Quasi-Hydrodynamical Systems

We wish to gain a perspective on circulations.  To that end we propose a unique, double-circuited, quasi-hydrodynamical system.



The planar diagram above represents the abstract interrelations between two circular flows in a unique closed system .  As shown by the arrowed side channels, the flows are mutually dependent upon one another. Each of the two circulation pipes contains internal flows, and each has transfer-flows to and from the other to satisfy some sort of dependency of each on the other.  There are no complications of viscosity, laminarity, homogeneity, compressibility, or curl in the fluid.  The acceleration equals zero, so a positive velocity is constant.

The system is a conservative system; it is characterized by a fixed total energy consisting of potential energy plus kinetic energy. (H = T + V; and L = T – V)


  • What is required for continuity within each of the two circulations?
  • What would constitute continuity and equilibrium of flows within and between the entire system of pipes?
  • What would be the effect if leaks were to occur in the system?


Let us postulate complexifying properties:

  • The system of flows is a non-conservative system.
  • Its flows are dualities; they are a product of two variables.
  • The flows are functional flows of productive activities rather than flows of water.  They are rates of activity; they are velocities of so much every so often. And they are implicitly-defined by their functional relations to one another.  They are mutually-definitive and mutually conditioning.
  • The top circuit’s functioning a) maintains both itself and the bottom circuit, and b) accelerates both itself and the bottom circuit.
  • It accelerates by means of improved technology.
  • The functional relations within and between the two flows are a unitary set of abstract, explanatory, primary relativities; they comprise a complete and coherent explanation of the unique or pure special relativity of the process.
  • Thus, they are objective relations of an objective process.  They are prior to, more fundamental than, and independent of the feelings, moods, expectations, or motivations of persons operating the controls.
  • Though not conservative, the system would be characterized at any instant by a total of potential energy plus kinetic energy.  However, the potential energy or the potentiality of the system can increase due to some improvement in the technology applicable to this system; and the incremental potential energy can be effected into increased kinetic energy until the potential energy is zero and the system is operating at its full capacity.
  • The fluidic content is now non-homogeneous.  Each of the two circulations may have different velocity and acceleration at different points within itself; i.e. there may be lags  within a circuit; each infinitesimal volume may fail to keep pace with all the others in its given circuit.
  • However, the top circuit’s (n’s) potential and potential  energy will increase first and become kinetic energy in that top circulation; it will first increase its own capabilities.  Its improvement may be in, say, the form of a logistic-growth or S-curve.
  • After accelerating its own capabilities, the top circuit, n, can then stimulate acceleration in the bottom circuit, n-1, according to the following rule:

kn[f’n(t-a)] = f”n-1(t)

where k is a multiplier, t-a and t are earlier and later intervals of time, f’ is the logistic increase in the top circulation, and (f”) is the lower circuit’s lagged logistic acceleration in response to the higher circuit’s stimulus.

  • Both circulations can expand in accord with the certain boundaried relations of their interdependence.  Thus, either circuit’s d2Q/dt2 may become d2(Q+ΔQ)/dt2.  The accelerations may be positive, zero, or negative, as the dualistic content becomes greater, constant, or negative.  Arrowheads indicate normal direction, which would apply in the cases of zero or positive accelerations.
  • Thus, the system is finite.  Its finitude is determined by the state of technology, natural and human resources, and culture – political, legal, and moral.

Let us make some assumptions:

  • Initially there is zero “slack” in the system; potential energy equals zero.
  • Again, new technology applicable to both circuits can increase the overall potential.

Queries to the serious reader:

  • What is required for continuity within each of the two circulations?
  • What would constitute continuity and equilibrium of flows within and between the entire system of pipes?
  • What would be the effect of leaks in the system?
  • What would be the effect of artificial bloating of either circuit?
  • Please comment on the necessity for concomitance at all flow points go each circuit in order to achieve equilibrium within and between the two circulations.
  • To gain a further understanding and appreciation of the dynamics of productive flows, please compare and contrast this representation of a simplified quasi-hydrodynamical system with Lonergan’s double-circuited, credit-centered Diagram of Rates of Pretio-Quantital Flows representing a purely relativistic macroeconomic field theory.
  • Please compare and contrast the quasi-hydrodynamical system above and Lonergan’s more nuanced treatment of the following macroeconomic phenomena:
    • circulation of Outlays, Incomes, Expenditures, and Receipts
    • invention; and widening and deepening of capital equipment,
    • increase of credit required for a geometric increase of surplus production
    • supplying credit through the productive supply vs. through the monetary demand
    • expansion in phases from a lower to a higher level of production
    • ever-shifting adjustments required for consuming vs. savings
    • interventions by monetary authorities
    • international trade imbalances and domestic fiscal imbalances
    • production time
    • price changes in pretty-quantital flows
    • magnitude and frequency of payments correlative to the magnitude and frequency of turnovers
    • depreciation and maintenance of capacity
    • human ignorance, especially as it applies to the implementation of the basic expansion
  • Please compare and contrast Lonergan’s lagged technical accelerator necessitating two circuits with the simplified formula above, which lacks treatment of depreciation and slack.

kn[f’n(t-a)] = f”n-1(t)

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1   [CWL 15, 37]

Rotating the image below by 45° counterclockwise places the accelerator circuit on top and the accelerated circuit on the bottom.