1. CEE 262A H YDRODYNAMICS Lecture 5 Conservation Laws Part I 1

2. Conservation laws V A The ultimate goal is to use Newton's 2nd law (F=ma) to relate forces on fluid parcels to their acceleration. A natural way to do this is to compute the equations of motion following a volume of fluid; We first begin with the governing equations for a solid, non-deformable object with volume V, surface area A, and density  Mass of object: Momentum of object:

3. Newton's 2nd law for the object is given by: Surface forces e.g. Friction Body forces e.g. Gravity Example: Block of mass m pushed with force F along surface with friction coefficient b  F m N g u

4. Which in 3D becomes: velocity of boundary The problem for fluids is that the volume is generally not fixed in time, and so mass and momentum may leave the volume unless a material volume is employed. The key to understanding what to do is Leibniz's Theorem: Consider a volume V(t) for which the bounding surface moves (but not necessarily at the fluid velocity)

5. Fixed volume–V F : Flow of fluid through system boundary (control surface) is non zero, but velocity of boundary is zero. For this case we get Material Volume–V M : Consists of same fluid particles and thus the bounding surface moves with the fluid velocity. Thus, the second term from the Leibnitz rule is now non- zero, so Using Gauss' theorem: This is Reynolds transport theorem, where D/Dt is the same as d/dt but implies a material volume.

6. ~ ~ Note that the Reynolds transport theorem is often written in the more general form which does not assume that the control volume is bounded by a material surface. Instead, the control volume is assumed to move at some velocity and that of the fluid is defined as relative to the control volume, such that In this case, V is not necessarily a material surface. If u r =0, then u b =u and we revert to the form on the previous page. ~

7. The general form of all conservation laws that we will use is: Quantity (F)Volume sources (D)Surface sources (C) Momentum 1 GravityStresses (pressure/viscous) HeatDissipationDiffusion/radiation Salt (scalar)NoneDiffusion Algae (e.g.)GrowthDiffusion 1 Note: momentum is a vector quantity

8. Conservation of Mass Mass is conserved (non-relativistic fluid mechanics) For any arbitrary material volume Since integral is zero for any volume, the integrand must be zero Process: We have taken an integral conservation law and used it to produce a differential balance for mass at any point

9. However, and Thus if the density of fluid particles changes, the velocity field must be divergent. Conversely, if fluid densities remain constant,

10. But Any other fluid property (scalar, vector,.. also drop triple integral)

11. Why is this important/useful? Because Newton’s 2 nd law: But from above: Rate of Change of Momentum = Net Applied Force Net Applied Force = Mass Acceleration Independent of volume type!

12. Some Observations 1. Incompressible [ No volumetric dilatation, fluid particle density conserved] Differential form of “Continuity”

13. 2. Slightly Compressible Typically found in stratified conditions where Boussinesq Approximation - Vertical scale of mean motion << scale height - or Allows us to treat fluid as if it were slightly incompressible Note: Sound and shock waves are not included ! Reference density (1000 kg/m 3 for water) Background variation (typ. 1- 10 kg/m 3 for water) Perturbation density due to motion (typ. 0.1-10 kg/m 3 for water)

14. Informal “Proof” If a fluid is slightly compressible then a small disturbance caused by a change in pressure,, will cause a change in density. This disturbance will propagate at celerity, c. If pressure in fluid is “hydrostatic” Now and [ Streamline curvature small]

15. Typically: g ≈10 m/s 2 ; c ≈ 1500 m/s ; w ~ 0.1m/s

16. Conservation of Momentum – Navier-Stokes We have: Two kinds of forces: Body forces Surface forces Two kinds of acceleration: Unsteady Advective (convective/nonlinear) Two kinds of surface forces: Those due to pressure Those due to viscous stresses Divergence of Stress Tensor

17. Plan for derivation of the Navier Stokes equation 1.Determine fluid accelerations from velocities etc. (done) 2.Decide on forces (done) 3.Determine how surface forces work : stress tensor 4.Split stress tensor into pressure part and viscous part 5.Convert surface forces to volume effect (Gauss' theorem) 6.Use integral theorem to get pointwise variable p.d.e. 7.Use constitutive relation to connect viscous stress tensor to strain rate tensor 8.Compute divergence of viscous stress tensor (incompressible fluid) 9.Result = Incompressible Navier Stokes equation

18. Forces acting on a fluid a) Body Forces: - distributed throughout the mass of the fluid and are expressed either per unit mass or per unit volume - can be conservative & non-conservative Force potential Examples: (1) force due to gravity (acts only in negative z direction) (2) force due to magnetic fields We only care about gravity

19. b) Surface forces: - are those that are exerted on an area element by the surroundings through direct contact - expressed per unit of area - normal and tangential components

20. c) Interfacial forces: - act at fluid interfaces, esp. phase discontinuities (air/water) - do not appear directly in equations of motion (appear as boundary conditions only) - e.g. surface tension – surfactants important - very important for multiphase flows (bubbles, droplets,. free surfaces!)

21. Very important deviation from text!!! CEE262a (and most others) Deviatoric (viscous) stress Full stress Kundu and Cohen Full stress Deviatoric (viscous) stress

22. Stress at a point (From K&C – remember difference in nomenclature,i.e.  ij ←  ij )

23. What is the force vector I need to apply at a face defined by the unit normal vector to equal that of the internal stresses? Consider a small (differential) 2-D element cut away

24. force component in x 1 direction [ has magnitude of 1] Defining the stress tensor to be And in general d

25. But [see Kundu p90] “ Surface force per unit area” (note this is a 2D area) Total, or net, force due to surface stresses

26. Conservation of momentum Dimensions: dx 1. dx 2. dx 3

27. Sum of surface forces in x 1 direction: Defining i component of surface force per unit volume to be

28. In general : “Cauchy’s equation of motion” Force = divergence of stress tensor

29. Important Note: This can also be derived from the Integral From of Newton’s 2nd Law for a Material Volume V M But and

30. Constitutive relation for a Newtonian fluid “Equation that linearly relates the stress to the rate of strain in a Newtonian Fluid Medium” (i)Static Fluid: - By definition cannot support a shear stress - still feels thermodynamic pressure (in compression) (ii) Moving Fluid: - develops additional components of stress (due to viscosity) Hypothesis Note difference from Kundu ! Deviatoric stress tensor [Viscous stress tensor]

31. If medium is isotropic and stress tensor is symmetric only 2 non-zero elements of Assume 4 th order tensor (81 components!) that depend on thermodynamic state of medium which gives or See derivation of in Kundu, p 100

32. Special cases (i) Incompressible (ii) Static In summary Cauchy's equation Constitutive relation for a compressible, Newtonian fluid.

33. Navier-Stokes equation The general form of the Navier-Stokes equation is given by substitution of the constitutive equation for a Newtonian fluid into the Cauchy equation of motion: Incompressible form (e kk =0):

34. Assuming where

35. If “Inviscid” Euler Equation Or in vector notation InertiaPressure gradient Gravity (buoyancy) Divergence of viscous stress (friction)