For the case of incompressible flow, write all 3 components of
the Navier-Stokes equations and the complete form of the
differential continuity equation.
Starting from the general expression of the Navier-Stokes
equations in cylindrical coordinates, provide the form of the
equations for an axisymmetric, steady flow. Explicitly write down
the continuity equation as well as the momentum equation in all
relevant directions in terms of partial derivatives. (Hint: How
much is uθ for this flow? Explain why. How much is ∂/∂θ ?
IMPORTANT NOTE: Please have the answer complete,
clear and computer generated!!
What is the closure problem, which occurs when applying
Reynolds-averaging to the Navier-
Stokes equations, and why is it a problem? How does one work around
it?
Use Cauchy-Riemann equations to show that the complex function
f(z) = f(x + iy) = z(x + iy) is nowhere differentiable except at
the origin z = 0.6 points) 2. Use Cauchy's theorem to evaluate the
complex integral ekz -dz, k E R. Use this result to prove the
identity 0"ck cos θ sin(k sin θ)de = 0
Show that every Pythagorean triple (x, y, z) with x, y, z having
no common factor d > 1 is of the form (r^2 - s^2, 2rs, r^2 +
s^2) for positive integers r > s having no common factor > 1;
that is
x = r^2 - s^2, y = 2rs, z = r^2 + s^2.