Reynolds number is defined as the ratio of dynamic pressure to shearing stress.It is a non dimensional parameter.Reynolds number predicts the extent of turbulence in a fluid, based on how fast the fluid is flowing, the geometry of flow(depth and wide etc…)  density  and viscosity of fluid.

Mathematically,

Reynolds number (Re) = Dynamic Pressure (ρ v²) / Shearing Stress (μ v / L)

Re   = ρ v² / (μ v / L)

Re = ρ v L / μ

where

Re = Reynolds Number (non-dimensional)

ρ = density

v = velocity

μ = dynamic viscosity

L= characteristic length

Dimensional Formula of Density (ρ) =M¹L^-3T⁰
Dimensional Formula of Velocity(v) = M⁰L¹T^-1
Dimensional Formula of Dynamic viscosity(μ) = M¹L^-1T^-1
Dimensional Formula of Length(L) = M⁰L¹T⁰

Putting these values in above equation we get,

Dimensional Formula of Reynolds Number = (M¹L^-3T⁰)(M⁰L¹T^-1)(M⁰L¹T⁰)/(M¹L^-1T^-1)

Dimensional Formula of Reynolds Number = M⁰L⁰T⁰
Reynolds Number has no units (non dimensional).

Rydberg formula expressed wave number in terms of constant common to all series of lines known as  Rydberg constant.

Rydberg formula defines the  relationship of frequencies in one series of lines.Rydberg formula: N = R [1/(n+a)² – 1/ (m + b)²] where N is wave number, R is Rydberg constant; m and n are integers, m is greater than n; and a and b are constants for particular series.

Mathematically,
Rydberg constant (R) = me⁴ / 8π  ε_0²  h²
where

m = electron mass
e = electron charge
ε₀ = permittivity of free space
h = Planck’s Constant

so putting all the dimensional formulas in the above equation we get

Dimensional formula of Rydberg constant = [M¹L⁰T⁰] [M⁰L⁰A¹T¹]⁴ / [(M^-1L^-3T⁴A²)²] [(M¹L²T^-1)²]  ]

Dimensional formula of Rydberg constant = M⁰L^-1T⁰
SI unit of Rydberg constant is m^-1

Linear Mass Density is defined as measure of mass per unit of length.

Mathematically,
Linear Mass Density = Mass / Length.

Dimensional Formula of Mass = M¹L⁰T⁰.
Dimensional Formula of Length = M⁰L¹T⁰.

Putting these values in above equation we get,

Dimensional Formula of Linear Mass Density = M¹L^-1T⁰.
SI unit of Linear Mass Density is kg m^-1.

Please note : Linear mass density is the quantity of mass per unit length. Here length is a simple metre quantity. Linear densities are basically used for long thin objects. Please do not get confused between Linear mass density and density.

Rate of flow is defined as the quantity of a fluid flowing per second through a section of a pipe or a channel.
Rate of flow is also known as Discharge Q.

Mathematically,
Rate of flow = Volume / time

Dimensional Formula of Volume = M⁰L³T⁰.
Dimensional Formula of time = M⁰L⁰T¹.

Putting these values in above equation we get,
Dimensional Formula of Rate of Flow= M⁰L³T^-1.
SI unit of Rate of flow is m³s^-1

The Planck’s constant (h) can be defined as a proportionality constant that relates the energy (E) of a photon to the frequency (ν) of its associated electromagnetic wave.

Mathematically,
Planck’s Constant (h) = Energy(E) / frequency (ν).

Dimensional Formula of energy = M¹L²T^-2.
Dimensional Formula of frequency = M⁰L⁰T^-1.

Putting these values in above equation we get,
Dimensional Formula of Planck’s constant = M¹L²T^-1.
SI unit of Planck’s constant is joule-seconds (j-s).