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Related Concept Videos

Velocity and Acceleration in Steady and Unsteady Flow01:11

Velocity and Acceleration in Steady and Unsteady Flow

In fluid mechanics, velocity and acceleration are key concepts for analyzing particle motion in both steady and unsteady flow. Consider a fluid particle moving along a pathline, where its velocity depends on its position and time. The particle's acceleration is obtained by differentiating the velocity with respect to time.
The acceleration can be generalized to any point in the flow, and expressed as components along three perpendicular directions, representing changes in velocity over time.
Velocity Potential01:20

Velocity Potential

In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:
Bernoulli's Equation for Flow Along a Streamline01:30

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:
Couette Flow01:22

Couette Flow

Couette flow represents the flow of fluid between two parallel plates, with one plate fixed and the other moving with a constant velocity. This configuration allows for a simplified analysis using the Navier-Stokes equations, which govern fluid motion under conditions of viscosity and incompressibility. For Couette flow, the assumptions include a steady, laminar, incompressible flow with a zero-pressure gradient in the flow direction. This flow type is beneficial for understanding shear-driven...
Applications of Integration to Find Blood Flow01:27

Applications of Integration to Find Blood Flow

Blood flow through a cylindrical blood vessel can be mathematically described using the principles of laminar flow, a regime in which fluid moves smoothly in parallel layers. In this model, the velocity of the blood is not uniform across the cross-section of the vessel; rather, it varies with the radial distance from the center. The maximum velocity occurs along the central axis, decreasing progressively toward the vessel walls, where it reaches zero due to viscous drag.Approximating Blood...
Steady, Laminar Flow Between Parallel Plates01:17

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.

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Related Experiment Video

Updated: Jul 17, 2026

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom
06:26

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom

Published on: February 25, 2022

Computationally efficient velocity profile solutions for cardiac haemodynamics.

C E Hann1, J G Chase, B W Smith

  • 1Dept. of Mech. Eng., Canterbury Univ., Christchurch, New Zealand.

Conference Proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference
|February 3, 2007
PubMed
Summary

This study presents a new linear model for cardiovascular systems, improving computational efficiency by 15x. This enhanced model accurately reflects time-varying resistance for better clinical applications in diagnosis and therapy.

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Last Updated: Jul 17, 2026

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In Silico Clinical Trials for Cardiovascular Disease

Published on: May 27, 2022

Area of Science:

  • Computational physiology
  • Mathematical modeling
  • Cardiovascular dynamics

Background:

  • Cardiovascular models often simplify resistance as constant, potentially impacting accuracy.
  • Time-varying resistance is physiologically relevant but computationally intensive.
  • Existing models face challenges in balancing accuracy and computational cost.

Purpose of the Study:

  • To reformulate nonlinear differential equations of cardiovascular models with time-varying resistance into a linear system.
  • To develop an analytical solution for improved computational efficiency.
  • To demonstrate the clinical utility of accurate cardiovascular modeling.

Main Methods:

  • Reformulation of nonlinear differential equations into a linear system.
  • Development of an analytical solution for the reformulated equations.
  • Comparison of cardiac output between time-varying and constant resistance models.

Main Results:

  • A 15x computational saving was achieved compared to previous methods.
  • A 17.5% difference in cardiac output was observed in a single-chamber model with time-varying resistance versus constant resistance.
  • The new formulation maintains physiological accuracy with minimal computational time.

Conclusions:

  • The new linear formulation offers significant computational savings for cardiovascular models.
  • Accurate representation of time-varying resistance is crucial for physiological fidelity.
  • This computationally efficient model can support clinical diagnosis and therapy selection.