The Navier-Stokes equations describe the motion of viscous fluid substances. In their standard form, they are written for vector-valued velocity fields $\mathbf{u} = (u_1, u_2, u_3)$ as a function of time $t$ and spatial coordinates $(x_1, x_2, x_3)$.

However, you are asking to express the Navier-Stokes equations in terms of a complex-valued function. This can be achieved by combining the velocity components into a complex-valued field, typically using two spatial dimensions for simplicity (though this can be extended to three dimensions). Let's focus on the 2D case for clarity.

Standard 2D Navier-Stokes Equations:

The 2D incompressible Navier-Stokes equations are typically written as: $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}$ where:

• $\mathbf{u} = (u_x, u_y)$ is the velocity vector field,
• $p$ is the pressure,
• $\nu$ is the kinematic viscosity,
• $\mathbf{f}$ is the external force (e.g., gravity),
• $\nabla \cdot \mathbf{u} = 0$ enforces incompressibility.

Complex-Valued Representation:

In two dimensions, we can combine the velocity components $u_x$ and $u_y$ into a single complex-valued function: $w(z, t) = u_x(x, y, t) + i u_y(x, y, t)$ where $z = x + iy$ is the complex spatial variable and $w(z, t)$ is the complex velocity field.

For incompressible flow, the divergence-free condition implies: $\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0$ which can be rewritten in terms of the complex function as: $\text{Re} \left( \frac{\partial w}{\partial \bar{z}} \right) = 0$ where $\bar{z} = x - iy$ is the complex conjugate of $z$, and the condition is that $w(z)$ is holomorphic (analytic) if there is no source or sink of the fluid.

Complex Form of the Navier-Stokes Equation:

The complex form of the Navier-Stokes equation involves rewriting the terms of the original equation using the complex velocity field $w(z,t)$. The equation becomes:

$\frac{\partial w}{\partial t} + w \frac{\partial w}{\partial z} = -\frac{\partial p}{\partial z} + \nu \frac{\partial^2 w}{\partial z \partial \bar{z}} + F(z, t)$ where:

• $F(z, t)$ is the complex force field (related to the external force $\mathbf{f}$),
• $\nu$ is the viscosity term, and
• $\frac{\partial^2}{\partial z \partial \bar{z}}$ is the Laplacian in complex coordinates.

This complex representation combines both velocity components into a single function and reflects the original dynamics of the 2D Navier-Stokes equations.

Key Points:

• The complex field $w(z,t)$ represents both components of the velocity field.
• The pressure $p$ and external force $F$ can also be expressed in terms of complex variables.
• This form is particularly useful in the study of potential flow and vortex dynamics, where the velocity field is often naturally expressed in terms of complex functions.

Would you like more details about any specific part of this transformation?

Further Questions:

1. How do boundary conditions affect the complex form of the Navier-Stokes equation?
2. What are the advantages of using the complex form for potential flow analysis?
3. Can we extend this method to three-dimensional fluid flow?
4. How do vorticity and circulation relate to the complex velocity function?
5. How does the pressure term behave in the complex plane?

Tip:

Using complex variables simplifies solving certain fluid dynamics problems, particularly in 2D, by reducing the system to a single equation for a complex field instead of two coupled equations for the velocity components.