Simplifying a Navier-Stokes Equation using Calculus

fullaclips

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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.
 
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.
For simplicity, let V=VΘ and V' = dV/dr, etc. If we ignore the asterisk, i.e treat r* as r, your initial equation is
(1) (r v')' - V/r = 0.
We will also need
(2) (V/r)' = V'/r - V/r2

Now (1) gives
(1') r V'' + V' - V/r = r [ V'' + V'/r - V/r2] = 0
or using (2)
(1'') r V'' + V' - V/r = r [ V'' + (V/r)'] = 0

Now, if r is not zero, one could divide by r.
 
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