Question

6.8 From the general equations of motion in cylindrical coordinates, prove that for incompressible, steady and axisymmetric flow $\frac{\partial^2 \phi}{\partial r^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{1}{r} \frac{\partial \phi}{\partial r} = 0$ $\frac{\partial^2 \psi}{\partial r^2} + \frac{\partial^2 \psi}{\partial x^2} - \frac{1}{r} \frac{\partial \psi}{\partial r} = 0$ State any other assumptions made in the derivation of the above equations.

          6.8 From the general equations of motion in cylindrical coordinates,
prove that for incompressible, steady and axisymmetric flow
$\frac{\partial^2 \phi}{\partial r^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{1}{r} \frac{\partial \phi}{\partial r} = 0$
$\frac{\partial^2 \psi}{\partial r^2} + \frac{\partial^2 \psi}{\partial x^2} - \frac{1}{r} \frac{\partial \psi}{\partial r} = 0$
State any other assumptions made in the derivation of the above
equations.

6.8 From the general equations of motion in cylindrical coordinates,
prove that for incompressible, steady and axisymmetric flow
(∂^2 ϕ)/(∂ r^2) + (∂^2 ϕ)/(∂ x^2) + (1)/(r)(∂ϕ)/(∂ r) = 0
(∂^2 ψ)/(∂ r^2) + (∂^2 ψ)/(∂ x^2) - (1)/(r)(∂ψ)/(∂ r) = 0
State any other assumptions made in the derivation of the above
equations.

Added by Donald S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Hubert Agamasu

Step 1

This means that the flow variables (velocity components u, v, w, pressure p, and density ρ) do not change with time and are only functions of the radial coordinate r and the axial coordinate z. ∂/∂t = 0 Also, for an axisymmetric flow, the velocity components do Show more…

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6.8 From the general equations of motion in cylindrical coordinates, prove that for incompressible, steady, and axisymmetric flow: âˆ‚Â²u/âˆ‚xÂ² + (1/r)(âˆ‚u/âˆ‚r) + (1/rÂ²)(âˆ‚Â²u/âˆ‚rÂ²) = 0 State any other assumptions made in the derivation of the above equations.