It depend what other parameters are constant. If delta P is const, the of course, the energy will be spent on drug. However, if Throughput(Q) is constant, then EK may be wrong. It is not obvious and need some math. It will be great if somebody can check the calcs below:

1. Let's assume that the speed at the wall is zero. (Standard assumption for non ideal liquids)

2. At radius Ro at center of tube speed(Vo) is maximum.

3. The speed profile is linear or, v(r) = Vo (1-r/Ro)

4. The throughput via the ring of size dR at radius r is:

dQ = v(r) * d(Pi*R^2) = 2*Pi*v(r)*r dR

5. Total integral from zero to Ro is:

Q=Integral(dQ)=Integral(2*Pi*v(r)*r ) dR

6. substitution v(r) from (3) to (5) gives:

Q=Integral( 2*Pi*Vo*(1-r/Ro) * r)dR

7. Taking the integral:

Q=2*Pi *Vo *(r^2/2 - r^3/3Ro)

8. Substituting limits (Ro,0) =>

Q = 2*Pi* Vo*(Ro^2/2 - Ro^2/3) = 2*Pi*Vo*Ro^2/6=Pi*Vo*Ro^2/3

9. From here Vo = 3*Q/(Pi*Ro^2)

10. Cool. the Ro^2 is in denumerator. It means that: Decreasing the radius will increase the speed of the liquid in the center quadratically.

For ideal liquid the speed increase is also quadratic. Indeed Q=v*Pi Ro^2 => V = Q/(Pi *Ro^2).

However, for non-ideal liquid the coefficient is higher.