AERE $3100-$ Midterm $1$ Bonus problem

Dute date: $10/12/2024$

Problem statement: Two opposing water jets $($density of ${}_j)$ with velocity of $V_j$ and a constant cross$-$sectional area of $A_j$ hold a plate assembly with total mass of $M$ in place as shown in the figure. The assembly includes a top plate connected at the bottom to a flat plate with length $L$ sitting on a thin layer of oil $($gap height of $H)$ with viscosity of $.$ At time zero, the plate assembly is suddenly moved to the right with an initial speed of $V_j.$ Determine the position of the plate as a function of time, $x(t)$ and the maximum displacement $x_{\text{m}\text{a}\text{x}}.$

You may assume the flow at any given time is quasi$-$steady and ignore the mass of water adhering to the plate. Assume motion is purely horizontal. The motion of top plate creates a quasit$-$steady motion within the oil gap where the flow is horizontal with a linear velocity profile given as $V\frac{y}{H}.$The shear stress at the surface of flat plate follows the Newtonian fluid model as $=\frac{du}{dy}.$