Laplace equation for irrotational flow

Irrotational flow exists in any region of a fluid flow field where the curl of the velocity is zero. That is when ∇ × v → = 0 {\displaystyle \nabla \times {\vec {v}}=0} Similarly, if it is assumed that the fluid is incompressible: ρ ( x , y , z , t ) = ρ (a constant) {\displaystyle \rho (x,y,z,t)=\rho {\text{ (a constant)}}} Then, starting with the continuity equation: ∂ ρ ∂ t + ∇ ⋅ ( ρ v → ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0} The condition of incompressibility means that the time derivative of the density is 0, and that the density can be pulled out of the divergence, and divided out, thus leaving the continuity equation for an incompressible system: ∇ ⋅ v → = 0 {\displaystyle \nabla \cdot {\vec {v}}=0} Now, the Helmholtz decomposition can be used to write the velocity as the sum of the gradient of a scalar potential and as the curl of a vector potential.

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