Upvote:4
If I understand correctly, you want to go back and fourth between the UK and Schengen area? As many might have told you already the decision really depends on the embassy so no one can guarantee whether you will get a multiple entry visa or just single visa. This largely depends on your supporting documents.
I can tell you from my past experience. I applied for a Schengen visa with French embassy for 5 days. I showed all the hotel booking and return tickets with enough money to cover the whole trip. But in the document I requested multiple entry for 6 months. I usually got it as I requested. I asked the embassy directly you can write in the form as many days as you want, maximum is 1 year. But the decision is still with the embassy.
You mentioned that you will be staying in the UK for 10 weeks that means your UK visa isn't valid more than 3 months so when you apply for Schengen visa you need to show your return ticket to your home country as well. They usually request that when they examine the documents at the embassy.