If I wanted to hire someone to do say, CFD on flow in an IC engine manifold, where would I look? I might hire someone with a strong background in gardening (if he somehow convinced me that he could do the work and wanted to do it), but it is not likely. I'd probably look for some one with an ME PhD in thermo/fluids who had done CFD research. Nothing you do will 100% exclude you from the job market, but that's not really the question, is it? I would think you would want to know what to do to maximize your prospects, not what to do to assure they are not exactly zero.
If a physics/math major were to apply for an engineering PhD program, they might very well be admitted, but subject to the condition that they take a prescribed list of undergraduate engineering courses as prerequisites. Without that, you would be absolutely destroyed on the qualifying exam.
When I was in graduate school (back before the last ice age), one of my class mates in an Advanced Dynamics class was a young woman who had a math degree but wanted to do work in Acoustics which was in the ME department. I know she was a good student, but I don't know if she ever made it through.
