User:Roscodono/sandbox

Explaining the requirement of elasticity for the Curvature Lift Theory and more general fluid flows
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Glossary of Terms 

$$ c $$                         ... the local speed of sound

$$ P $$                          ... the local static pressure

$$ V $$                          ... the local velocity

$$ \rho $$                      ... the local density of the fluid

$$ dl $$                        ... the length element normal to the streamline

$$ a_{radial}$$            ... the local radial acceleration of the fluid particle

$$ k $$                         ... the local curvature of the streamline

$$ M $$                         ... the local Mach Number

$$ Ca $$                       ... the local Cauchy Number

$$ \frac{d\epsilon}{dl} \  \ $$         ... the local radial dilatoric strain gradient

The relationship between flow curvature and fluid elasticity

It is assumed the flow is steady-state, isentropic and only inertial and elastic forces are applicable.

Note: Compressive strain is positive

Proof From .... $$ \     \  c   =  \sqrt{ \frac{dP}{d\rho}} $$

$$ c^2 \  \ =  \  \ \frac{dP}{d\rho}\ $$

$$\frac{dP}{dl}\ \ \ =  \  \ c^2 \  \  \frac{d\rho}{dl}\ $$

$$ - \rho \ \ a_{radial} \ \ =  \  \  c^2\   \  \frac{d\rho}{dl}\ $$

$$ a_{radial} \ \ = \ \ -\  \   c^2   \ \left (\   \frac{1}{\rho}\   \frac{d\rho}{dl}\  \right )     $$

$$ \ k\ \ V^2 \ = \ \ -\  \   c^2   \ \left (\   \frac{1}{\rho}\   \frac{d\rho}{dl}\  \right )     $$

$$ \ k \  = \ \ -\   \left (\  \frac{c^2}{V^2}\  \right ) \ \  \left (\   \frac{1}{\rho}\   \frac{d\rho}{dl}\  \right )     $$

$$ \ k \  = \ \ -\   \left (\  \frac{1}{M^2}\  \right ) \ \  \left (\   \frac{1}{\rho}\   \frac{d\rho}{dl}\  \right )     $$

$$ \ k \  =  \ \ -\   \left (\  \frac{1}{Ca}\  \right ) \ \  \left (\   \frac{1}{\rho}\   \frac{d\rho}{dl}\  \right )     $$

$$ \ k \  =  \ \ -\   \left (\  \frac{1}{Ca}\  \right ) \ \  \left (\  \frac{d\rho/\rho}{dl}    \  \right )     $$

$$ \ k \  =  \ \ -\   \left (\  \frac{1}{Ca}\  \right ) \ \  \left (\  \frac{d\epsilon}{dl}    \  \right )     $$

Where k is the local curvature of the streamline, Ca is the local Cauchy Number, and $$ \frac{d\epsilon}{dl} \   \ $$  is the local radial dilatoric strain gradient

CONCLUSION

1. General fluid flows must be considered as compressible since flows create elastic strain gradients in order to bend.

2. The presumption of incompressibility in a fluid is a mathematically convenient falsehood.

For instance, in the Bernoulli Theory, it obscures the understanding of inertia and elasticity interacting to make the fluid flow.

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