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

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. 1 st year 2 nd 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 Poi s euil l e’s Flow 5 L apla c e Law 6 Yo u ng - Lapla c e Law 7 The one-dimensional vertical flow of water 8 Reference

3. 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.

4. 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 ( P b − P a ), 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

5. 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.

6. 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)

7. Poi s euil l e 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 ( a 2  x 2 ) 2  G can be gravitational acceleration or (linear) pressure gradient (P in – x = x = a u(x) out P ) /L

8. Poi s euil l e Flow S.GOKALTUN Florida International University

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

10. 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. L apla c e Law

11. L apla c e 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

12. L apla c e Law  P =  /r →  =  P/r, which is linear in 1/r (a.k.a. curvature) r P in P out

13. Yo u ng - Lapla c e 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   

14. Yo u ng - Lapla c e Law Zero contact angle means perfect wetting;  P =  cos  /r

15. 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 [L 2 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

16. 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 > 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- ,

17. 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