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Navier–Stokes Equations01:28

Navier–Stokes Equations

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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Steady, Laminar Flow Between Parallel Plates01:17

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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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Bernoulli's Equation for Flow Along a Streamline01:30

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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:
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Velocity and Acceleration of a Wave00:51

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A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which is not constant. In addition, the velocity of the medium is perpendicular to the velocity of the wave. The variable speed of the particles of the medium implies that there must be acceleration associated with it. 
The velocity of the particles can be obtained by taking the partial derivative of the position equation with respect to time....
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Traveling Waves: Lossless Lines01:27

Traveling Waves: Lossless Lines

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The provided content explores the behavior of traveling waves on single-phase lossless transmission lines. It begins with a single-phase two-wire lossless transmission line of length Δx, characterized by a loop inductance LH/m and a line-to-line capacitance C F/m. These parameters result in a series inductance LΔx  and a shunt capacitance CΔx.
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Modeling with Differential Equations01:25

Modeling with Differential Equations

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Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
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Related Experiment Video

Updated: Mar 6, 2026

Visualization of Flow Field Around a Vibrating Pipeline Within an Equilibrium Scour Hole
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Visualization of Flow Field Around a Vibrating Pipeline Within an Equilibrium Scour Hole

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Modeling digital pulse waveforms by solving one-dimensional Navier-stokes equations.

Aleksandr A Fedotov, Anna S Akulova, Sergey A Akulov

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    |March 9, 2017
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    Summary
    This summary is machine-generated.

    Mathematical modeling reveals distal arterial pulse waves in upper limbs are formed by forward and reflected waves. This study uses electrical transmission line analogies and Navier-Stokes equations to analyze pulse wave propagation dynamics.

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    Area of Science:

    • Biomedical Engineering
    • Fluid Dynamics
    • Cardiovascular Physiology

    Background:

    • Understanding arterial pulse wave propagation is crucial for diagnosing cardiovascular conditions.
    • Distal arterial pulse waves are complex phenomena influenced by vessel geometry and blood flow dynamics.

    Purpose of the Study:

    • To develop a mathematical model for the composition of distal arterial pulse waves in upper limb vasculature.
    • To investigate the formation of pulse waves as a combination of forward and reflected wave components.

    Main Methods:

    • Mathematical modeling of pulse wave formation and propagation.
    • Establishing a formal analogy between arterial pulse wave propagation and electrical transmission line oscillations.
    • Solving one-dimensional Navier-Stokes equations for specific arterial configurations.

    Main Results:

    • The distal arterial pulse wave is modeled as a composition of forward and reflected waves.
    • A validated analogy exists between arterial pulse wave propagation and electrical transmission line dynamics.
    • Derived dependencies for pulse wave propagation using Navier-Stokes equations.

    Conclusions:

    • The proposed mathematical model accurately represents distal arterial pulse wave formation.
    • The electrical transmission line analogy provides a useful framework for analyzing arterial hemodynamics.
    • The study offers insights into the physics governing pulse wave propagation in the human arterial system.