Math Problem Statement

convert the navier stokes equation into a complex valued function

Solution

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Math Problem Analysis

Mathematical Concepts

Fluid Dynamics
Partial Differential Equations
Complex Analysis

Formulas

Navier-Stokes equation in 2D: ∂u/∂t + (u·∇)u = -∇p + ν∇²u + f
Complex velocity field: w(z, t) = u_x(x, y, t) + i u_y(x, y, t)
Complex Navier-Stokes equation: ∂w/∂t + w∂w/∂z = -∂p/∂z + ν∂²w/∂z∂z̄ + F(z, t)

Theorems

Incompressibility condition (∇·u = 0)
Holomorphic function (complex analysis)

Suitable Grade Level

Undergraduate/Graduate (Advanced Physics and Mathematics)