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I was told that the total integral of the stress over the surface of a swimmer (i.e. the total force exerted by the swimmer on the fluid) always vanishes, because there are no external forces applied on it. That seems fair by the 3rd Newton Law.

But, how does it take into account the effects of the 2nd Newton's Law? If, for example the swimmer starts from a stationary state, and at the time $t_1$ reaches a velocity $v_1$, where does the force generating the acceleration $v_1/t_1$ come from?

I was told that this can happen just because in the case of a swimmer with density different than that of the fluid, so gravitational forces are present. But it seems bulls**t to me. Could you help me to clarify all this?

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Let's imagine a swimmer in a swimming pool and approximate the Earth as an inertial frame. The swimmer can certainly accelerate relative to the Earth frame (in the direction parallel to the Earth's surface); we see this happen all the time in real life. It follows from Newton's second Law, as you point out, that the net external force on the swimmer is nonzero. The only object that can possibly exert a horizontal force on the swimmer is the water in the pool (the swimmer is making physical contact with nothing else, and the force due to gravity cannot affect his/her horizontal acceleration). It follows that the net horizontal force of the water on the swimmer is nonzero.

If you had a blob of water floating around in outer space, and a swimmer inside, and if the blob of water and swimmer began stationary relative to an inertial frame, then the swimmer attempting to swim would cause some of the water to move backward, and this would propel the swimmer forward relative to the inertial frame. However, in this case the total external force on the blob+swimmer system would be zero, so the center of mass of the water+blob would remain stationary, but even in this case, the swimmer could accelerate relative to the inertial frame.

Essentially, the swimmer is doing the same thing that a torpedo would do; he/she expels water backward, and this propels him/her forward.

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