How to Convert Cartesian Navier-Stokes to Cylindrical/Spherical Coordinates?

[sam2]

Hi,

I'm trying to understand how to convert the cartesian form of the N-S equation to cylinderical/spherical form. Rather than re-derive the equation for spherical/cylindrical systems, I am trying to directly convert the cartesian PDE.

I'm ok with converting the d/dx and d2/dx2 terms. What I am struggling with a little, is the v_x, v_y and v_z terms which represent velocity in the x, y and z directions respectively.

Start simple with cylindrical...

Any idea on how to represent v_x and v_y in terms of v_r and v_theta?

I make v_r to be v_x / cos(theta). But can't see how to find v_theta. Any help is much appreciated.

Regards,

Sam

 

[sam2]

Any ideas at all?

Thanks,

Sam

 

[M98Ranger]

Hi Sam,

Converting the cartesian form of the Navier-Stokes equation to cylindrical or spherical form can be a bit tricky, but it is definitely doable. First, let's start with the cylindrical form. The x, y, and z directions in cartesian coordinates correspond to r, θ, and z directions in cylindrical coordinates. So, we can represent v_x and v_y in terms of v_r and v_θ as follows:

v_x = v_r cos(θ) - v_θ sin(θ)
v_y = v_r sin(θ) + v_θ cos(θ)

To understand why this is the case, think about the components of velocity in the x and y directions. The x component (v_x) is related to the radial direction (v_r) and the tangential direction (v_θ). Similarly, the y component (v_y) is related to the radial direction and the tangential direction, but with a rotation of 90 degrees. This is why we have the cos(θ) and sin(θ) terms in the equations above.

For the spherical form, the x, y, and z directions correspond to r, θ, and φ directions. Again, we can represent v_x and v_y in terms of v_r, v_θ, and v_φ as follows:

v_x = v_r sin(θ) cos(φ) - v_θ cos(θ) cos(φ) + v_φ sin(φ)
v_y = v_r sin(θ) sin(φ) - v_θ cos(θ) sin(φ) - v_φ cos(φ)

To understand this representation, think about the components of velocity in the x and y directions in spherical coordinates. The x component (v_x) is related to the radial direction (v_r), the meridional direction (v_θ), and the azimuthal direction (v_φ). Similarly, the y component (v_y) is related to these three directions, but with different combinations of sin and cos functions to account for the different orientations.

I hope this helps you understand how to convert the cartesian form of the Navier-Stokes equation to cylindrical and spherical forms. Good luck with your studies!