[Kummi]

Hello Foamers,
I have 2 equations to solve for 2 variables - pressure (p) and velocity (V)
Used --> del - nabla operator, dow - partial derivative term

Continuity:: del.(rho*V) = RRg --> Eq. (1)
Ergun:: del(p2) = (f1+f2*(rho*V))(rho*V) --> Eq .(2)

By manipulating Eqs. (1) and (2), the variables pressure and velocity can be obtained.
By applying the operator (del X) cross product on Eq. (2),

(del X V) - ( V X del (ln(f1+f2*V)) = 0 --> Eq. (3)

• taking (del X) on Eq. (2), the pressure term will be eliminated, since V vector is the only dependent variable.
• From Eq. (3), velocity can be computed.
• **Eq. (3) is verified with the below link: (explaining the Product Curl and the Curl) http://math.mit.edu/classes/18.013A/...section04.html

Next By applying (del) on Eq. (2),

del2(p2) = -(f1+f2*(rho*V)) * del.(rho*V) - (rho*V) . del(f1+f2*(rho*V)) --> Eq. (4)

• After computing velocity from Eq. (3), the pressure can be computed from Eq. (4)
• **Eq. (4) is verified with the below link: (explaining the Product Curl and the Divergence) http://math.mit.edu/classes/18.013A/...section02.html

Questions: 2D problem

(1) When expanding Eq. (3), [First term + Second term]

L.H.S - First term, (del X V) = Curl = [ (dow vy/dow x) - (dow vx/dow y) ]
L.H.S - Second term, it can be expanded in 2 ways,

MATRIX FORM: ( V X del (ln(f1+f2*V)) =dow /dow y [ ln(f1+f2*vy) ]*vx - dow /dow x [ ln(f1+f2*vx) ]*vy --> Eq. (5)
GEOMETRIC FORM: ( V X del (ln(f1+f2*V)) = (V) ( del (ln(f1+f2*V) ) sin (theta)

In which of the two forms, second term can be expanded ?
However, when expanding Eq. (3) as above first and second terms, it looks like a SINGLE EQUATION. So, how come its possible to obtain velocities vx and vy (2 variables) from one equation ?

(2) When expanding Eq. (4),

In R.H.S - First term, del.(rho*V) =RRg (substituted from CONTINUITY Eq. (1))
In R.H.S - Second term, Dot product is used - so cos (0)= 1 (Along x) and cos (90) = 0 (Along y)

(dow2p2/ dowx2) = -(f1+f2*(rho*vx)) RRg - (rho*vx)* dow/dowx (f1+f2*(rho*vx)) //partial derivative
(dow2p2/ dowy2) = -(f1+f2*(rho*vy)) RRg, //here second term is zero since cos 90 = 0

Pressure is a scalar quantity, from above 2 equations of (dow2p2/dowx2) and (dow2p2/dowy2), one equation is enough to calculate pressure. So, which one equation from 2 can be chosen optimistically ?
My above questions might be rather trivial, but for some reason I can't seem to get this work done. Kindly correct me if I'm wrong somewhere in expanding equations.
Kindly someone explain the physical reality of above numerical equations..
Thank you

**The above equations are from the following reference:
https://aiche.onlinelibrary.wiley.co.../aic.690200519