Please see the attached file for the fully formatted problems.
Z(x, y) = ln(sin y/sin x)
is a solution to the minimal surface equation.
(1 + Z)Z1 + 2ZXZZX + (1 + Z)Z = 0,
in the region 0 < x < ir, 0 < y < pi. What happens on the boundary of this region? Suppose we consider a constant multiple of Z(x, y) ? is it still a solution of the PDE?
Please see the attached file for the complete solution.
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A minimum surface equation is investigated. The solution is detailed and well-presented.
Laminar Boundary Layer Flow Problem
Text Book : Viscous Fluid Flow by Tasos C. Papanastasiou
Download link for the book
8.1. Water approaches an infinitely long and thin plate with uniform velocity.
(a) Determine the velocity distribution ux in the boundary layer given that
ux(x, y) = a(x)y² + b(x)y + c(x) .
(b) What is the flux of mass (per unit length of plate) across the boundary layer?
(c) Calculate the magnitude and the direction of the force needed to keep the plate in place.