In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, a flow with periodic variations is known as pulsatile flow, or as Womersley flow. The flow profiles was first derived by
John R. Womersley
John Ronald Womersley (20 June 1907 – 7 March 1958) was a British mathematician and computer scientist who made important contributions to computer development, and hemodynamics. Nowadays he is principally remembered for his contribution ...
(1907–1958) in his work with blood flow in
arteries
An artery (plural arteries) () is a blood vessel in humans and most animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). Most arteries carry oxygenated blood; the two exceptions are the pul ...
. The
cardiovascular
The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, tha ...
system of
chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in
engines
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power ...
and
hydraulic system
Hydraulics (from Greek language, Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is th ...
s, as a result of
rotating
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
mechanisms pumping the fluid.
Equation
![Womersley Flow Profile](https://upload.wikimedia.org/wikipedia/commons/d/d0/Womersley_Flow_Profile.svg)
The pulsatile flow profile is given in a straight pipe by
:
where:
:
Properties
Womersley number
The pulsatile flow profile changes its shape depending on the Womersley number
:
For
, viscous forces dominate the flow, and the pulse is considered
quasi-static with a parabolic profile.
For
, the inertial forces are dominant in the central core, whereas viscous forces dominate near the boundary layer. Thus, the velocity profile gets flattened, and
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
between the pressure and velocity waves gets shifted towards the core.
Function limits
Lower limit
The Bessel function at its lower
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
becomes
[
]
:
which converges to the
Hagen-Poiseuille flow profile for steady flow for
:
or to a
quasi-static pulse with parabolic profile when
:
In this case, the function is real, because the pressure and velocity waves are in phase.
Upper limit
The Bessel function at its upper limit becomes
[
:
which converges to
:
This is highly reminiscent of the Stokes layer on an oscillating flat plate, or the skin-depth penetration of an alternating magnetic field into an electrical conductor.
On the surface , but the exponential term becomes negligible once becomes large, the velocity profile becomes almost constant and independent of the viscosity. Thus, the flow simply oscillates as a plug profile in time according to the pressure gradient,
:
However, close to the walls, in a layer of thickness , the velocity adjusts rapidly to zero. Furthermore, the phase of the time oscillation varies quickly with position across the layer. The exponential decay of the higher frequencies is faster.
]
Derivation
For deriving the analytical solution of this non-stationary flow velocity profile, the following assumptions are taken:
* Fluid is homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
and Newtonian;
* Tube wall is rigid and circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circula ...
;
* Motion is laminar
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
* Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "band ...
, axisymmetric
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
and parallel to the tube's axis;
* Boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
are: axisymmetry at the centre, and no-slip condition on the wall;
* Pressure gradient is a periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
that drives the fluid;
* Gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
has no effect on the fluid.
Thus, the Navier-Stokes equation and the continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
are simplified as
:
and
:
respectively. The pressure gradient driving the pulsatile flow is decomposed in Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
,
:
where is the imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
, is the angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the first harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
(i.e., ), and are the amplitudes
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of each harmonic . Note that, (standing for ) is the steady-state pressure gradient, whose sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
is opposed to the steady-state velocity (i.e., a negative pressure gradient yields positive flow). Similarly, the velocity profile is also decomposed in Fourier series in phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
with the pressure gradient, because the fluid is incompressible,
:
where are the amplitudes of each harmonic of the periodic function, and the steady component () is simply Poiseuille flow
The poiseuille (symbol Pl) has been proposed as a derived SI unit of dynamic viscosity, named after the French physicist Jean Léonard Marie Poiseuille (1797–1869).
In practice the unit has never been widely accepted and most international s ...
:
Thus, the Navier-Stokes equation for each harmonic reads as
:
With the boundary conditions satisfied, the general solution of this ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
for the oscillatory part () is
:
where is the Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of first kind and order zero, is the Bessel function of second kind and order zero, and are arbitrary constants, and is the dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
Womersley number
The Womersley number (\alpha or \text) is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley ...
. The axisymetic boundary condition () is applied to show that for the derivative of above equation to be valid, as the derivatives and approach infinity. Next, the wall non-slip boundary condition () yields . Hence, the amplitudes of the velocity profile of the harmonic becomes
:
where is used for simplification.
The velocity profile itself is obtained by taking the real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
part of the complex function
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
resulted from the summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
of all harmonics of the pulse,
:
Flow rate
Flow rate
Flow rate may refer to:
* Flow measurement, a quantification of bulk fluid movement
* Mass flow rate, the mass of a substance which passes per unit of time
* Volumetric flow rate, the volume of fluid which passes per unit time
** Discharge (hydrol ...
is obtained by integrating the velocity field on the cross-section. Since,
:
then
:
Velocity profile
To compare the shape of the velocity profile, it can be assumed that
:
where
:
is the shape function.[
]
It is important to notice that this formulation ignores the inertial effects. The velocity profile approximates a parabolic profile or a plug profile, for low or high Womersley numbers, respectively.
Wall shear stress
For straight pipes, wall shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the oth ...
is
:
The derivative of a Bessel function is
:
Hence,
:
Centre line velocity
If the pressure gradient is not measured, it can still be obtained by measuring the velocity at the centre line. The measured velocity has only the real part of the full expression in the form of
:
Noting that , the full physical expression becomes
:
at the centre line. The measured velocity is compared with the full expression by applying some properties of complex number. For any product of complex numbers (), the amplitude and phase have the relations and , respectively. Hence,
:
and
:
which finally yield
:
See also
* Cardiovascular system
The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, tha ...
* Hemodynamics
Hemodynamics or haemodynamics are the dynamics of blood flow. The circulatory system is controlled by homeostatic mechanisms of autoregulation, just as hydraulic circuits are controlled by control systems. The hemodynamic response continuously mo ...
* Womersley number
The Womersley number (\alpha or \text) is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley ...
* Fluid hammer
References
{{Reflist
Biological engineering
Cardiovascular physiology
Fluid dynamics