Navier-Stokes Equations, Length & Velocity Scale Along With Non-Dimensional Quantity & Reynolds Number - Science & Maths Assignment Help

Assignment Task

Navier-Stokes Equations

1. (a) We discussed Navier-Stokes equations in the class. Assuming that the problem in hand has a characteristic length scale L, a velocity scale U and a time scale T, non-dimensionalise the Navier Stokes equations. You may assume that pressure scales like viscous stresses. Identify the non-dimensional quantity that appears along with unsteady term as Stokes number and convective term as Reynolds number. 
(b) See that, if the characteristic time scale T = L/U then Stokes number is same as Reynolds number. (c) Take the curl of the Navier stokes equations to replace velocity with vorticity. What are the terms that are not expressible in terms of vorticity? What happens to these terms in the limit Re → 0? 
2. Considering the relative motion between two neighbouring points explain the mathematical definitions and physical interpretations of the concepts of (i) linear strain rate, (ii) shear strain rate and (iii) vorticity. 
3. A spherical particle is settling near a wall in an otherwise quiescent fluid, and in an unbounded domain as shown below: 
Without solving the detailed hydrodynamic field, predict whether the particle will move towards or away from the wall as it settles under gravity in Stokes flow approximation. 
4. (a) Write down the solution of Laplace’s equation, both growing and decaying harmonics upto 5th order (n = 4). (b) Show that the growing harmonics obtained as r2n+1× decaying harmonics are solutions of Laplace’s equation by explicit substitution for n = 0, 1. 
5. An important property of Stokes  its uniqueness. Proving it is an interesting exercise. I will first outline the basic manipulations involved: 
 

where v, p and e are the velocity, pressure and rate of strain tensor associated with an incompressible Stokes flow field in a fluid domain V , bounded by S. Note that we have used ∂ivi = 0. The logic goes into each manipulation is written in brackets. If you are able to follow the above procedure, you can easily prove the uniqueness of Stokes solutions in the following way. 
Step 1: For a Newtonian fluid, the rate of energy dissipation due to viscosity may be calculated as  where µ is the viscosity and e is the rate of strain tensor in a fluid of region V . 
Step 2: Now assume that v and v0 are two solutions of the Stokes equations that satisfy the same boundary conditions (v = v0 on S). The rate of energy dissipated by the difference field, v0 − v may be calculated using the above equation 6. 
Step 3: Apply manipulations used in equations 1-5 on this difference field, v − v0. Do you find that the total dissipation associated with this difference field vanishes? 
Step 4: If you find that the total dissipation associated with the difference field (v − v0) vanishes, then the only way that can happen is by having v = v0. That completes the proof of uniqueness of the solutions of Stokes equations .
 

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