Integrating Compressible Flow Equations for V as a Function of x in MATLAB

[yangshi]

Homework Statement

( V^1.4 A^.4 C1 - (1/V) ) dV = dA / A
C1 is a constant, V=f(x), A=.25*pi*(.022² - x²)
I'm trying to simplify the equation into a form with no integrals or derivatives, so I can put it into MATLAB to spit out an expression for V as a function of x. Or is it possible to put all this into MATLAB (r2011a)? Sorry about the notation; I'm new to this.

Homework Equations

The Attempt at a Solution

Derived this expression from Euler's, energy, ideal gas, continuity, and calorically perfect equations for compressible, isentropic flow. If I integrate both sides, I have trouble integrating:
V^1.4 A^.4 dV
It seems I have to know V(x) in the first place to integrate with respect to V.

 

[Ray Vickson]

Homework Statement

( V^1.4 A^.4 C1 - (1/V) ) dV = dA / A
C1 is a constant, V=f(x), A=.25*pi*(.022² - x²)
I'm trying to simplify the equation into a form with no integrals or derivatives, so I can put it into MATLAB to spit out an expression for V as a function of x. Or is it possible to put all this into MATLAB (r2011a)? Sorry about the notation; I'm new to this.

Homework Equations

The Attempt at a Solution

Derived this expression from Euler's, energy, ideal gas, continuity, and calorically perfect equations for compressible, isentropic flow. If I integrate both sides, I have trouble integrating:
V^1.4 A^.4 dV
It seems I have to know V(x) in the first place to integrate with respect to V.

Using your definitions, we have
V^{1.4} A^{.4} dV = f(x)^{1.4} (a(b^2 - x^2))^{.4} d f(x),
where $a = \pi/4, \: b = 0.22$. You can write $df(x) = f'(x) \, dx$, so your integral has the form
\int f'(x) f(x)^{1.4} (a b^2 - a x^2)^{.4} \, dx
Whether or not this is "doable" depends on the form of the function $f(x)$. You have not told us what is $f(x)$.

 

[yangshi]

I'm actually trying to figure out an expression for V=f(x). I know V=f(x) decreases when A increases, though I expect V=f(x) to be some ridiculous function. I'm trying to approach the problem analytically instead of numerically finding V, though it may be possible to input the whole function into MATLAB since everything's in terms of x. Thanks!