Navier-Stokes Equations in Cartesian Coordinates

In three-dimensional Cartesian coordinates (\(x\), \(y\), and \(z\)), the Navier-Stokes equations can be expressed separately for each direction. For a Newtonian, incompressible fluid, the conservation of momentum in each direction includes contributions from pressure, viscous, and external forces.

Let’s assume:

• \( \mathbf{V} = (u, v, w) \), where \( u \), \( v \), and \( w \) are the velocity components in the \( x \), \( y \), and \( z \) directions, respectively.

• \( \rho \) is the fluid density, and \( \mu \) is the dynamic viscosity.

The Navier-Stokes equations in each direction are as follows:

1. \( x \)-Direction

The Navier-Stokes equation in the \( x \) direction is:

\[ \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + f_x \]

where:

• \( \frac{\partial u}{\partial t} \) is the local acceleration in the \( x \) direction,

• \( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \) represents convective acceleration,

• \( -\frac{\partial p}{\partial x} \) is the pressure gradient in the \( x \) direction,

• \( \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \) is the viscous term (Laplacian of \( u \)),

• \( f_x \) is any external body force in the \( x \) direction.

2. \( y \)-Direction

The Navier-Stokes equation in the \( y \) direction is:

\[ \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + f_y \]

where:

• \( \frac{\partial v}{\partial t} \) is the local acceleration in the \( y \) direction,

• \( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \) represents convective acceleration,

• \( -\frac{\partial p}{\partial y} \) is the pressure gradient in the \( y \) direction,

• \( \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) \) is the viscous term (Laplacian of \( v \)),

• \( f_y \) is any external body force in the \( y \) direction.

3. \( z \)-Direction

The Navier-Stokes equation in the \( z \) direction is:

\[ \rho \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \right) = -\frac{\partial p}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) + f_z \]

where:

• \( \frac{\partial w}{\partial t} \) is the local acceleration in the \( z \) direction,

• \( u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \) represents convective acceleration,

• \( -\frac{\partial p}{\partial z} \) is the pressure gradient in the \( z \) direction,

• \( \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \) is the viscous term (Laplacian of \( w \)),

• \( f_z \) is any external body force in the \( z \) direction.

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Summary

The Navier-Stokes equations in Cartesian coordinates provide a breakdown of the momentum equations in each spatial direction, highlighting how forces (pressure, viscous, and external) contribute to acceleration in the \( x \), \( y \), and \( z \) directions. These component forms are particularly useful for solving problems in specific geometries, where each direction can be analyzed separately.