of this element as well as the rate at which the mass or the element is increasi
ID: 1847896 • Letter: O
Question
of this element as well as the rate at which the mass or the element is increasing. The resulting equation should be expressed in terms of the cylindrical coordinates (r, theta, omega, t), the cylindrical velocity components (ur, u theta, u omega), and the fluid density rho. 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 = R cos theta y = R sin theta z = z u = uR cos theta - u theta sin theta v = uR sin theta + u theta cos theta w = uzExplanation / Answer
in cartesian coordinate, continuity equation is
du/dx + dv/dy + dw/dz = 0, where d is partial derivative
dx = -Rsin d
dy = Rcos d
dz = dz
du/dx = [-Ur(sin)d + dUrcos - Ucosd - dUsin ]/-Rsind = [(-Ur/R) - (dUr/d)cot/R + (U/R)cot + (dU/d)/R]
dv/dx = [Urcos d + dUr sin - Usind + dUcos]/Rcosd = [(Ur/R) + (tan/R)(dUr/d) - Utan + (1/R)(dU/d)
dw/dz = dUz/dz
adding all these and equationg to 0
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