Sure, let's write a test case generator for the given problem statement. This generator will create random test cases within the specified constraints.

Here's the Python code for the test case generator:

```python
import random

def tcgen():
    # Randomly generate the dimensions of the grid H and W
    H = random.randint(1, 10)
    W = random.randint(1, 10)
    
    # Randomly generate the grid with 'H' rows and 'W' columns consisting of '.' and '#'
    grid = []
    for _ in range(H):
        row = ''.join(random.choice(['.', '#']) for _ in range(W))
        grid.append(row)
    
    return H, W, grid

# Generate a test case and print it
H, W, grid = tcgen()
print(H, W)
for row in grid:
    print(row)
```

This generator will produce random grids within the constraints, ensuring that each grid is between 1 and 10 rows and columns, and each cell in the grid is either a '.' (white) or '#' (black).

Let's walk through the generated test case output:

```text
H W
A_1
A_2
...
A_H
```

For example, if the generated grid has dimensions 3x4, the output might look like this:

```text
3 4
.#..
##.#
...#
```

This output is ready to be used as an input for the problem statement. You can run the `tcgen` function multiple times to generate different test cases for extensive testing.