A satellite in low, circular orbit around the Earth experiences drag (friction) and slowly spirals into Earth's atmosphere. It then enters the Earth's atmosphere, heats up catastrophically and burns up.
I'm trying to understand the forces that act on the satellite that secure this outcome.
Let's take the case where the drag force acts only briefly. Intuition tells us that the drag force $\mathbf{F_D}$ reduces tangential speed $\mathbf{v}$ and the centripetal force $\mathbf{F_c}$ (the gravitational force) then 'pulls' the satellite into a lower orbit, i.e. of smaller radius $r$.
But there's a snake in the grass: the tangential velocity $v$ is given by:
$$v=\sqrt{\frac{GM}{r}}\tag{1}$$
So, as is well known, smaller orbits run at higher tangential speeds, not lower!
Or take another scenario, in which a thruster on the satellite briefly exerts a force parallel and in the same direction as $\mathbf{F_c}$, thereby 'pushing' the satellite inwards. In accordance with $(1)$ we'd expect $v$ to increase. But where is the force that causes this tangential acceleration?
Can anything be gleaned from energy conservation? Call $T$ the total energy of the system, $U$ its potential energy and $K$ its kinetic energy:
$$T=U+K$$
For a stable, circular orbit:
$$T=-\frac{GMm}{r}+\frac12 \frac{GMm}{r}=-\frac12 \frac{GMm}{r}$$
Assume we do an amount of work $W$ on the initial system $T_0$:
$$T_0+W=T_1$$
$$-\frac12 \frac{GMm}{r_0}+W=-\frac12 \frac{GMm}{r_1}$$
$$W=\frac12 \frac{GMm}{r_0}-\frac12 \frac{GMm}{r_1}$$
$$W=\frac{GMm}{2}\Big(\frac{1}{r_0}-\frac{1}{r_1}\Big)$$
$$r_0>r_1 \Rightarrow W<0$$
Which fits because in the case of the drag force:
$$\mathbf{d}W=\mathbf{F_D}.\mathbf{ds}=F_D\mathbf{d}s\cos\pi=-F_d\mathbf{d}s $$
But it doesn't enlighten much.
I think due to the friction the orbit becomes elliptical:
This way the attractive force $\frac{GMm}{r^2}$ can be decomposed into a Normal component and a tangential component.
But it remains unclear what's the dynamic (forces) that causes the orbit to transition from a higher, circular orbit to a lower elliptical one?