I have seen that sometimes the Navier-Stokes equations are written with the term ( v ⋅ ∇ ) v expressed as v ⋅ ∇ v . However, is it true in general the following equality for any vector field v ? ( v ⋅ ∇ ) v = ? v ⋅ ( ∇ v )A couple of examples: in Batchelor's An Introduction to Fluid Dynamics, equation (2.1.2) defines the mass derivative of the velocity field (which is the LHS of the NS equation) as ∂ u ∂ t + u ⋅ ∇ u Whereas, in Landau-Lifschitz Fluid Mechanics, 2nd edition, the Navier-Stokes equation is written in equation (15.7) as ∂ v ∂ t + ( v ⋅ ∇ ) v = − 1 ρ ∇ p + η ρ Δ v

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I have seen that sometimes the Navier-Stokes equations are written with the term   (      v    ⋅  ∇  )      v   expressed as       v    ⋅  ∇      v  . However, is it true in general the following equality for any vector field       v  ?  (      v    ⋅  ∇  )      v                      =                    ?                  v    ⋅  (  ∇      v    )A couple of examples: in Batchelor&#039;s An Introduction to Fluid Dynamics, equation (2.1.2) defines the mass derivative of the velocity field (which is the LHS of the NS equation) as            ∂              u                    ∂      t        +      u    ⋅  ∇      u  Whereas, in Landau-Lifschitz Fluid Mechanics, 2nd edition, the Navier-Stokes equation is written in equation (15.7) as            ∂              v                    ∂      t        +  (      v    ⋅  ∇  )      v    =  −      1    ρ    ∇  p  +      η    ρ    Δ      v

Adelaide Barr · Accepted Answer

The equality is true because both sides in index notation are      v    μ        ∇    μ        v    ν      .On the level of components this &quot;multiplication&quot; is associative. Then we sum over the repeated index   μ. What this describes is the directional derivative of the vector field v into the direction of v.

Joyce Sharp · Accepted Answer

According to the NRL Plasma Formulary,  ∇  ⋅  (      A    B    )  =  (  ∇  ⋅      A    )      B    +  (      A    ⋅  ∇  )      B  For       A    =    B    =    v  , we have  ∇  ⋅  (      v    v    )  =  (  ∇  ⋅      v    )      v    +  (      v    ⋅  ∇  )      v  I don&#039;t think I&#039;ve ever seen the Navier-Stokes equations expressed as other than       v    ⋅    ∇    v  , and clearly, from the above, this is not equal to   (      v    ⋅  ∇  )      v

Is it true that (b *grad)v=v(grad v)?

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