∂u}{∂y}+w\frac{∂u}{∂z}=\frac{−1}{ρ}\frac{∂P}{∂x}+gx+ν(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2})$$

Why would someone use a form of this equation where both sides are not divided by the density? Are there any cases where the above equation cannot be used because the equation has been divided by the density?

• Density is nonzero everywhere so they're entirely equivalent. – eyeballfrog May 30 at 18:50
• Apparently if $\rho=0$. – Qmechanic May 30 at 19:39
• What are v and w? I assume u to be a component of the velocity field. – DanielC May 30 at 20:06
• @ DanielC (u, v, w) are commonly used to denote the (x, y, z) velocity components – D. Halsey May 30 at 22:18
• @Qmechanic would you mind expanding on that? – Chris May 31 at 15:39