1. The problem describes the steady motion of an incompressible viscous fluid between two parallel planes: the lower plane at rest (z = 0) and the upper plane moving with constant velocity $V$ at $z = h$. 2. The velocity vector in the fluid is given by $\mathbf{q} = v(z) \mathbf{j}$ where $v$ depends only on $z$ due to incompressibility and no slip at the boundaries. 3. The Navier-Stokes equation for steady, incompressible, viscous flow without body forces simplifies to: $$0 = -\nabla p + \mu \nabla^2 \mathbf{q}$$ 4. Since the pressure gradient in $x$ and $z$ directions are zero and pressure varies only in $y$, the governing equation reduces to: $$\frac{dp}{dy} = \mu \frac{d^2v}{dz^2}$$ where $\frac{dp}{dy} = -P$ with $P > 0$ indicating pressure decreases in the positive $y$ direction. 5. Integrating the velocity equation twice: $$v(z) = A + Bz - \frac{P}{2\mu} z^2$$ 6. Applying boundary conditions: - At $z = 0$, $v = 0$ gives $A = 0$. - At $z = h$, $v = V$ gives: $$V = Bh - \frac{P}{2\mu} h^2 \implies B = \frac{V}{h} + \frac{Ph}{2\mu}$$ 7. Hence the velocity profile is: $$v(z) = \left( \frac{V}{h} + \frac{Ph}{2\mu} \right) z - \frac{P}{2\mu} z^2$$ 8. The profile is parabolic, reflecting the combined effect of the moving plane and the pressure gradient. 9. The volumetric flow rate per unit breadth in the $x$ direction is: $$Q = \int_0^h v(z) dz = \frac{1}{2} V h + \frac{P h^3}{12 \mu}$$ 10. Tangential shear stress, $\tau = \mu \frac{dv}{dz}$, at any $z$ is: $$\tau = \mu \left( \frac{V}{h} + \frac{Ph}{2\mu} - \frac{P}{\mu} z \right) = \mu \frac{V}{h} + \frac{P h}{2} - P z$$ 11. The drag per unit area on the lower plane ($z=0$) is: $$\tau_0 = \mu \frac{V}{h} + \frac{P h}{2}$$ 12. The drag per unit area on the upper plane ($z=h$) is: $$\tau_h = \mu \frac{V}{h} - \frac{P h}{2}$$ \textbf{Final Results:} - Velocity profile: $$v(z) = \left( \frac{V}{h} + \frac{Ph}{2\mu} \right) z - \frac{P}{2\mu} z^2$$ - Volumetric flow rate per unit breadth: $$Q = \frac{1}{2} V h + \frac{P h^3}{12 \mu}$$ - Shear stress at boundaries: $$\tau_0 = \mu \frac{V}{h} + \frac{P h}{2}, \quad \tau_h = \mu \frac{V}{h} - \frac{P h}{2}$$