### Navier - Stokes Eqn on Fluid Behaviour

**Navier Stokes** (NS) Equations tries to define a method of fluid (liquid) behaviour.

__Limitations of NS__

- continuous vol of fluid on a 2D rectangular domain
- no interaction between air / sloshing water

__Maths Background__

Behaviour of fluids : Velocity Eqns -> determines how the fluid moves itself & the things that are on it.

Nature of fluids : Velocity varies on both time and space -> represent using a 'vector field'

__Vector Field__

Assumptions : -> on a 2D cartesian of M by N (M columns by N rows)

- For every position vector
**x**= (*x,y*) on vector field - Associated Velocity at time
*t ->***u**(**x**,t) = [**u**(**x**,t),**v**(**x**,t),**w**(**x**,t)]

Key is to think in **time **steps -> if we are able to calculate the **u **at the current velocity field, then we can move objects and smoke densities.

We assume an incompressible(volume constant in time), homogeneous(density rho constant in space) fluid.

__Variables__

**x**(x,y) : vector for 2d coordinate (cartesian grid)**u**(**x**,t) : velocity field*p*(**x**,t) : pressure/scalar field

At time t = 0, if **u** and *p* are known,

**Fluid behaviour** = **Advection** (velocity of e field carrying itself & others along the flow) + **Pressure** (molecules interaction with each other) + **Diffusion** (viscosity - how resistive a fluid is to flow -> resistance = diffusion of momentum) + **External Forces** (e.g. gravity)

*References taken from GPU GEMS*

## 1 Comments:

At 11:30 AM, Anonymous said…

wah u still blog !

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