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For the case of a sphere in a uniform far field flow, it is advantageous to use a cylindrical coordinate system (r, φ, z). The z –axis is through the centre of the sphere and aligned with the mean flow direction, while r is the radius as measured perpendicular to the z –axis. The origin is at the sphere centre.
Shown is a sphere in Stokes flow, at very low Reynolds number. Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, [1] is a type of fluid flow where advective inertial forces are small compared with viscous forces. [2] The Reynolds number is low, i.e. . This is a typical situation in flows where the ...
Drag coefficients in fluids with Reynolds number approximately 10 4 [1] [2] Shapes are depicted with the same projected frontal area. In fluid dynamics, the drag coefficient (commonly denoted as: , or ) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water.
In turbulent flow the flow rate is proportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressure gradient in laminar flow. Using the definition of the Reynolds number we can see that a large diameter with rapid flow, where the density of the blood is high, tends towards turbulence.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
Streamlines around a sphere in axisymmetric Stokes flow.At terminal velocity the drag force F d balances the force F g propelling the object.. In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry.
A shift in the position of the reference point effectively adds a constant (for steady flow) or a function solely of time (for nonsteady flow) to the stream function at every point . The shift in the stream function, Δ ψ {\displaystyle \Delta \psi } , is equal to the total volumetric flux, per unit thickness, through the continuous surface ...
In fluid mechanics, the Rayleigh number (Ra, after Lord Rayleigh [1]) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. [2] [3] [4] It characterises the fluid's flow regime: [5] a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow.