Obtain the continuity equation in cylindrical coordinates by expanding the vecto
ID: 1861145 • Letter: O
Question
Obtain the continuity equation in cylindrical coordinates by expanding the vector form in cylindrical coordinates. To do this, make use of the following relationships connecting the coordinates and the velocity components in cartesian and cylindrical coordinates:
x = Rcos(theta)
y = Rsin(theta)
z = z
u = u_(R)cos(theta) - u_(theta)sin(theta)
v = u_(R)sin(theta) + u_(theta)cos(theta)
w = u_(z)
***Key: "u_(r)" is a letter u with a subscript r. "u_(theta)" is a letter u with a subscript theta.
Explanation / Answer
x = r Cos theta
y = r sin theta
r = sqrt(x^2 + y^2)
theta = atan (y/x)
del r / del x = x/sqrt(x^2 + y^2)
del r / del y = y/sqrt(x^2 + y^2)
del theta / del x = -y/(x^2 + y^2)
del theta / del y = x/(x^2 + y^2)
Thus,
del r / del x = Cos theta
del r / del y = Sin theta
del theta / del x = -(Sin theta)/r
del theta / del y = (Cos theta)/r
Further,
del / delx = (del / delr)(delr / delx) + (del / del theta)(del theta / del x)
del / dely = (del / delr)(delr / dely) + (del / del theta)(del theta / del y)
Thus,
del / del x = (Cos theta)(del / del r) - ((Sin theta)/r)(del / del theta)
del/ del y = (Sin theta) (del / del r) + ((Cos theta)/r)(del / del theta)
del u / del x = (Cos theta)(del (u_(R)cos(theta) - u_(theta)sin(theta)) / del r) - ((Sin theta)/r)(del (u_(R)cos(theta) - u_(theta)sin(theta)) / del theta)
del u / del x = (del u_r / del r) + u_r/r
del v / del y = (Sin theta) (del (u_(R)sin(theta) + u_(theta)cos(theta)) / del r) + ((Cos theta)/r)(del (u_(R)sin(theta) + u_(theta)cos(theta)) / del theta)
del v / del y = (1/r) (del u_theta / del theta)
del w / del z
Continuity eqn: del u / del x + del v / del y + del w / del z = 0
Thus,
(del u_r / del r) + u_r/r + (1/r) (del u_theta / del theta) + (del w / del z ) = 0
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.