Water Emulsion Droplets to Settle in a Container - Science Assignment Help

Assignment Task:

Task:

1. How long would it take to sediment 1μm and 10μm oil in water emulsion droplets to settle in a container of height 0.1 m. The oil has a density 3 times that of water and a large viscosity that the drop motion can be approximated with Stokes law of a translating particle. What is the ratio of thermal energy to the potential energy gained by the emulsion drops?
2. In production operations in petroleum industry, Water in oil emulsions are very important and the water content can be as hight as 60% in volume. The table shows the viscosity of crude oil water emulsions at various temperatures.

(a) Plot the relative viscosity of the emulsion as a function of volume fraction at any two different temperatures.
(b) How does the plot compare with the predictions of
i. Einstein’s equation for viscosity of suspension of hard spheres
ii. Bachelor’s expression for viscosity of suspensions of hydrodynamically interacting hard spheres and
iii. Dougherty - Krieger semiempirical equation?
3. (a) We learned several new terms associated with complex fluids in the class. List 3-5 terms that you found conceptually interesting along with a brief (very brief) description of it.
4. Show that a·(b∧c) vanishes if two of the three vectors are proportional to each one another using manipulation of indices.
5. Prove that (a ∧ b) ∧ (c ∧ d) = [c · (d ∧ a)] b − [c · (d ∧ b)] a using manipulation of indices.
6. An antisymmetric tensor has only 3 independent components and therefore, this quantity should be related to a vector. Consider a vector ω whose components are [ω1, ω2, ω3]. Now let us construct an antisymmetric tensor Ωij = ijkωk. Write down the components of this tensor explicitly and make sure that this tensor is antisymmetric. This formulation gives as a way of writing a vector as an antisymmetric tensor. Can you now invert this relationship to express ω in terms of Ω ? (It will be useful to know that ω is the vorticity that we obtain by taking curl of the velocity field in the context of fluid flow. On the other hand Ω will be the antisymmetric part of the velocity gradient tensor).
7. (a) Write down a 3 × 3 matrix (yes, with any elements you like). Now express it as a sum of a symmetric and an antisymmetric matrix.
(b) Now calculate the matrix product of this symmetric and antisymmetric matrix. What do you find?
(c) Take the trace of this product matrix. What do you find?
8. If a is a vector variable, show that (using index manipulations)
(a) ∇ · (∇ ∧ a) = 0. (Note how the generalisation of the above question no: (6) is useful in getting this identity proved.)
(b) ∇ ∧ (∇ ∧ a) = ∇(∇ · a) − ∇2a
9. Calculate R ninjnknlnmnndS on a unit sphere. Write it down along with the results that we obtained in the class for similar integrals, R ninjdS and R ninjnknldS. Do you find a pattern for the coefficients that you get for these integrals ? Can you predict what the coefficient of next bigger integral in this series would be?
10. Write down the following equations in index notation.
1) Advection diffusion equation for a scalar field (like temperature).
2) Navier Stokes equations - Continuity and momentum balance
In the Navier Stokes equations (momentum balance), on the LHS the inertial term has the form u · ∇u. Using manipulation of indices, show that u · ∇u = 1 2∇(u) 2 − u ∧ ω where ω is the vorticity, given as the curl of the velocity field, ω = ∇ ∧ u.
[Start simplifying u ∧ ∇ ∧ u.]

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