Suppose I have a tank filled and there is no slip at the walls. If the tank is filled with a Newtonian fluid and is in static equilibrium, we know that the pressure is defined as $p = \rho g z$.
But what if the tank is filled with a yield stress fluid. For example, the Bingham plastic model defines viscous stress as:
$$\tau = \tau_0 + \mu \gamma$$
(or if it is more clear, the definition of shear rate) $$ \gamma = \begin{cases}0 & |\tau|\leq\tau_0 \\ \frac{1}{\mu}\left(\tau-\tau_0\right) & |\tau|>\tau_0\end{cases} $$
where $\tau_0$ is the yield stress of the fluid, $\mu$ is the plastic viscosity, and $\gamma$ is the shear rate. The material acts as a solid below the yield stress, and then as a Newtonian fluid above it. Intuitively, I feel as though the static pressure is reduced relative to the yield stress and that the pressure distribution is not a continuous function. Is my thinking correct? How do I calculate the static pressure in a yield stress fluid?