For the 2D duct below, determine the stream function using 4x = 4y = 0.1 m. 3 m s/wz 5m 2m 1m

To determine the stream function for the given 2D duct flow, you can use the finite difference method with the given grid spacing of  The stream function satisfies the 2D continuity equation:

In your case, the flow is steady (time-independent), so the continuity equation is simplified to:

Now, let's solve for  using the finite difference method. We'll need to discretize the domain and apply the finite difference formula to approximate the Laplacian. Given the dimensions of the duct and the flow velocity, let's set up a grid. We'll use subscripts i and j to denote grid points in the x and y directions, respectively. The grid spacing is 

in the x-direction, We have 5 m of length, so there are 50 grid points 

In the y-direction, we have 2 m of height, so there are 20 grid points  

Now, you can use the finite difference method to solve for  Here are the steps:

1. Initialize  at the boundary conditions.

2. Apply the finite difference formula to calculate  at each interior grid point (excluding the boundaries) using the Laplacian equation:

3. Repeat the calculation until  converges (changes become negligible).

4. Once you have  at all grid points, you've determined the stream function for the given flow. This process will yield the stream function  for the 2D duct flow with the specified grid spacing and boundary conditions.