The Thrust Of The Universe

Work-in-progress: Writing and editing on Saturday, July 22, 2017

• Measuring an Expanding Universe Using Planck Units (a rough draft)
• The Thrust of the Universe: What is it? (this page, a 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 was introduced to the work of Princeton’s William Bialek. He is the John Archibald Wheeler/Battelle Professor of Physics. Like John Wheeler whose chair he holds, Bialek is one of those rare scholars who searches for 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; one of his projects 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, in this period of their work (July 2017), 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 working document. We will continue to write to anybody and everybody making scholarly contributions within this subject area and we’ll update this article appropriately. Of course, one of our earliest emails was sent to William Bailek and Bob Brown.

Our initial speculative, unconstructed ideas, all possibilities for the deep thrust within the universe:

  1. The finite-infinite transformations. The beginning of this thrust (and a thrust of life) perhaps will be informed by these transformations.
  2. Planck charge. This charge is one of the first mathematically-defined places for thrust. For\ us, this 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 one or many of these finite-infinite transformations.  Planck Charge
  3. 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.
  4. Quantum fluctuations.  If that which cannot be measured by standard processes today is dark energy, perhaps David Bohm 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 1038 ergs, which is equal to that which would be liberated by fission of about 1010 tons of uranium.”
  5. Other analogies. There are many thrusts described within mathematics 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.

– – – – – – – – – – – MORE TO COME – – – – – – – – – – WORKING DRAFT – – – – – – – – – –

We will be trying to reconcile the Planck Length multiples with each of the other Planck units. All appear to be deeply inter-related; and in some measure, each of us constantly participates within multiple notations.

Our first, and the most simple, observation is  of 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. One might conclude that notations 67 to 134  define the human scale. However, Planck Time is necessarily defined by Planck Length and the speed of light within every notation. Given we live and die within notation 202, it is as if Jung was right about archetypes and all these human scale notations are in some manner of speaking 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. Hypothetically our functional parts for perception and beingness may be defined by their Planck Length notation, an instantaneous systems integration brings together all the notations within the Now of the 202nd notation.

[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:

Period Doubling Bifurcation.

Coupling constant

Point processes:

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 V = \frac{ n_d - 1 }{ n_F - n_C } optics (dispersion in optical materials)
Activity coefficient \gamma  \gamma= \frac {{a}}{{x}} chemistry (Proportion of “active” molecules or atoms)
Albedo α ( \alpha )
\alpha= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha} climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar  \mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2} fluid mechanics (motion of fluids due to density differences)
Arrhenius number \alpha \alpha = \frac{E_a}{RT} 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 \mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba \mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu} 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 \mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha} fluid mechanics (dimensionless pressure drop along a channel)[3]
Bejan number
Be \mathrm{Be} = \frac{\dot S'_{\mathrm{gen},\, \Delta T}}{\dot S'_{\mathrm{gen},\, \Delta T}+ \dot S'_{\mathrm{gen},\, \Delta p}} thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[4]
Bingham number Bm \mathrm{Bm} = \frac{ \tau_y L }{ \mu V } fluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot number Bi \mathrm{Bi} = \frac{h L_C}{k_b} heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B \mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D} geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bodenstein number Bo or Bd \mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc} chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[5]
Bond number Bo \mathrm{Bo} = \frac{\rho a L^2}{\gamma} geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [6]
Brinkman number Br  \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)} heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK \mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} fluid mechanics (combination of capillary number and Bond number) [7]
Capillary number Ca \mathrm{Ca} = \frac{\mu V}{\gamma} porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number Q  \mathrm{Q} = \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda} magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factors JM, JH, JD turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Coefficient of kinetic friction \mu _{k} mechanics (friction of solid bodies in translational motion)
Coefficient of static friction \mu _{s} mechanics (friction of solid bodies at rest)
Coefficient of determination R^{2} statistics (proportion of variance explained by a statistical model)
Coefficient of variation {\frac {\sigma }{\mu }} {\frac {\sigma }{\mu }} statistics (ratio of standard deviation to expectation)
Cohesion number Coh {\displaystyle Coh={\frac {1}{\rho g}}\left({\frac {\Gamma ^{5}}{{E^{*}}^{2}{R^{*}}^{8}}}\right)^{\frac {1}{3}}} Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)
Correlation ρ or r {\displaystyle {\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}} statistics (measure of linear dependence)
Cost of transport COT {\displaystyle \mathrm {COT} ={\frac {E}{mgd}}} energy efficiency, economics (ratio of energy input to kinetic motion)
Courant–Friedrich–Levy number C or 𝜈 C = \frac {u\,\Delta t} {\Delta x} mathematics (numerical solutions of hyperbolic PDEs)[8]
Damkohler number Da  \mathrm{Da} = k \tau chemistry (reaction time scales vs. residence time)
Damping ratio \zeta  \zeta = \frac{c}{2 \sqrt{km}} mechanics (the level of damping in a system)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy number Da  \mathrm{Da} = \frac{K}{d^2} porous media (ratio of permeability to cross-sectional area)
Dean number D \mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2} turbulent flow (vortices in curved ducts)
Deborah number De  \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}} rheology (viscoelastic fluids)
Decibel dB acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficient cd c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, , aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number Du  \mathrm{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a} colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert number Ec  \mathrm{Ec} = \frac{V^2}{c_p\Delta T}  convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number Ek \mathrm{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi} geophysics (viscous versus Coriolis forces)
E E_{x,y} = \frac{\partial \ln(x)}{\partial \ln(y)} = \frac{\partial x}{\partial y}\frac{y}{x} economics (response of demand or supply to price changes)
Eötvös number Eo \mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma} fluid mechanics (shape of bubbles or drops)
Ericksen number Er \mathrm{Er}=\frac{\mu v L}{K} fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu  \mathrm{Eu}=\frac{\Delta{}p}{\rho V^2} hydrodynamics (stream pressure versus inertia forces)
Euler’s number e e =  \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} \approx 2.71828 mathematics (base of the natural logarithm)
Excess temperature coefficient \Theta_r \Theta_r = \frac{c_p (T-T_e)}{U_e^2/2} 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 \alpha , \delta \alpha \approx 2.50290,\ \delta \approx 4.66920 chaos theory (period doubling)[11]
Fine structure constant \alpha {\displaystyle \alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}hc}}} quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-number f  f = \frac {{\ell}}{{D}} optics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán number \gamma \gamma = \frac{Y r^2}{\kappa} virology, solid mechanics (thin-shell buckling)
Fourier number Fo \mathrm{Fo} = \frac{\alpha t}{L^2} heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel number F \mathit{F} = \frac{a^{2}}{L \lambda} optics (slit diffraction)[12]
Froude number Fr \mathrm{Fr} = \frac{v}{\sqrt{g\ell}} 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 \mathrm{Ga} = \frac{g\, L^3}{\nu^2} fluid mechanics (gravitational over viscous forces)
Golden ratio \varphi \varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803 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}} \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} } \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}}}}  \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}} \alpha _{G} α G = G m e 2 ℏ c {\displaystyle \alpha _{G}={\frac {Gm_{e}^{2}}{\hbar c}}} \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} }}}} \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}}}}  \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}} {\displaystyle P_{H}} P H = Z d n d n i {\displaystyle P_{H}={\frac {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}} {\displaystyle Z_{d}} carried by the dust particles d {\displaystyle d} d to the charge carried by the ions i {\displaystyle i} i, with n {\displaystyle n} 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} }}} 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}}}}} \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} }}}} \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}} \mathrm{Ka} = k t_c turbulent combustion (characteristic flow time times flame stretch rate)
Keulegan–Carpenter number KC K C = V T L {\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}} \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}}} \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}}}} \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}}}} \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} }}} \mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}} heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL C L = L q S {\displaystyle C_{\mathrm {L} }={\frac {L}{q\,S}}} 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 } \chi χ = m ℓ m g ρ g ρ ℓ {\displaystyle \chi ={\frac {m_{\ell }}{m_{g}}}{\sqrt {\frac {\rho _{g}}{\rho _{\ell }}}}} \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 }}} S = \frac{\mu_0LV_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} }}}}  \mathrm{M} = \frac{{v}}{{v_\mathrm{sound}}} gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number Rm R m = U L η {\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}} \mathrm{R}_\mathrm{m} = \frac{U L}{\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 }}} \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}}}} {\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}}}} \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}}} \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} }}}  \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} } \mathrm{Pe} =  \frac{du\rho c_p}{k} = \mathrm{Re}\, \mathrm{Pr} heat transfer (advectiondiffusion problems; total momentum transfer to molecular heat transfer)
Peel number NP N P = Restoring force Adhesive force {\displaystyle N_{\mathrm {P} }={\frac {\text{Restoring force}}{\text{Adhesive force}}}} 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})} {K} = \frac{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e) charged particle transport (measure of the strength of space charge in a charged particle beam)
pH p H {\displaystyle \mathrm {pH} } \mathrm{pH} p H = − log 10 ⁡ ( a H + ) {\displaystyle \mathrm {pH} =-\log _{10}(a_{{\textrm {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 } \pi π = C d ≈ 3.14159 {\displaystyle \pi ={\frac {C}{d}}\approx 3.14159} \pi = \frac{C}{d} \approx 3.14159 mathematics (ratio of a circle‘s circumference to its diameter)
Pierce parameter C {\displaystyle C} C C 3 = Z c I K 4 V K {\displaystyle C^{3}={\frac {Z_{c}I_{K}}{4V_{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 } \beta β = n k B T B 2 / 2 μ 0 {\displaystyle \beta ={\frac {nk_{B}T}{B^{2}/2\mu _{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 } \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} }}}} \nu = -\frac{\mathrm{d}\varepsilon_\mathrm{trans}}{\mathrm{d}\varepsilon_\mathrm{axial}} elasticity (strain in transverse and longitudinal direction)
Porosity ϕ {\displaystyle \phi } \phi ϕ = V V V T {\displaystyle \phi ={\frac {V_{\mathrm {V} }}{V_{\mathrm {T} }}}} \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}}} {\displaystyle pf={\frac {P}{S}}} electrical (real power to apparent power)
Power number Np N p = P ρ n 3 d 5 {\displaystyle N_{p}={P \over \rho n^{3}d^{5}}}  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}}} \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}}}} \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 CP C p = p − p ∞ 1 2 ρ ∞ V ∞ 2 {\displaystyle C_{p}={p-p_{\infty } \over {\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}} 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}}}} 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}}} \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}} \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 heat transfer (buoyancy versus viscous forces in free convection)
Refractive index n n = c v {\displaystyle n={\frac {c}{v}}} 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} }}}} 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}} \mu _{r} μ r = μ μ 0 {\displaystyle \mu _{r}={\frac {\mu }{\mu _{0}}}} \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}} \varepsilon _{r} ε r = C x C 0 {\displaystyle \varepsilon _{r}={\frac {C_{x}}{C_{0}}}} \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 }}} \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}}}}  \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 Crr C r r = F N f {\displaystyle C_{rr}={\frac {F}{N_{f}}}} 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} }  \mathrm{Ro} = {f L^{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}}} \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_{*}}}} \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}}} \mathrm{Sc} = \frac{\nu}{D} mass transfer (viscous over molecular diffusion rate)[21]
Shape factor H H = δ ∗ θ {\displaystyle H={\frac {\delta ^{*}}{\theta }}} 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}}} \mathrm{Sh} = \frac{K L}{D} mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter τ ∗ {\displaystyle \tau _{*}} \tau_* or θ {\displaystyle \theta } \theta τ ∗ = τ ( ρ s − ρ ) g D {\displaystyle \tau _{\ast }={\frac {\tau }{(\rho _{s}-\rho )gD}}} \tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D} 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}}}  \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} }}} \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}}} \mathrm{Ste} = \frac{c_p \Delta T}{L} phase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes number Stk or Sk S t k = τ U o d c {\displaystyle \mathrm {Stk} ={\frac {\tau U_{o}}{d_{c}}}} \mathrm{Stk} = \frac{\tau U_o}{d_c} particles suspensions (ratio of characteristic time of particle to time of flow)
Strain ϵ {\displaystyle \epsilon } \epsilon ϵ = ∂ F ∂ X − 1 {\displaystyle \epsilon ={\cfrac {\partial {F}}{\partial {X}}}-1} \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}} \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} }}}  \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}}}}  \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}}}} 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}}}} \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} }}} \mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}} porous media (governs the effects of porosity ϕ {\displaystyle \phi } \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)}  i = 1 + \alpha (n - 1) quantitative analysis (Kf and Kb)
Wallis parameter j* j ∗ = R ( ω ρ μ ) 1 2 {\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}} 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} \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 }}} \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 } \mathrm{Wi} = \dot{\gamma} \lambda viscoelastic flows (shear rate times the relaxation time)[27]
Womersley number α {\displaystyle \alpha } \alpha α = R ( ω ρ μ ) 1 2 {\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}} \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 } \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}}}} {\displaystyle \beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}} fluid dynamics, Combustion (Measure of activation energy)

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