Work-in-progress: Writing and editing on Thursday, June 22, 2017
• Measuring an Expanding Universe Using Planck Units (work in progress)
• The Thrust of the Universe: What is it? (work in progress)
• Visualizing the Universe (work in progress)
Search. The words, “thrust of the universe”, were entered (with the quotes) into Google on Sunday, June 4, 2017; only seven (7) results were returned. One was an article on cosmological inflation that I wrote on July 12, 2016. The first entry was by David Birnbaum of NYU. In time each of those six other entries will be examined in depth.
Background. In and around 1975, I first heard the term, thrust of the universe within a classroom with John Niemeyer Findlay, a Rhodes scholar and expert on both Plato and Hegel. On occasion he lectured about the fabric of life, an energy and direction, an abiding thrust, to make things better. I can still see that ever-so-dear professor with a small smile and all-knowing twinkle in his eye.
Center for Science Information (CSoI). At one time, this center made “thrust” a focal point of their work. Sponsored by the National Science Foundation, this consortium of universities and scholars is based within Purdue University and it was here that I discovered the thrust of William Bialek’s work. Bialek is the John Archibald Wheeler/Battelle Professor of Physics at Princeton. Like John Wheeler whose chair he holds, Bialek is one of those rare scholars who is always in search of abiding principles. I thought that he just might find our work using base-2 exponentiation to be of interest so I was learning as much as possible about his research, particularly about his work to develop a mathematically-sophisticated introduction to the natural sciences. He was carrying on the Wheeler spirit and tradition; this project resulted in a special Freshman class seminar called Integrated Science. Wheeler once said, “Behind it all is surely an idea that is so simple, so beautiful, that when we grasp it — in a decade, a century, or a millennium — we will all say to each other, how could it have been otherwise?”^{1}
Footnote #1: John Archibald Wheeler, 1911-2008, physicist, How Come the Quantum? from New Techniques and Ideas in Quantum Measurement Theory, Annals of the New York Academy of Sciences, Vol. 480, Dec. 1986 (p. 304, 304–316), DOI: 10.1111/j.1749-6632.1986.tb12434.x
When I first discovered the Center for Science of Information, they had a clear focus and mandate from the NSF to grapple with thrust from large-scale to small-scale and throughout all of life. According to Bob Brown, the Managing Director of the center, none of their 50+ scholars are currently focused on the cosmological thrust of the universe. That’s unfortunate.
There are many possibilities to consider to find the deepest sources of thrust within this universe. Because this article will be under constant pressure to be updated with the latest and greatest insights within emergent scholarship, we will consider it to be an open document. We will continue writing to anybody and everybody making scholarly contributions within this subject area and we’ll update this article appropriately. Of course, email was sent to William Bailek and Bob Brown.
– – – – – – – – – – – MORE TO COME – – – – – – – – – – WORKING DRAFT – – – – – – – – – –
Possibilities for the deep thrust within the universe. Our initial speculative, unconstructed ideas:
- The finite (infinite) transformations. The beginning of this thrust and a thrust of life perhaps will be best described by this transformation.
- Planck charge. This charge is one of the first mathematically-defined places for thrust. Analysis is difficult; it becomes quite complicated rather quickly. Planck charge is defined by c, the speed of light in the vacuum, ℏ (hbar, the reduced Planck constant), ϵ (the permittivity of free space), e (the elementary charge) and α (alpha, the fine structure constant). Following the Planck charge as it progresses up the notation is easy; understanding the processes involved is not. It seems that the successive doublings of Planck Charge is the natural extension of the finite-infinite transformation.
- The first 67 doublings or notations. Prior to the Big Board-little universe project, we found no discussions or articles about the first 67 notations, i.e. the Planck scale to the CERN-scale. We postulate that which cannot be measured by standard processes today is the long disputed dark energy making up 68.3% of the total energy in the universe.
- Quantum fluctuations. That which cannot be measured by standard processes today is dark energy and David Bohm may have described it best in 1957 (Causality & Chance in Modern Physics pages 163-164): “Thus, in the last century only mechanical, chemical, thermal, electrical, luminous, and gravitational energy were known. Now, we know of nuclear energy, which constitute a much larger reservoir. But the infinite substructure of matter very probably contains energies that are as far beyond nuclear energies as nuclear energies are beyond chemical energies. Indeed, there is already some evidence in favour of this idea. Thus, if one computes the “zero point” energy due to quantum-mechanical fluctuations on even one cubic centimetre of space, one comes out with something of the order of 10^{38} ergs, which is equal to that which would be liberated by fission of about 10^{10} tons of uranium.”
- Other analogies. There are many thrusts described within mathematiocs and many of these are summarized on line 11 within the chart of the universe. A simple, obvious thrust is with cellular development, especially from inception to birth and then throughout life.
We will be trying to reconcile the Planck Length multiples with each of the other Planck units. All are deeply inter-related and it appears that in some measure, each of us constantly participate within multiple notations. There is a natural notational size, i.e. a virus is within the size defined by Notation 87, bacteria at notation 97, cells from 101 to 103.
Notations 67 to 134 begin to define the human scale. However, the Planck Time is necessarily defined by Planck Length and the speed of light. It is as if Jung was right about archetypes and all these human scale notations are archetypal.
What might be a good word to develop that would further take the concept of non-locality and would allow us to have an identity that spans many notations? Perhaps our mind and sleep are within notations with the 40s and 50s. Our functional parts for perception are within the “correct” Planck Length notation, our systems integrations are within all those notations that are defined within a Planck length multiple, and of course, our fully functional integration is within the Now of the 202nd notation.
[This is a key discussion and it is conceptually at the very edge of our current insights.]
Ratio analysis. Consider Max Planck’s formula. There certainly is power in light, c. There is power in pi. There is plenty of thrust within that not-so-simple formula that Planck gave us in 1900. Does anybody know why this combination of facets works to define a base unit for charge? We will be asking many experts throughout the world.
Point process theory: Nagel and Mecke
There is something about dimensionless constants. The first simple bridge between the finite and infinite may well be with their non-ending, non-repeating numbers. Are all dimensionless constants never-ending and non-repeating? That area of scholarship is another vibrant area of scholarship to consult with the experts.
Naive Assumptions. Within our initial analysis, it seems that our fundamental assumptions are:
1. The infinite is the source of the finite, its structure and energy (thrust).
2. The primary bridges between the infinite and finite are the dimensionless constants.
3. A well-studied mechanism for thrust can be found within cellular division.
4. The structures of the small-scale universe give rise to the CERN-scale, human-scale, and large-scale structures.
5. The mathematics of period doubling bifurcation, pi, the coupling constant, point processes may hold clues to study.
All five assumptions will be woven into the fabric of this study.
One of the most simple-but-complex places to start is within pi.
Power of Pi. The most commonplace, best known, and longest studied of all the dimensionless constants, there is a special place for pi in this website. Since the very first writing about this project in December 2011, there have been several discussions about pi ( π ). The focus will be to build on just two of those discussions: (1) an early-draft article from January 2016 about numbers and (2) a very current working article about visualization.
###
Working Notes for Further Inspiration:
Point processes: https://arxiv.org/pdf/1703.10000.pdf
There are many books and articles about these constants, however, our primary reference is the 2006 article by Tegmark, Aquirre, Rees, Wilczek (TARW), “Dimensionless constants, cosmology and other dark matters” where they identify 31 dimensionless physical constants (PDF). The Planck Length (space) and Planck Time are two of their 31.
There are well over 100 additional dimensionless quantities and, in time, all of these that follow will be analyzed in light of our base-2 exponentiation numbers and the thrust of the universe.
Wikipedia’s Dimensionless Quantities
Name | Standard symbol | Definition | Field of application |
---|---|---|---|
Abbe number | V | optics (dispersion in optical materials) | |
Activity coefficient | chemistry (Proportion of “active” molecules or atoms) | ||
Albedo | α ( ) |
climatology, astronomy (reflectivity of surfaces or bodies) | |
Archimedes number | Ar | fluid mechanics (motion of fluids due to density differences) | |
Arrhenius number | chemistry (ratio of activation energy to thermal energy)^{[1]} | ||
Atomic weight | M | chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u) | |
Atwood number | A | fluid mechanics (onset of instabilities in fluid mixtures due to density differences) | |
Bagnold number | Ba | fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)^{[2]} | |
Bejan number (fluid mechanics) |
Be | fluid mechanics (dimensionless pressure drop along a channel)^{[3]} | |
Bejan number (thermodynamics) |
Be | thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)^{[4]} | |
Bingham number | Bm | fluid mechanics, rheology (ratio of yield stress to viscous stress)^{[1]} | |
Biot number | Bi | heat transfer (surface vs. volume conductivity of solids) | |
Blake number | Bl or B | geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media) | |
Bodenstein number | Bo or Bd | chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)^{[5]} | |
Bond number | Bo | geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) ^{[6]} | |
Brinkman number | Br | heat transfer, fluid mechanics (conduction from a wall to a viscous fluid) | |
Brownell–Katz number | N_{BK} | fluid mechanics (combination of capillary number and Bond number) ^{[7]} | |
Capillary number | Ca | porous media, fluid mechanics (viscous forces versus surface tension) | |
Chandrasekhar number | Q | magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection) | |
Colburn J factors | J_{M}, J_{H}, J_{D} | turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients) | |
Coefficient of kinetic friction | mechanics (friction of solid bodies in translational motion) | ||
Coefficient of static friction | mechanics (friction of solid bodies at rest) | ||
Coefficient of determination | statistics (proportion of variance explained by a statistical model) | ||
Coefficient of variation | statistics (ratio of standard deviation to expectation) | ||
Cohesion number | Coh | Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles) | |
Correlation | ρ or r | statistics (measure of linear dependence) | |
Cost of transport | COT | energy efficiency, economics (ratio of energy input to kinetic motion) | |
Courant–Friedrich–Levy number | C or 𝜈 | mathematics (numerical solutions of hyperbolic PDEs)^{[8]} | |
Damkohler number | Da | chemistry (reaction time scales vs. residence time) | |
Damping ratio | mechanics (the level of damping in a system) | ||
Darcy friction factor | C_{f} or f_{D} | fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor) | |
Darcy number | Da | porous media (ratio of permeability to cross-sectional area) | |
Dean number | D | turbulent flow (vortices in curved ducts) | |
Deborah number | De | rheology (viscoelastic fluids) | |
Decibel | dB | acoustics, electronics, control theory (ratio of two intensities or powers of a wave) | |
Drag coefficient | c_{d} | aeronautics, fluid dynamics (resistance to fluid motion) | |
Dukhin number | Du | colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems) | |
Eckert number | Ec | convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy) | |
Ekman number | Ek | geophysics (viscous versus Coriolis forces) | |
Elasticity (economics) |
E | economics (response of demand or supply to price changes) | |
Eötvös number | Eo | fluid mechanics (shape of bubbles or drops) | |
Ericksen number | Er | fluid dynamics (liquid crystal flow behavior; viscous over elastic forces) | |
Euler number | Eu | hydrodynamics (stream pressure versus inertia forces) | |
Euler’s number | e | mathematics (base of the natural logarithm) | |
Excess temperature coefficient | heat transfer, fluid dynamics (change in internal energy versus kinetic energy)^{[9]} | ||
Fanning friction factor | f | fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)^{[10]} | |
Feigenbaum constants | , | chaos theory (period doubling)^{[11]} | |
Fine structure constant | quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction) | ||
f-number | f | optics, photography (ratio of focal length to diameter of aperture) | |
Föppl–von Kármán number | virology, solid mechanics (thin-shell buckling) | ||
Fourier number | Fo | heat transfer, mass transfer (ratio of diffusive rate versus storage rate) | |
Fresnel number | F | optics (slit diffraction)^{[12]} | |
Froude number | Fr | fluid mechanics (wave and surface behaviour; ratio of a body’s inertia to gravitational forces) | |
Gain | – | electronics (signal output to signal input) | |
Gain ratio | – | bicycling (system of representing gearing; length traveled over length pedaled)^{[13]} | |
Galilei number | Ga | fluid mechanics (gravitational over viscous forces) | |
Golden ratio | mathematics, aesthetics (long side length of self-similar rectangle) | ||
Hereinafter the formatting instructions (code) to generate the mathematical formulas will be left in place so we all can learn the code that generates each of these exquisite formula. | |||
Görtler number | G | G = U e θ ν ( θ R ) 1 / 2 {\displaystyle \mathrm {G} ={\frac {U_{e}\theta }{\nu }}\left({\frac {\theta }{R}}\right)^{1/2}} | fluid dynamics (boundary layer flow along a concave wall) |
Graetz number | Gz | G z = D H L R e P r {\displaystyle \mathrm {Gz} ={D_{H} \over L}\mathrm {Re} \,\mathrm {Pr} } | heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer) |
Grashof number | Gr | G r L = g β ( T s − T ∞ ) L 3 ν 2 {\displaystyle \mathrm {Gr} _{L}={\frac {g\beta (T_{s}-T_{\infty })L^{3}}{\nu ^{2}}}} | heat transfer, natural convection (ratio of the buoyancy to viscous force) |
Gravitational coupling constant | α G {\displaystyle \alpha _{G}} | α G = G m e 2 ℏ c {\displaystyle \alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}} | gravitation (attraction between two massy elementary particles; analogous to the Fine structure constant) |
Hatta number | Ha | H a = N A 0 N A 0 p h y s {\displaystyle \mathrm {Ha} ={\frac {N_{\mathrm {A} 0}}{N_{\mathrm {A} 0}^{\mathrm {phys} }}}} | chemical engineering (adsorption enhancement due to chemical reaction) |
Hagen number | Hg | H g = − 1 ρ d p d x L 3 ν 2 {\displaystyle \mathrm {Hg} =-{\frac {1}{\rho }}{\frac {\mathrm {d} p}{\mathrm {d} x}}{\frac {L^{3}}{\nu ^{2}}}} | heat transfer (ratio of the buoyancy to viscous force in forced convection) |
Havnes parameter | P H {\displaystyle P_{H}} | P H = Z d n d n i {\displaystyle P_{H}={\frac {Z_{d}n_{d}}{n_{i}}}} | In Dusty plasma physics, ratio of the total charge Z d {\displaystyle Z_{d}} carried by the dust particles d {\displaystyle d} to the charge carried by the ions i {\displaystyle i} , with n {\displaystyle n} the number density of particles |
Hydraulic gradient | i | i = d h d l = h 2 − h 1 l e n g t h {\displaystyle i={\frac {\mathrm {d} h}{\mathrm {d} l}}={\frac {h_{2}-h_{1}}{\mathrm {length} }}} | fluid mechanics, groundwater flow (pressure head over distance) |
Iribarren number | Ir | I r = tan α H / L 0 {\displaystyle \mathrm {Ir} ={\frac {\tan \alpha }{\sqrt {H/L_{0}}}}} | wave mechanics (breaking surface gravity waves on a slope) |
Jakob number | Ja | J a = c p ( T s − T s a t ) Δ H f {\displaystyle \mathrm {Ja} ={\frac {c_{p}(T_{\mathrm {s} }-T_{\mathrm {sat} })}{\Delta H_{\mathrm {f} }}}} | chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)^{[14]} |
Karlovitz number | Ka | K a = k t c {\displaystyle \mathrm {Ka} =kt_{c}} | turbulent combustion (characteristic flow time times flame stretch rate) |
Keulegan–Carpenter number | K_{C} | K C = V T L {\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}} | fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow) |
Knudsen number | Kn | K n = λ L {\displaystyle \mathrm {Kn} ={\frac {\lambda }{L}}} | gas dynamics (ratio of the molecular mean free path length to a representative physical length scale) |
Kt/V | Kt/V | medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time) | |
Kutateladze number | Ku | K u = U h ρ g 1 / 2 ( σ g ( ρ l − ρ g ) ) 1 / 4 {\displaystyle \mathrm {Ku} ={\frac {U_{h}\rho _{g}^{1/2}}{\left({\sigma g(\rho _{l}-\rho _{g})}\right)^{1/4}}}} | fluid mechanics (counter-current two-phase flow)^{[15]} |
Laplace number | La | L a = σ ρ L μ 2 {\displaystyle \mathrm {La} ={\frac {\sigma \rho L}{\mu ^{2}}}} | fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport) |
Lewis number | Le | L e = α D = S c P r {\displaystyle \mathrm {Le} ={\frac {\alpha }{D}}={\frac {\mathrm {Sc} }{\mathrm {Pr} }}} | heat and mass transfer (ratio of thermal to mass diffusivity) |
Lift coefficient | C_{L} | C L = L q S {\displaystyle C_{\mathrm {L} }={\frac {L}{q\,S}}} | aerodynamics (lift available from an airfoil at a given angle of attack) |
Lockhart–Martinelli parameter | χ {\displaystyle \chi } | χ = m ℓ m g ρ g ρ ℓ {\displaystyle \chi ={\frac {m_{\ell }}{m_{g}}}{\sqrt {\frac {\rho _{g}}{\rho _{\ell }}}}} | two-phase flow (flow of wet gases; liquid fraction)^{[16]} |
Love numbers | h, k, l | geophysics (solidity of earth and other planets) | |
Lundquist number | S | S = μ 0 L V A η {\displaystyle S={\frac {\mu _{0}LV_{A}}{\eta }}} | plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma) |
Mach number | M or Ma | M = v v s o u n d {\displaystyle \mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}} | gas dynamics (compressible flow; dimensionless velocity) |
Magnetic Reynolds number | R_{m} | R m = U L η {\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}} | magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion) |
Manning roughness coefficient | n | open channel flow (flow driven by gravity)^{[17]} | |
Marangoni number | Mg | M g = − d σ d T L Δ T η α {\displaystyle \mathrm {Mg} =-{\frac {\mathrm {d} \sigma }{\mathrm {d} T}}{\frac {L\Delta T}{\eta \alpha }}} | fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces) |
Markstein number | Ma | M a = L b l f {\displaystyle \mathrm {Ma} ={\frac {{\mathcal {L}}_{b}}{l_{f}}}} | fluid dynamics, combustion (turbulent combustion flames) |
Morton number | Mo | M o = g μ c 4 Δ ρ ρ c 2 σ 3 {\displaystyle \mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}}} | fluid dynamics (determination of bubble/drop shape) |
Nusselt number | Nu | N u = h d k {\displaystyle \mathrm {Nu} ={\frac {hd}{k}}} | heat transfer (forced convection; ratio of convective to conductive heat transfer) |
Ohnesorge number | Oh | O h = μ ρ σ L = W e R e {\displaystyle \mathrm {Oh} ={\frac {\mu }{\sqrt {\rho \sigma L}}}={\frac {\sqrt {\mathrm {We} }}{\mathrm {Re} }}} | fluid dynamics (atomization of liquids, Marangoni flow) |
Péclet number | Pe | P e = d u ρ c p k = R e P r {\displaystyle \mathrm {Pe} ={\frac {du\rho c_{p}}{k}}=\mathrm {Re} \,\mathrm {Pr} } | heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer) |
Peel number | N_{P} | N P = Restoring force Adhesive force {\displaystyle N_{\mathrm {P} }={\frac {\text{Restoring force}}{\text{Adhesive force}}}} | coating (adhesion of microstructures with substrate)^{[18]} |
Perveance | K | K = I I 0 2 β 3 γ 3 ( 1 − γ 2 f e ) {\displaystyle {K}={\frac {I}{I_{0}}}\,{\frac {2}{{\beta }^{3}{\gamma }^{3}}}(1-\gamma ^{2}f_{e})} | charged particle transport (measure of the strength of space charge in a charged particle beam) |
pH | p H {\displaystyle \mathrm {pH} } | p H = − log 10 ( a H + ) {\displaystyle \mathrm {pH} =-\log _{10}(a_{{\textrm {H}}^{+}})} | chemistry (the measure of the acidity or basicity of an aqueous solution) |
Pi | π {\displaystyle \pi } | π = C d ≈ 3.14159 {\displaystyle \pi ={\frac {C}{d}}\approx 3.14159} | mathematics (ratio of a circle‘s circumference to its diameter) |
Pierce parameter | C {\displaystyle C} | C 3 = Z c I K 4 V K {\displaystyle C^{3}={\frac {Z_{c}I_{K}}{4V_{K}}}} | Traveling wave tube |
Pixel | px | digital imaging (smallest addressable unit) | |
Beta (plasma physics) | β {\displaystyle \beta } | β = n k B T B 2 / 2 μ 0 {\displaystyle \beta ={\frac {nk_{B}T}{B^{2}/2\mu _{0}}}} | Plasma (physics) and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma. |
Poisson’s ratio | ν {\displaystyle \nu } | ν = − d ε t r a n s d ε a x i a l {\displaystyle \nu =-{\frac {\mathrm {d} \varepsilon _{\mathrm {trans} }}{\mathrm {d} \varepsilon _{\mathrm {axial} }}}} | elasticity (strain in transverse and longitudinal direction) |
Porosity | ϕ {\displaystyle \phi } | ϕ = V V V T {\displaystyle \phi ={\frac {V_{\mathrm {V} }}{V_{\mathrm {T} }}}} | geology, porous media (void fraction of the medium) |
Power factor | pf | p f = P S {\displaystyle pf={\frac {P}{S}}} | electrical (real power to apparent power) |
Power number | N_{p} | N p = P ρ n 3 d 5 {\displaystyle N_{p}={P \over \rho n^{3}d^{5}}} | electronics (power consumption by agitators; resistance force versus inertia force) |
Prandtl number | Pr | P r = ν α = c p μ k {\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}} | heat transfer (ratio of viscous diffusion rate over thermal diffusion rate) |
Prater number | β | β = − Δ H r D T A e C A S λ e T s {\displaystyle \beta ={\frac {-\Delta H_{r}D_{TA}^{e}C_{AS}}{\lambda ^{e}T_{s}}}} | reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)^{[19]} |
Pressure coefficient | C_{P} | C p = p − p ∞ 1 2 ρ ∞ V ∞ 2 {\displaystyle C_{p}={p-p_{\infty } \over {\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}} | aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable) |
Q factor | Q | Q = 2 π f r Energy Stored Power Loss {\displaystyle Q=2\pi f_{r}{\frac {\text{Energy Stored}}{\text{Power Loss}}}} | physics, engineering (damping of oscillator or resonator; energy stored versus energy lost) |
Radian measure | rad | arc length / radius {\displaystyle {\text{arc length}}/{\text{radius}}} | mathematics (measurement of planar angles, 1 radian = 180/π degrees) |
Rayleigh number | Ra | R a x = g β ν α ( T s − T ∞ ) x 3 {\displaystyle \mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}} | heat transfer (buoyancy versus viscous forces in free convection) |
Refractive index | n | n = c v {\displaystyle n={\frac {c}{v}}} | electromagnetism, optics (speed of light in a vacuum over speed of light in a material) |
Relative density | RD | R D = ρ s u b s t a n c e ρ r e f e r e n c e {\displaystyle RD={\frac {\rho _{\mathrm {substance} }}{\rho _{\mathrm {reference} }}}} | hydrometers, material comparisons (ratio of density of a material to a reference material—usually water) |
Relative permeability | μ r {\displaystyle \mu _{r}} | μ r = μ μ 0 {\displaystyle \mu _{r}={\frac {\mu }{\mu _{0}}}} | magnetostatics (ratio of the permeability of a specific medium to free space) |
Relative permittivity | ε r {\displaystyle \varepsilon _{r}} | ε r = C x C 0 {\displaystyle \varepsilon _{r}={\frac {C_{x}}{C_{0}}}} | electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum) |
Reynolds number | Re | R e = v L ρ μ {\displaystyle \mathrm {Re} ={\frac {vL\rho }{\mu }}} | fluid mechanics (ratio of fluid inertial and viscous forces)^{[1]} |
Richardson number | Ri | R i = g h u 2 = 1 F r 2 {\displaystyle \mathrm {Ri} ={\frac {gh}{u^{2}}}={\frac {1}{\mathrm {Fr} ^{2}}}} | fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)^{[20]} |
Rockwell scale | – | mechanical hardness (indentation hardness of a material) | |
Rolling resistance coefficient | C_{rr} | C r r = F N f {\displaystyle C_{rr}={\frac {F}{N_{f}}}} | vehicle dynamics (ratio of force needed for motion of a wheel over the normal force) |
Roshko number | Ro | R o = f L 2 ν = S t R e {\displaystyle \mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re} } | fluid dynamics (oscillating flow, vortex shedding) |
Rossby number | Ro | R o = U L f {\displaystyle \mathrm {Ro} ={\frac {U}{Lf}}} | geophysics (ratio of inertial to Coriolis force) |
Rouse number | P or Z | P = w s κ u ∗ {\displaystyle \mathrm {P} ={\frac {w_{s}}{\kappa u_{*}}}} | sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain) |
Schmidt number | Sc | S c = ν D {\displaystyle \mathrm {Sc} ={\frac {\nu }{D}}} | mass transfer (viscous over molecular diffusion rate)^{[21]} |
Shape factor | H | H = δ ∗ θ {\displaystyle H={\frac {\delta ^{*}}{\theta }}} | boundary layer flow (ratio of displacement thickness to momentum thickness) |
Sherwood number | Sh | S h = K L D {\displaystyle \mathrm {Sh} ={\frac {KL}{D}}} | mass transfer (forced convection; ratio of convective to diffusive mass transport) |
Shields parameter | τ ∗ {\displaystyle \tau _{*}} or θ {\displaystyle \theta } | τ ∗ = τ ( ρ s − ρ ) g D {\displaystyle \tau _{\ast }={\frac {\tau }{(\rho _{s}-\rho )gD}}} | sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress) |
Sommerfeld number | S | S = ( r c ) 2 μ N P {\displaystyle \mathrm {S} =\left({\frac {r}{c}}\right)^{2}{\frac {\mu N}{P}}} | hydrodynamic lubrication (boundary lubrication)^{[22]} |
Specific gravity | SG | (same as Relative density) | |
Stanton number | St | S t = h c p ρ V = N u R e P r {\displaystyle \mathrm {St} ={\frac {h}{c_{p}\rho V}}={\frac {\mathrm {Nu} }{\mathrm {Re} \,\mathrm {Pr} }}} | heat transfer and fluid dynamics (forced convection) |
Stefan number | Ste | S t e = c p Δ T L {\displaystyle \mathrm {Ste} ={\frac {c_{p}\Delta T}{L}}} | phase change, thermodynamics (ratio of sensible heat to latent heat) |
Stokes number | Stk or S_{k} | S t k = τ U o d c {\displaystyle \mathrm {Stk} ={\frac {\tau U_{o}}{d_{c}}}} | particles suspensions (ratio of characteristic time of particle to time of flow) |
Strain | ϵ {\displaystyle \epsilon } | ϵ = ∂ F ∂ X − 1 {\displaystyle \epsilon ={\cfrac {\partial {F}}{\partial {X}}}-1} | materials science, elasticity (displacement between particles in the body relative to a reference length) |
Strouhal number | St or Sr | S t = ω L v {\displaystyle \mathrm {St} ={\omega L \over v}} | fluid dynamics (continuous and pulsating flow; nondimensional frequency)^{[23]} |
Stuart number | N | N = B 2 L c σ ρ U = H a 2 R e {\displaystyle \mathrm {N} ={\frac {B^{2}L_{c}\sigma }{\rho U}}={\frac {\mathrm {Ha} ^{2}}{\mathrm {Re} }}} | magnetohydrodynamics (ratio of electromagnetic to inertial forces) |
Taylor number | Ta | T a = 4 Ω 2 R 4 ν 2 {\displaystyle \mathrm {Ta} ={\frac {4\Omega ^{2}R^{4}}{\nu ^{2}}}} | fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces) |
Transmittance | T | T = I I 0 {\displaystyle T={\frac {I}{I_{0}}}} | optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample) |
Ursell number | U | U = H λ 2 h 3 {\displaystyle \mathrm {U} ={\frac {H\,\lambda ^{2}}{h^{3}}}} | wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer) |
Vadasz number | Va | V a = ϕ P r D a {\displaystyle \mathrm {Va} ={\frac {\phi \,\mathrm {Pr} }{\mathrm {Da} }}} | porous media (governs the effects of porosity ϕ {\displaystyle \phi } , the Prandtl number and the Darcy number on flow in a porous medium) ^{[24]} |
van ‘t Hoff factor | i | i = 1 + α ( n − 1 ) {\displaystyle i=1+\alpha (n-1)} | quantitative analysis (K_{f} and K_{b}) |
Wallis parameter | j^{*} | j ∗ = R ( ω ρ μ ) 1 2 {\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}} | multiphase flows (nondimensional superficial velocity)^{[25]} |
Weaver flame speed number | Wea | W e a = w w H 100 {\displaystyle \mathrm {Wea} ={\frac {w}{w_{\mathrm {H} }}}100} | combustion (laminar burning velocity relative to hydrogen gas)^{[26]} |
Weber number | We | W e = ρ v 2 l σ {\displaystyle \mathrm {We} ={\frac {\rho v^{2}l}{\sigma }}} | multiphase flow (strongly curved surfaces; ratio of inertia to surface tension) |
Weissenberg number | Wi | W i = γ ˙ λ {\displaystyle \mathrm {Wi} ={\dot {\gamma }}\lambda } | viscoelastic flows (shear rate times the relaxation time)^{[27]} |
Womersley number | α {\displaystyle \alpha } | α = R ( ω ρ μ ) 1 2 {\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}} | biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)^{[28]} |
Zel’dovich number | β {\displaystyle \beta } | β = E R T f T f − T o T f {\displaystyle \beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}} | fluid dynamics, Combustion (Measure of activation energy) |