Fundamentals of fluid flow, Darcy's law, Unsaturated Condition, Reynolds Number, Poiseuille’s Flow, Laplace Law, The one-dimensional vertical flow of water

1. ASSIGNMENT ON
Fundamentals of fluid flow, Darcy's law, Unsaturated Condition, Reynolds
Number, Poiseuille’s Flow, Laplace Law, The one-dimensional vertical flow of
water
SUBMITTED TO- Dr. K. TEDIA
Head of Department
Department of Soil Science and
Agricultural Chemistry
IGKV, RAIPUR
SUBMITTED BY- DEEPIKA SAHU
Ph.D. 1st year 2nd sem.
Department- Soil Science and
Agricultural Chemistry
College of Agriculture, Raipur
INDIRA GANDHI KRISHI VISHWAVIDYALAYA, RAIPUR

2. CONTENT
S. No. Topic Page No.
1 Darcy's law
2 Darcy's law Unsaturated Condition
3 Reynolds Number
4 Poiseuille’s Flow
5 Laplace Law
6 Young-Laplace Law
7 The one-dimensional vertical flow of water
8 Reference

4. In fluid dynamics and hydrology, Darcy's law is a
phenomenological derived constitutive equation that
describes the flow of a fluid through a porous
medium. The law was formulated by Henry Darcy
based on the results of experiments (published 1856)[
on the flow of water through beds of sand. It also
forms the scientific basis of fluid permeability used
in the earth sciences.
Diagram showing definitions and directions for
Darcy's law.

5. Darcy's law is a simple proportional relationship between the
instantaneous discharge rate through a porous medium, the
viscosity of the fluid and the pressure drop over a given
distance.
The total discharge, Q (units of volume per time, e.g., ft³/s or m³/s) is equal
to the product of the permeability (κ units of area, e.g. m²) of the medium,
the cross-sectional area (A) to flow, and the pressure drop (Pb − Pa), all
divided by the dynamic viscosity μ (in SI units e.g. kg/(m·s) or Pa·s), and the
length L the pressure drop is taking place over. The negative sign is needed
because fluids flow from high pressure to low pressure. So if the change in
pressure is negative (in the x-direction) then the flow will be positive (in the
x-direction). Dividing both sides of the equation by the area and using more
general notation leads to

6. where q is the filtration velocity or Darcy flux (discharge per unit area, with
units of length per time, m/s) and is the pressure gradient vector. This
value of the filtration velocity (Darcy flux), is not the velocity which the
water traveling through the pores is experiencing
The pore (interstitial) velocity (v) is related to the Darcy flux (q) by the
porosity (φ). The flux is divided by porosity to account for the fact that
only a fraction of the total formation volume is available for flow. The pore
velocity would be the velocity a conservative tracer would experience if
carried by the fluid through the formation.

10. Reynolds Number
• The Reynolds Number (Re) is a non-dimensional
number that reflects the balance between viscous and
inertial forces and hence relates to flow instability (i.e.,
the onset of turbulence)
• Re = v L/
• L is a characteristic length in the system
• Dominance of viscous force leads to laminar flow (low
velocity, high viscosity, confined fluid)
• Dominance of inertial force leads to turbulent flow (high
velocity, low viscosity, unconfined fluid)

11. Poiseuille Flow
• In a slit or pipe, the velocities at the walls are 0
(no-slip boundaries) and the velocity reaches its
maximum in the middle
• The velocity profile in a slit is parabolic and
given by:
u(x) 
G
(a2
 x2
)
2
• G can be gravitational acceleration
or (linear) pressure gradient (Pin –
x = 0 x = a
u(x)
out
P )/L

12. Poiseuille Flow
S.GOKALTUN
Florida International University

13. Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.

14. The solution of Laplace equation gives
two sets of curves perpendicular to
each other. One set is known as flow
lines and other set is known as
equipotential lines. The flow lines
indicate the direction of flow and
equipotential lines are the lines joining
the points with same total potential or
elevation head.
Laplace Law

15. Laplace Law
• There is a pressure difference between
the inside and outside of bubbles and
drops
• The pressure is always higher on the
inside of a bubble or drop (concave side)
– just as in a balloon
• The pressure difference depends on the
radius of curvature and the surface
tension for the fluid pair of interest: P = /
r

16. Laplace Law
P = /r →  = P/r,
which is linear in 1/r (a.k.a. curvature)
r
Pin Pout

17. Young-Laplace Law
• With solid surfaces, in addition to the
fluid1/fluid2 interface – where the interaction is
given by the interfacial tension  -- we have
interfaces between each fluid and the surface
S1 and S2
• Often one of the fluids preferentially ‘wets’ the
surface
• This phenomenon is captured by the contact
angle
• cos  = (S2 - S1 

18. Young-Laplace Law
• Zero contact angle means perfect wetting;
• P =  cos /r

19. The one-dimensional vertical flow of water in variably saturated porous
media is described by the equation
The corresponding equation of mass transport of conservative solutes can be
expressed as
where: h is the pressure head [L]; K ia the hydraulic conductivity [L T '] i h is the
specific moisture capacity [L ']; r is the vertical coordinate (positive down) [L]; t is
the time [T].
where: C is the concentration of solute [M L "); D is the dispersion coefficient [L2 T‘'];
O• is the volumetric moisture content [L' L ']; q is the volumetric flux or Darcy velocity
[L T*']. Thia equation can be converted to a more convenient form, suitable to finite
element discretization (Huyakorn et al., 1985).
Using the continuity equation of water flow
𝑳𝒒 𝒉 =
𝝏
𝝏𝒛
𝑲
𝒃𝒉
𝝏𝒛
− 𝑲 − 𝑪𝒉
𝝏𝒉
𝝏𝒕
= 𝟎

20. and expanding the advective and mass accumulation terma of eqn. (2), the following
equation ia obtained:
The dispersion coefhcient (D) in eqna. (2) and (4), according to Biggar and Nielsen (1976) and
Bear (1979), can be expressed as
where: Do is the molecular diffusion coefficient [L'T*']; r is the tortuosity factor; 2 is the
diapersivity [L]; n is a constant; V(= q/O-) ia the average pore-water velocity [L T*'].
In the case of infiltration of salt-containing water in porous media, the initial and boundary
conditions are as follows:
Initial condition
fi(z, 0) = fi, or O-(z, 0) = O-, C(z, 0) = C
Boundary condition at the soil surface
K bh
bz + K —— —— .' >
— OD$g + qC —— qt Ct z -- 0, t > 0
or
/i(0, t) = At or O-(0, t) = O-t
c(o,') - c,
Boundary condition at the soil bottom
/i(1, I) = h or O-(1, i) = O-,

21. http://hays.outcrop.org/images/groundwater/press4e/figu
http://en.wikipedia.org/wiki/Darcy%27s_law
•Tan, Kim. H. 2017. Principles of Soil Chemistry, CRC Press
Taylor & Francis Group, fourth edition.
•Brady, Nyle. C. and Weil, Ray. R.,2019, The Nature and
Properties of Soils, fourteenth edition.
•Sanyal, Saroj Kumar.,2018. A Textbook of Soil Chemistry, Daya
Publishing House A division of Astral International Pvt. Ltd.
•Das, D.K.1996. Introductory Soil Science. Kalyani Publishers,
New Delhi.
•Brady, Nyle. C. and Weil, Ray. R., 2019, The Nature and
Properties of Soils, fourteenth edition.
REFERENCES