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Tutorial 3 (ME332): Exact Solutions Instructors: Prof. Nadeem Hasan and Dr. Syed Fahad Anwer

1. A moving belt is inclined at an angle α to the horizontal. The lower end of this belt is immersed in a pool of liquid and the belt drags some of the liquid with it as it moves upward and out of the liquid. The liquid may be assumed to be viscous, but incompressible. (a) Using the configuration shown in Fig. Q1(a), neglecting pressure gradient, solve the

Navier - Stokes equations for the following quantities: (i) The velocity distribution in the liquid layer (ii) The volumetric flow rate of liquid in the x direction (iii) The angle α for which the volumetric flow rate is zero

(b) Repeat part (a) of this question for the configuration shown in Fig. Q1(b).

2. Fig. Q2 shows a conduit whose cross section is the shape of an equilateral triangle. For the coordinate system shown in the figure, the equations of the three sides are

𝑧 +𝑏2 3

= 0

𝑧 + 3𝑦 −𝑏3= 0

𝑧 − 3𝑦 −𝑏3= 0

Look for a solution for the velocity distribution in this conduit of the following form:

𝑢 𝑦, 𝑧 = 𝛼 𝑧 +𝑏2 3

𝑧 + 3𝑦 −𝑏3

𝑧 − 3𝑦 −𝑏3

Determine the value of the constant α such that the assumed form of solution is exact, the value of this constant being expressed in terms of the applied pressure gradient.

Fig. Q1 – Liquid layer between a moving surface and (a) solid surface, (b) free surface

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Fig. Q2 – Conduit with cross-sectional shape of an equilateral triangle

3. Two concentric circular cylinders enclose a viscous fluid. If the inner cylinder is at rest and the outer cylinder rotates at a constant angular velocity, calculate the torque required to rotate the outer cylinder and that required to hold the inner cylinder at rest.

4. Obtain the velocity distribution for the modified Stokes second problem consisting of a fluid that is contained between two infinite parallel surfaces separated by a distance h. The upper surface is held fixed, while the lower surface oscillates in its own plane with velocity Ucos(wt).

5. For potential flow due to a line vortex the vorticity is concentrated along the axis of the vortex. Assume axisymmetric and planar flows. Thus, the problem to be solved for the decay of a line vortex due to the viscosity of the fluid is as follows:

𝜕𝜔𝜕𝑡 = 𝜈∇!𝜔

𝜔 𝑟, 0 = 0 𝑓𝑜𝑟 𝑟 > 0∞ 𝑓𝑜𝑟 𝑟 = 0

Here ω (r, t) is the vorticity, and the circulation Γ 𝑟, 0 = Γ! = 2𝜋 𝜔(𝑟, 0) 𝑟𝑑𝑟!!

around the vortex. (a) Apply Stokes theorem to transform the above governing equation into Circulation

form by considering a circular circuit of arbitrary radius ‘r’ enclosing the decaying line vortex. Show that circulation Γ(r, t) is governed by:

𝜕Γ𝜕𝑡 = 𝜈

𝜕!Γ𝜕𝑟! −

1𝑟𝜕Γ𝜕𝑟

(b) Transform the above boundary conditions to show that line vortex is bound by Γ 𝑟, 0 = Γ! Γ 0, 𝑡 = 0 Γ ∞, 𝑡 = Γ!

(c) Use similar methodology as adopted in solving stokes first problem to obtain expressions for Γ 𝑟, 𝑡 , ω (r, t), velocity uθ(r, t) and the pressure p(r, t) in the fluid.

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6. The following flow field satisfies the continuity equation everywhere except at R = 0, where a singularity exists:

𝑢! = −𝑎𝑅

𝑢! = 𝐾𝑅

𝑢! = 2𝑎𝑧

Show that this flow field also satisfies the Navier-Stokes equations everywhere except at R = 0, and find the pressure distribution in the flow field.

Modify the foregoing expressions to the following:

𝑢! = −𝑎𝑅

𝑢! = 𝐾𝑅 𝑓(𝑅)

𝑢! = 2𝑎𝑧

Determine the function f(R) such that the modified expression satisfies the governing equations for a viscous, incompressible flow and such that the original flow field is recovered for R → ∞.

7. Consider the problem of Couette flow between parallel plates, Fig. Q7, for a power-law

Non-Newtonian fluid, τxy=K(du/dy)n, where n≠1. Assuming constant pressure, solve for the velocity distribution u(y) between the plates of (a) n<1 and (b) n>1, and compare with the Newtonian solution, 𝑢 = !

!(1+ !

!). Comment on the results.

Fig. Q7

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8. Consider the axial Couette flow of Fig. Q8 with both cylinders moving. Find the velocity distribution u(r) and plot it for (a) U1 = Uo, (b) U1 = -Uo, and (c) U1 = 2Uo. Comment on the results.

Fig. Q8

9. A circular cylinder of radius R is rotating at steady angular rate ω in an infinite fluid of

constant ρ and µ. Assuming purely circular streamlines, find the velocity and pressure distribution in the fluid and compare with the flow field of an inviscid “potential” vortex.

10. Assuming that the velocity distribution between rotating concentric cylinders is known

from, 𝑢! = 𝑟!𝜔!!! ! ! ! !!!! !! ! !! !!

+ 𝑟!𝜔!! !!! !! !!! !!! !! !!

(see, Fig. 5), find the pressure distribution

p(r) if the pressure is po at the inner cylinder. 11. Air at 20oC and 1atm is driven between two parallel plates 1cm apart by an imposed

pressure gradient (dp/dx) and by the upper plate moving at 20 cm/s. Find (a) the volume flow rate (in cm3/s per meter of width) if dp/dx= -0.3 Pa/m and (b) the value of (dp/dx) (in Pa/m) which causes the shear stress at the lower plate to be zero.

12. Derive the solution u(y,z) for flow through an elliptical duct, Fig. Q9, by solving !!!!!!

+ !!!!!!

= !!

!"!"= 𝑐𝑜𝑛𝑠𝑡. Begin with a guessed quadratic solution,

𝑢 = 𝐴 + 𝐵𝑦! + 𝐶𝑧!, and work your way through to the exact solution.

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Fig. Q9

13. Consider a wide liquid film of constant thickness h flowing steadily due to gravity down an inclined plane at angle θ, as shown in Fig. Q13. The atmosphere exerts constant pressure and negligible shear on the free surface. Show that the velocity distribution is given by

𝑢 = 𝜌𝑔 sin𝜃2𝜇 𝑦(2ℎ − 𝑦)

and that the volume flow rate per unit width is 𝑄 = 𝜌𝑔ℎ! 𝑠𝑖𝑛𝜃/3𝜇. Compare this result with flow between parallel plates, (i.e. equations given below)

𝑢 = 𝑢!"# 1− 𝑦!

ℎ!

𝑢!"# = − 𝑑𝑝𝑑𝑥

ℎ!

2𝜇

𝑄 = 𝑢 𝑑𝑦!!

!!= 43 ℎ𝑢!"#

ū = 𝑄2ℎ =

23 𝑢!"#

Fig. Q13

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14. Consider a double layer of immiscible fluids 1 and 2, flowing steadily down an inclined plane, as in Fig. Q14. The atmosphere exerts no shear stress on the surface and is at constant pressure. Find the laminar velocity distribution in the two layers.

Fig. Q14