fullaclips
New member
- Joined
- Oct 7, 2016
- Messages
- 3
In my fluid dynamics course we are simplifying the Navier-Stokes equations in cylindrical coordinates for a Couette Flow between rotating, concentric cylinders.
When applying all assumptions, the equation for Θ momentum simplifies to:
(d/dr)(r*(dVΘ/dr))-(VΘ/r)=0
But the book gives the simplified form of this equation as
(d2VΘ)/(dr2)+(d/dr)(VΘ/r)=0
without giving any explanation.
I am interested in understanding how to go from the first to the second equation.
I assume that you would use the product rule to simplify (d/dr)(r*(dVΘ/dr)) which would lead to
r*(d2VΘ)/(dr2)+(dVΘ/dr)*(dr)-(VΘ/r)=0
but I am not sure how to proceed from there. Also I am not sure if the lone (dr) in (dVΘ/dr)*(dr) is necessary. If it is, would the two dr's cancel out? I am able to solve the simplified ODE that the book provides. All I want to understand is the simplification process. Also, I am certain the first equation is correct.
Thank you in advance for any help you can offer.
When applying all assumptions, the equation for Θ momentum simplifies to:
(d/dr)(r*(dVΘ/dr))-(VΘ/r)=0
But the book gives the simplified form of this equation as
(d2VΘ)/(dr2)+(d/dr)(VΘ/r)=0
without giving any explanation.
I am interested in understanding how to go from the first to the second equation.
I assume that you would use the product rule to simplify (d/dr)(r*(dVΘ/dr)) which would lead to
r*(d2VΘ)/(dr2)+(dVΘ/dr)*(dr)-(VΘ/r)=0
but I am not sure how to proceed from there. Also I am not sure if the lone (dr) in (dVΘ/dr)*(dr) is necessary. If it is, would the two dr's cancel out? I am able to solve the simplified ODE that the book provides. All I want to understand is the simplification process. Also, I am certain the first equation is correct.
Thank you in advance for any help you can offer.