Petroleum was formed from organisms living in the sea. As these organisms died, their bodies settled down at the bottom of the sea and got covered with sand and clay. The absence of air, high pressure and high temperature for over millions of years transformed the dead organisms into petroleum and natural gas. The petroleum deposits are usually found mixed with salt water. The petroleum is lighter than salt water, and hence, floats over it.
An oil droplet is to be mobilized by injecting water through a pore throat. The oil–water interface has the rear radius of curvature $r_A = 25\times10^{-6}\ \text{m}$ and the forward radius of curvature $r_B = 5\times10^{-6}\ \text{m}$. The pore is completely water-wet (contact angle $=0^\circ$) and interfacial tension is $\sigma = 0.025\ \text{N/m}$. The minimum pressure drop required to mobilize the trapped oil droplet is ________ N/m$^2$ (nearest integer).
A non-Newtonian drilling fluid (Bingham plastic) is between two flat parallel rectangular plates of area $10\ \mathrm{cm^2}$ each, separated by $1\ \mathrm{cm}$. A force of $300$ dyne is required to initiate motion of the upper plate. A force of $600$ dyne keeps the plate moving at a constant velocity of $10\ \mathrm{cm/s}$. The constitutive law is \[ \tau_{yx} = \mu_p \dot{\gamma} + \tau^o_{yx}. \] Find the Bingham plastic viscosity $\mu_p$ in dyne$\cdot$s/cm$^2$ (rounded to the nearest integer).