Model 2D Incompressible Fluid w/ Navier-Stokes Equations

In summary, the conversation discusses modeling a 2D problem using the Navier-Stokes equations, where an incompressible, non-viscous fluid is subject to a constant force applied at a specific location. The speaker suggests using the static regime, setting ρ=1 and assuming a time-independent solution. They also mention using a pressure gradient or a force term in the equation to model the force. The problem is further complicated by the need for boundary conditions, which are specified as fixed walls at certain points. The speaker also provides a system of equations for the velocity and pressure, but it is unclear how to set the initial force or pressure.
  • #1
Zhivago
26
1
Hello everyone

I'm for the first time trying to model using the Navier-Stokes equations.
I want to model a 2D problem where I have an incompressible, non viscous fluid. I have a region (a segment of line) where a force is applied to the fluid.
For example: a rectangular box with size 2L x L. In a line at x=L (or some other point) with height (L/4) a constant force is applied to the fluid.
How could I model this?

I suppose I could use the static regime, and model this force as a pressure gradient at the line?
But how?

setting /rho = 1, and assuming the solution doesn't depend on time,

v . grad(v) = - grad (p)

is this a good way to go? what can I do with this equation? Should I use the force term in the equation instead of a pressure gradient?
I would say this problem is similar to having 2 close conducting plates (or lines) with opposite charges and finding the electric field / current density.

Any thoughts / ideas?
 
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  • #2
The problem is incomplete without specifying boundary conditions: i.e. what lies outside of the region you're considering.
 
  • #3
Oh yes, of course. The walls are fixed at x = 0, 2L and y = 0, L

The walls are fixed, no fluid going through them. All velocities start at 0.

If the velocity is (u, v), I should have a system of equations

[tex]
u u_x + v u_y = -p_x
[/tex]

[tex]
u v_x + v v_y = -p_y
[/tex]

[tex]
u_x + v_y = 0
[/tex]

the indices represent derivative

(the last equation from div(velocity) = 0)

Would this tackle the problem somehow? But how to set the initial force / pressure?
 
Last edited:

Related to Model 2D Incompressible Fluid w/ Navier-Stokes Equations

1. What is a 2D incompressible fluid?

A 2D incompressible fluid is a type of fluid that is assumed to have zero thickness in one direction and thus can be modeled in two dimensions. This means that the fluid is not able to be compressed, and the velocity and density of the fluid remain constant throughout the domain.

2. What are Navier-Stokes equations?

Navier-Stokes equations are a set of partial differential equations that describe the motion of a fluid by taking into account its velocity, pressure, density, and viscosity. They are used to model the behavior of fluids, including 2D incompressible fluids, and are essential in fluid dynamics and engineering applications.

3. How are Navier-Stokes equations used in modeling 2D incompressible fluids?

Navier-Stokes equations are used to solve for the velocity and pressure fields of a 2D incompressible fluid. By solving these equations, we can predict the behavior of the fluid, such as its flow patterns and pressure distribution, under different conditions.

4. What are some applications of modeling 2D incompressible fluids with Navier-Stokes equations?

Some common applications of modeling 2D incompressible fluids with Navier-Stokes equations include aerodynamics, weather prediction, and fluid flow in pipes and channels. These equations are also used in the design and analysis of aircraft, ships, and other vehicles that interact with fluids.

5. What are the limitations of modeling 2D incompressible fluids with Navier-Stokes equations?

While Navier-Stokes equations are widely used in fluid dynamics, they have some limitations when it comes to modeling 2D incompressible fluids. These equations assume that fluids are continuous and do not take into account certain factors such as surface tension and turbulence, which can be important in some applications. Additionally, they can be computationally expensive to solve, making them difficult to apply in complex systems.

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