/* Isothermal compressible capillary flow and ideal Rankine method. */
kill(all)$
display2d:false$

C : %pi*a^4*(p1^2-p2^2)/(16*eta*l)$
Q2 : C/p2$
eta_outlet : %pi*a^4*(p1^2-p2^2)*t/(16*l*p2*V2)$

print("Outlet-flow viscosity residual (must be 0):",
  ratsimp(ev(Q2,eta=eta_outlet)-V2/t))$
print("Pressure-factorization residual (must be 0):",
  expand(p1^2-p2^2-(p1-p2)*(p1+p2)))$

pm : (p1+p2)/2$
Qm : ratsimp(C/pm)$
Qm_expected : %pi*a^4*(p1-p2)/(8*eta*l)$
print("Mean-pressure flow residual (must be 0):",
  ratsimp(Qm-Qm_expected))$

Q2_drop : ratsimp(ev(Q2,p1=p2+d))$
Q2_drop_expected : %pi*a^4*d/(8*eta*l)*(1+d/(2*p2))$
print("Small-drop expansion residual (must be 0):",
  ratsimp(Q2_drop-Q2_drop_expected))$

/* Exact closed-loop Rankine method. */
pL : P-dpr+dpr*VL/Vtot$
pH : P+dpr*VL/Vtot$
print("Rankine pressure-difference residual (must be 0):",
  ratsimp(pH-pL-dpr))$
print("Rankine gas-content residual (must be 0):",
  ratsimp(pL*VL+pH*(Vtot-VL)-P*Vtot))$

throughput : %pi*a^4*(pH^2-pL^2)/(16*eta*l)$
storage_derivative : diff(pL*VL,VL)$
dt_dVL : ratsimp(storage_derivative/throughput)$
dt_dVL_expected : 16*eta*l/(%pi*a^4*dpr)
  *(P-dpr+2*dpr*VL/Vtot)/(2*P-dpr+2*dpr*VL/Vtot)$
print("Rankine differential-time residual (must be 0):",
  ratsimp(dt_dVL-dt_dVL_expected))$

Iprimitive : VL-P*Vtot/(2*dpr)
  *log(2*P-dpr+2*dpr*VL/Vtot)$
ratio_integrand : (P-dpr+2*dpr*VL/Vtot)
  /(2*P-dpr+2*dpr*VL/Vtot)$
print("Rankine antiderivative residual (must be 0):",
  ratsimp(diff(Iprimitive,VL)-ratio_integrand))$

Iab : (Vb-Va)-P*Vtot/(2*dpr)
  *log((2*P-dpr+2*dpr*Vb/Vtot)
    /(2*P-dpr+2*dpr*Va/Vtot))$
print("Rankine small-pressure integral residual (must be 0):",
  ratsimp(limit(Iab,dpr,0)-(Vb-Va)/2))$

eta_rankine_small : %pi*a^4*(m*g/A)*t/(16*l*((Vb-Va)/2))$
print("Rankine quoted-limit residual (must be 0):",
  ratsimp(ev(eta_rankine_small,Vb-Va=Omega)
    -%pi*a^4*m*g*t/(8*l*A*Omega)))$

quit();