Using MATLAB's symbolic tool to find functional relationships

Question

I am working with MATLAB's symbolic tools to rearrange some equations to find the functional relationship between the three.

My equations are...

, ,

All the terms in the above equations are related by

and where dP/dx is constant and my goal is to get an answer in the form .

This is the start of my code.

syms V u v H mu rho dPdx tau % u and v replace V1 and V2
eqn1 = Pi_1 == tau/(rho*V^2);
eqn2 = Pi_2 == (rho*V*H)/mu;
eqn3 = Pi_3 == (dPdx*H)/(rho*V^2);

eqnA = V == (u+v)/2 - H^2/(12*mu)*dPdx;
eqnB = tau == (H/2)*dPdx + mu*(u-v)/2;

A = solve(eqn1,tau);
B = solve(eqn2,V);
C = solve(eqn3,dPdx);

eqnA2 = subs(eqnA,[V,dPdx],[B,subs(C,V,B)]);
eqnB2 = subs(subs(eqnB,dPdx,C),V,B);

eqns = [eqnA2;eqnB2];

S = solve(eqns);
u = S.u;   % V1
v = S.v;   % V2

But now if I take that solution for u and v and plug it into eqnB and solve for Pi_1 I get Pi_1 = Pi_1 which is not what I'm looking for.