Depending on the shape, and thereby shading, it may be quite easy to use 3D effect(s) to spout and handle, too. I am still with 10, without 3D.
Otherwise, you may use simple blends, as follows:
1) Give the full shape the colour to be at the edges;
2) Create a smaller shape following the brightest part in between and give it the brightest colour to be;
3) Make a smooth blend aligned to path (se Blend Options).
Corresponding edges of the paths to be blended should have matching anchor points.
In some cases, the smaller shape may be created using a one step blend between opposite parts of the path of the full shape (you may cut them as single paths from a copy), outlining the stroke, and possibly adjusting the width (this may apply to the spout) and ends.
An image posted here, using the Camera icon to avoid queueing, may help the helpers.
Thanks for your input. However, I found blends didn't work very well. I will try once more.
The mesh tool would be the easiest because I can apply an effect at the mesh points. However, the mesh does not follow the path properly. On a S-shaped object it fails to make the first inflection properly and instead traces across the shape. Is there a way to force the mesh to follow the shape? I tried starting with a rectangle, creating the mesh and then dragging anchor points to create the S-shape but found that not to work very well.
Any additional thoughts or suggestions would be most welcomed!