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1)Derive the vorticity–velocity formulation of the N–S equation and discuss the advantages and disadvantages of using this formulation

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1)Derive the vorticity–velocity formulation of the N–S equation and discuss the advantages and disadvantages of using this formulation. Hint: take the cross product of the Navier– Stokes equation and simplify the resulting equation using standard vector identities.

2)Verify the steps leading to the vorticity–streamfunction formulation of the N–S equation for 2D flow given in Section 5.6.1. If the flow has no vorticity it is called irrotational flow. What form does this equation take for irrotational flow?

3)Determine the flow rate of water at 25 ?C in a 3000-m-long pipe of diameter 20 cm under a pressure gradient of 20 kPa. Assume a relative roughness parameter of 2.3×10−4. Use ν = 0.916 × 10−6 m2/s for the kinematic viscosity.

4)What form does the Colebrook–White equation take for a smooth pipe? Compare this with the Prandtl formula by plotting the values on the same graph.Verify the expression (6.28) for torque for flow between two cylinders with the outer cylinder rotating. What would the corresponding result be if the inner cylinder were rotating?

5)Write the stress vs. rate-of-strain relations in cylindrical and spherical coordinates for Newtonian fluids.Simplify the Navier–Stokes equation for flows in a circular pipe with only the axial velocity vz as the non-vanishing component. How does it compare with the equations developed in Section 2.6.3?

6)Write the equation of motion in cylindrical coordinates using the various vector operations. Also write the equation of continuity. These equations together are the needed equations for the solution of flow problems posed in cylindrical coordinates.Repeat for spherical coordinates.

7)Write the divergence of the dyad ρvv in index notation. Expand the derivatives using the chain rule. Write the continuity equation in index notation and use this in the expanded expression for the divergence of the above dyad. Simplify and show that the result is (v · ∇)v. Hence verify that Eq. (5.5) is the same as Eq. (5.2).

8)Calculate the settling velocity of a spherical particle of diameter 2.2 cm with a density of 2620 kg/m3 in a liquid of density of 1590 and a viscosity of 9.58 millipoise. What is the regime under which the particle is settling?

9)Consider a circular viewing port on an aquarium. This window has a radius of R, and the center of this port is at a depth d + R from the water surface. Find the force and the center of pressure by direct integration of the differential pressure force.

10)Consider a lighter solid of density ρs floating on the surface of a liquid of density ρl. Derive an expression for the volume fraction for the solid submerged inside the liquid by using the vector calculus. The result will verify the Archimedes principle for floating solids.

11)The average ocean depth is 2 km. Compute the pressure at this point. Assume a constant density. The change in density of water with pressure is small, and can be represented using the bulk modulus K defined by the following equation: K = ρ ∂P ∂ρ T Recalculate the pressure now using the density variation. K ≈ 2.2 × 109 Pa for water.

12)A stress tensor in two dimensions has the following components at a given point: τx,x = 3, τx,y = 2, and τy,y =2 Find the stress vector on a plane that is inclined at an angle of 60? with the x-axis. The plane is located at the same point where the stress tensor has the above reported values. Repeat for a plane oriented at 45?.

13)Derive expressions for the gradient of velocity in cylindrical and spherical coordinates. Hint: define the nabla operator in cylindrical/spherical coordinates and let it act on a vector. Use the chain rule, taking care to differentiate some of the unit vectors. Rearrange the results into a componentwise form. Using this and the definition of E˜, verify the expressions in Sections 3.12.2 and 3.12.3.

14)Show that for a 2D flow or plane confined to the (x, y) plane only the z-component of the vorticity is non-zero. This component, ωz, is simply abbreviated as ω and treated like a scalar. Also verify by direct substitution that −ω = ∇2ψ for 2D flow, where ∇2 is the Laplacian operator with only x and y terms included. Here ψ is the streamfunction for 2D flows. Also derive the corresponding relations for axisymmetric flows.

15)Prove that ∇ · (∇ × A) = 0; i.e., the divergence of the curl of a vector is zero. Use this result to prove that the vector potential of velocity satisfies the continuity equation.Show that the necessary and sufficient condition for the flow to be irrotational is the existence of a scalar velocity potential.

16) Derive an expression for Dv/Dt in cylindrical coordinates. Hint: define the nabla operator in cylindrical coordinates and let it be dotted with a vector. Expand and use the chain rule, taking care to differentiate some of the unit vectors. The components in Table 3.1 would be obtained.