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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 (1907–1958) for his work with blood flow in arteries. The Womersley number is important in keeping dynamic similarity when scaling an experiment. An example of this is scaling up the vascular system for experimental study. The Womersley number is also important in determining the thickness of the boundary layer to see if entrance effects can be ignored. The square of this number is also referred to as the Stokes number, \text=^2, due to the pioneering work done by Sir George Stokes on the Stokes second problem. Derivation The Womersley number, usually denoted \alpha, is defined by the relation \alpha^2 = \frac = \frac = \frac = \frac \, , where L is an appropriate length scale (for example the radius of a pipe), \omega is the ang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensionless Number
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined Unit of measurement, units. For instance, alcohol by volume (ABV) represents a volumetric ratio; its value remains independent of the specific Unit of volume, units of volume used, such as in milliliters per milliliter (mL/mL). The 1, number one is recognized as a dimensionless Base unit of measurement, base quantity. Radians serve as dimensionless units for Angle, angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference. Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinematic Viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per metre squared, or pascal-seconds. Viscosity quantifies the internal friction, frictional force between adjacent layers of fluid that are in relative motion. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls. Experiments show that some stress (physics), stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biomechanics
Biomechanics is the study of the structure, function and motion of the mechanical aspects of biological systems, at any level from whole organisms to Organ (anatomy), organs, Cell (biology), cells and cell organelles, using the methods of mechanics. Biomechanics is a branch of biophysics. Etymology The word "biomechanics" (1899) and the related "biomechanical" (1856) come from the Ancient Greek βίος ''bios'' "life" and μηχανική, ''mēchanikē'' "mechanics", to refer to the study of the mechanical principles of living organisms, particularly their movement and structure. Subfields Biofluid mechanics Biological fluid mechanics, or biofluid mechanics, is the study of both gas and liquid fluid flows in or around biological organisms. An often studied liquid biofluid problem is that of blood flow in the human cardiovascular system. Under certain mathematical circumstances, blood flow can be modeled by the Navier–Stokes equations. ''In vivo'' whole blood is assum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microcirculation
The microcirculation is the circulation of the blood in the smallest blood vessels, the microvessels of the microvasculature present within organ tissues. The microvessels include terminal arterioles, metarterioles, capillaries, and venules. Arterioles carry oxygenated blood to the capillaries, and blood flows out of the capillaries through venules into veins. In addition to these blood vessels, the microcirculation also includes lymphatic capillaries and collecting ducts. The main functions of the microcirculation are the delivery of oxygen and nutrients and the removal of carbon dioxide (CO2). It also serves to regulate blood flow and tissue perfusion, thereby affecting blood pressure and responses to inflammation which can include edema (swelling). Most vessels of the microcirculation are lined by flattened cells of the endothelium and many of them are surrounded by contractile cells called pericytes. The endothelium provides a smooth surface for the flow of blood and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stokes Number
The Stokes number (Stk), named after Sir George Stokes, 1st Baronet, George Gabriel Stokes, is a dimensionless number characterising the behavior of particles Suspension (chemistry), suspended in a fluid flow. The Stokes number is defined as the ratio of the characteristic time of a particle (or Drop (liquid), droplet) to a characteristic time of the flow or of an obstacle, or \mathrm = \frac where t_0 is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), u_0 is the fluid velocity of the flow well away from the obstacle, and l_0 is the characteristic dimension of the obstacle (typically its diameter) or a characteristic length scale in the flow (like boundary layer thickness). A particle with a low Stokes number follows fluid streamlines (perfect advection), while a particle with a large Stokes number is dominated by its inertia and continues along its initial trajectory. In the case of Stokes flow, which is whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strouhal Number
In dimensional analysis, the Strouhal number (St, or sometimes Sr to avoid the conflict with the Stanton number) is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind. The Strouhal number is an integral part of the fundamentals of fluid mechanics. The Strouhal number is often given as : \text = \frac, where ''f'' is the frequency of vortex shedding in Hertz, ''L'' is the characteristic length (for example, hydraulic diameter or the airfoil thickness) and ''U'' is the flow velocity. In certain cases, like heaving (plunging) flight, this characteristic length is the amplitude of oscillation. This selection of characteristic length can be used to present a distinction between Strouhal number and reduced frequency: : \text = \frac, where ''k'' is the reduced frequency, and ''A'' is amplitude of the heaving oscilla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reynolds Number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar flow, laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulence, turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (Eddy (fluid dynamics), eddy currents). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar–turbulent transition, laminar to turbulent flow and is used in the scaling of similar but different-sized fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Length
In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by the construction of a dimensionless quantity, in the general framework of dimensional analysis and in particular applications such as fluid mechanics. In computational mechanics, a characteristic length is defined to force localization of a stress softening constitutive equation. The length is associated with an integration point. For 2D analysis, it is calculated by taking the square root of the area. For 3D analysis, it is calculated by taking the cubic root of the volume associated to the integration point. Examples A characteristic length is usually the volume of a system divided by its surface: L_c = \frac For example, it is used to calculate flow through circular and non-circular tubes in order to examine flow conditions (i.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Viscosity
Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per metre squared, or pascal-seconds. Viscosity quantifies the internal frictional force between adjacent layers of fluid that are in relative motion. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls. Experiments show that some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube with a constant rate of flow, the strengt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
