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2003 Dodge Grand Caravan - sound from steering wheel

I have a 2003 Dodge Grand Caravan Sport. For the past few months, at different times but more often in the morning shortly after startup, I hear a slight whirring sound only when I turn the steering wheel or apply the brakes firmly. It “sounds” like it has something to do with the steering but yet it also will start to make the noise when I apply the brakes. It seems to go away, or at least not be as noticeable, after I gain some speed. I did just take it to my mechanic who, after I verbally explained it to him, said that it was likely that it needed the power steering cooler hose replaced with a new longer power steering hose (as per a Technical Service Bulletin that he was aware of for this vehicle). He did this but it did not fix the problem at all. Any ideas on what this my be ? Thanks.