Solution (source code)

= Solution

\b[The <heart> supplies the principal continuing mechanical input to <blood circulation>; arterial recoil, respiratory movements and the action of <skeletal muscles> redistribute and supplement that input.] They have different sources of <energy> and should not all be called independent propulsive powers. The <pressure gradient> remaining after passage through the <capillaries> is still a consequence of the upstream pump.

\b[Arterial elasticity.] During ventricular <systole>, ejected <blood> expands the <aorta> and large <arteries>, storing <elastic potential energy> in their walls. During <diastole>, recoil continues to drive <blood> into the peripheral vessels while closure of the <aortic valve> limits return towards the <heart>. This reduces the intermittency of flow downstream. The recoil is a temporary store of cardiac <work>, rather than a fresh source of <energy>. A vessel that is too rigid provides less of this smoothing; a compliant vessel with an open backward outlet would not provide the same useful result.

\b[The skeletal-muscle pump.] Contracting <skeletal muscles> compress adjacent <veins>. <Venous valves> make the displacement preferentially towards the <heart>; when compression ceases, refilling permits another stroke. Walking therefore aids return from dependent limbs. The source of the added <work> is <muscle contraction>, and the valves guide it without producing <energy>. Merely possessing valves cannot make a stationary column circulate.

\b[The respiratory pump.] During inspiration, falling intrathoracic <pressure> and displacement of the abdominal contents favour a <pressure gradient> from abdominal and peripheral <veins> towards the thoracic <veins> and right side of the <heart>. The <respiratory diaphragm> thus participates indirectly in <venous return> as well as <pulmonary ventilation>. The exact flow varies with posture, vessel collapse and the phase of the respiratory cycle; positive-pressure inflation does not reproduce the same mechanical circumstances as spontaneous inspiration.<Anatomy and Physiology 2e by J. Gordon Betts and others/20.2 Blood Flow, Blood Pressure, and Resistance>[Venous System; Skeletal Muscle Pump; Respiratory Pump]{ref}

\b[Venous tone and cardiac filling.] <Sympathetic> contraction of venous <smooth muscle> can reduce venous storage capacity and transfer <blood> towards the central circulation. This alters the volume available for filling rather than requiring a wave of venous peristalsis. Cardiac relaxation, recoil and movement of the atrioventricular plane also affect filling <pressure>. Some traditionally described cardiac suction is therefore a feature of the cardiac cycle and mechanical coupling, not a separate organ pulling <blood> through every vessel.<Anatomy and Physiology 2e by J. Gordon Betts and others/20.4 Homeostatic Regulation of the Vascular System>[Neural Regulation of Blood Flow; sympathetic effects on vessels and venous return]{ref}

\b[Gravity and supposed capillary attraction.] <Gravity> assists a downward column but opposes an upward one. In a complete circuit returning to its original height it supplies no net gravitational <work>; its important practical effects include hydrostatic <pressure> differences and dependent venous pooling. Ordinary <capillary action> in an open narrow tube, driven by a liquid meniscus, is not a general pump in a fluid-filled <capillary> network. Nor does a vague attraction of <blood> by <tissues> replace a measurable <pressure gradient>. Local <vasodilation> can increase <perfusion> by lowering <vascular resistance>, but that change does not manufacture the necessary <pressure> source.

For a simple mean-flow description, with $Q$ the volume flow rate, $\Delta P$ the driving <pressure gradient> and $R$ the effective <vascular resistance>,
$$
\boxed{Q=\frac{\Delta P}{R}}.
$$
This relation distinguishes a source of driving <pressure> from a change of resistance. In a lumped compliant arterial reservoir with <vascular compliance> $C$ and no inflow between beats, volume balance gives
$$
C\frac{dP}{dt}=-\frac{P-P_v}{R},\qquad
P(t)-P_v=(P(0)-P_v)e^{-t/(RC)},
$$
where $P_v$ is the downstream <pressure>. The <exponential decay> follows from stored volume leaving through the resistance. It explains continued flow during <diastole> without claiming that the idealization describes every reflected arterial wave.

A physical demonstration can compare a pulsatile pump feeding a narrow outlet through rigid tubing with the same pump feeding it through an elastic reservoir. Collecting the outflow over the pumping and resting phases reveals the reservoir's smoothing action; a side-arm <manometer> records its falling <pressure>. Separately compressing a valved flexible tube shows the role of a peripheral <muscle pump>. Keeping pump stroke and outlet resistance comparable prevents the apparatus from conflating a smoother flow with creation of extra <energy>.