SonographyAcademy

Curriculum · Doppler & Hemodynamics

Hemodynamics: Poiseuille, Continuity & Bernoulli

The flow physics behind Doppler interpretation: laminar vs turbulent flow and the Reynolds number, Poiseuille's r⁴ law, the continuity equation, and the simplified Bernoulli equation (ΔP = 4v²) with its limits.

~40 min · level: advanced · SPIVascularAdult Echo draft — pending clinical review

Learning objectives

  • Differentiate laminar from turbulent flow and interpret the Reynolds number.
  • Apply Poiseuille's law and explain the dominance of the r⁴ term.
  • Use the continuity equation to relate stenosis area to velocity (and compute valve area).
  • Apply the simplified Bernoulli equation, derive its constant, and know when the full form is required.

Laminar flow is orderly, with parallel streamlines (a blunt "plug" profile in large vessels, a parabolic profile in long straight tubes). Turbulent flow is chaotic with eddies; it dissipates energy and produces the spectral broadening, bruits, and post-stenotic color mosaic that flag disease. The transition is predicted by the dimensionless Reynolds number:

Re=ρ v DηRe = \frac{\rho\,v\,D}{\eta}
Reynolds number: ρ density, v velocity, D diameter, η dynamic viscosity.

For steady flow, Re<2000Re < 2000 is laminar and Re≳2300Re \gtrsim 2300 favors turbulence. Blood density ρ≈1060 kg/m3\rho \approx 1060\text{ kg/m}^3 and viscosity η≈0.0035 Pa\cdotps\eta \approx 0.0035\text{ Pa·s}.

Resistance to steady laminar flow is given by Poiseuille's law:

Q=ΔP π r48 η L⟺R=8 η Lπ r4Q = \frac{\Delta P\,\pi\,r^{4}}{8\,\eta\,L} \qquad\Longleftrightarrow\qquad R = \frac{8\,\eta\,L}{\pi\,r^{4}}
Volume flow Q and resistance R for a rigid tube.

Conservation of mass gives the continuity equation, the physical basis for grading stenosis by velocity:

Q=A1v1=A2v2⟹v2=v1 A1A2Q = A_1 v_1 = A_2 v_2 \quad\Longrightarrow\quad v_2 = v_1\,\frac{A_1}{A_2}
Where cross-sectional area falls (a stenosis), velocity must rise.

The same principle yields the echocardiographic aortic valve area from the LVOT, since a small valve drives a high jet velocity:

AVA=ALVOT⋅VTILVOTVTIAV,ALVOT=π(DLVOT2)2AVA = \frac{A_{LVOT}\cdot VTI_{LVOT}}{VTI_{AV}}, \qquad A_{LVOT} = \pi\left(\frac{D_{LVOT}}{2}\right)^2
Continuity-equation valve area (VTI = velocity–time integral).

Finally, the simplified Bernoulli equation converts a measured velocity into a pressure gradient — the workhorse of valvular and vascular quantification:

ΔP=4(v22−v12)  ⇒  ΔP=4v2(v1 small; mmHg, m/s)\Delta P = 4(v_2^{2}-v_1^{2})\;\Rightarrow\;\boxed{\Delta P = 4v^{2}}\quad(v_1\ \text{small; mmHg, m/s})
Full and simplified Bernoulli; the simplified form drops the proximal velocity.
MeasurementRelationshipYields

Aortic stenosis peak gradient

ΔP=4 VAV2\Delta P = 4\,V_{AV}^2

Peak LV–Ao gradient

RVSP / pulmonary pressure

RVSP=4 (TRVmax)2+RAPRVSP = 4\,(TR_{Vmax})^2 + RAP

PA systolic pressure

Mitral stenosis

mean of 4v24v^2 across diastole

Transmitral gradient

VSD / PDA

ΔP=4v2\Delta P = 4v^2

Inter-chamber pressure difference

Common clinical applications of the Bernoulli relationship.
Worked example — pulmonary pressure

A tricuspid regurgitation jet peaks at 3.0 m/s; estimated right atrial pressure is 8 mmHg. Estimate the RV systolic (≈ PA systolic) pressure.

Solution.

RVSP=4(3.0)2+8=4(9)+8=36+8=44 mmHgRVSP = 4(3.0)^2 + 8 = 4(9) + 8 = 36 + 8 = 44\text{ mmHg}. This non-invasive estimate of pulmonary artery systolic pressure is a routine, high-value echo measurement.

Key takeaways

  • Poiseuille's law makes radius the dominant determinant of flow because resistance varies with the fourth power of radius, so halving the radius raises resistance 16-fold and cuts flow to 1/16.
  • The continuity equation (Q = A1v1 = A2v2) means a smaller cross-sectional area forces a higher velocity, which is the physical basis for grading stenosis severity by peak velocity.
  • The simplified Bernoulli equation, deltaP = 4v^2, converts a measured velocity (m/s) into a pressure gradient (mmHg), and the constant 4 bundles blood density and the Pa-to-mmHg unit conversion.
  • Simplified Bernoulli assumes proximal velocity is negligible, so when v1 exceeds about 1.0-1.5 m/s you must use the full form deltaP = 4(v2^2 - v1^2) to avoid overestimating the gradient.
  • Estimate pulmonary artery systolic pressure with RVSP = 4(TR Vmax)^2 + RAP, and recall steady flow is laminar at Re < 2000 with turbulence favored above Re ~2300.

Check your understanding

Registry-style items with worked rationales.

1If the radius of a vessel is reduced by half, resistance to laminar flow (by Poiseuille's law) increases by a factor of:application

2An aortic stenosis jet measures 4 m/s with a negligible proximal velocity. The estimated peak gradient is:application

3The continuity equation predicts that as blood passes through a stenosis, velocity will:analysis

Go deeper — trusted free resources

Hand-picked, verified links to authoritative open resources. Opens in a new tab.

Radiopaedia
Bernoulli equation (physics)

Explains the full and simplified/modified Bernoulli equation and how a measured velocity is converted to a pressure gradient across a stenotic valve.

Open →
StatPearls
Physiology, Cardiovascular Hemodynamics

Foundational overview of blood flow physics, including the Poiseuille relationship Flow = (P1 - P2)/R and the determinants of vascular resistance.

Open →
StatPearls
Basic Concepts of Echocardiography Hemodynamics

Applies the continuity equation and modified Bernoulli equation to Doppler echo, connecting conservation of mass and energy to stenosis velocity criteria.

Open →
StatPearls
Physiology, Peripheral Vascular Resistance

Detailed treatment of the Hagen-Poiseuille equation, emphasizing how vessel radius to the fourth power dominates resistance to blood flow.

Open →

References

  1. Kremkau FW. Sonography Principles and Instruments. 9th ed. Elsevier; 2016.
  2. American Society of Echocardiography guidelines (valve quantification, diastolic function).
  3. Grant EG, et al. Carotid artery stenosis: gray-scale and Doppler US diagnosis — SRU Consensus. Radiology. 2003.
  4. IAC Updated Recommendations for Carotid Stenosis Interpretation Criteria (2023).
  5. Rethinking the simplified Bernoulli for transvalvular gradients. Med Biol Eng Comput. 2020.