Drop a heavy incompressible object into water and it would splash and then presumably reach a certain terminal velocity where acceleration is nill.
If you dropped a heavy incompressible object into a sufficiently deep thixotropic, or shear thinning fluid, that enabled movement of the object in the first place, what would happen? Would it theoretically be possible for the object to just continue accelerating?
Asking out of pure curiosity, this is obviously not a real scenario.
There's quite abundant research about drag force in thixotropic materials like this one or this. Basically, object (depending on density) can stop at once, reach terminal velocity or gradually decrease in speed.
Also, drag coefficient form has more complex form in thixotropic fluids than compared to Newtonian fluids. For Newtonian fluids drag coefficient takes form of $$ c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}} \tag 1 $$
And in thixotropic fluid it may take form, like (authors give it in first paper) :
$$ c_s = \frac {D}{6\pi~\eta~a~U \cdot ~K(a,R)} \tag 2 $$
where $a$ is falling sphere radius, $U$ fluid flow velocity, $\eta$ - overall fluid viscosity which is composed from structural viscosity and residual viscosity, i.e. $\eta = \eta_\infty + \eta_{str}$. And last term $K(a,R)$ is non-linear series expansion over $a/R$ ratio ($R$ is tube where sphere is falling, diameter), so called "tube wall factor",- which is what makes this expression really complex.
In any case, I'm pretty sure that complex drag in thixotropic fluid will not help to achieve constant acceleration, because drag coefficient is still there. Which means that sooner or later object will reach constant velocity, because even in non-Newtonian fluids still $F_d \propto u^\gamma$.
