1996/1 Fluid Dynamics II - University of · PDF file1996/1 Fluid Dynamics II Viscous...

1996/1 Fluid Dynamics II

Viscous incompressible fluid is contained in the annulus between twoconcentric circular cylinders r = a, b (a < b). The inner cylinder rotates withangular velocity Ω about its axis and the outer cylinder is at rest. The velocityfield u in the fluid has components

u = (0, v(r), 0)

in cylindrical polar coordinates (r, θ, z). Show that the θ-component of the steadyNavier–Stokes equation reduces to

d2v

dr2+

1

r

dv

dr− v

r2= 0 .

Hence find v(r), and show that the torque G per unit length that must be appliedto the inner cylinder to maintain its motion is

G =4πµΩa2b2

b2 − a2,

where µ is the dynamic viscosity of the fluid.[The rθ-component of the rate-of-strain tensor is erθ =

1

2rd

dr

(v(r)

r

).

]

Suppose now that the Reynolds number is small and that the outer cylinderis replaced by a cylinder of square cross-section of side 2c (> 2a) centred on r = 0.By considering circles that are inscribed and escribed to the square and using theminimum dissipation theorem (which may be stated without proof), find lower andupper bounds for the torque G that is now required to maintain the rotation of theinner cylinder.

[TURN OVER


A solid sphere of radius a falls through viscous incompressible fluid towardsa plane horizontal boundary.

Consider the final stage of sedimentation when the gap h0(t) between thelowest point of the sphere and the plane is small, i.e.

h0(t) ≪ a .

In the neighbourhood of this gap, the distance between the sphere and the planemay be approximated, at distance r from the axis of symmetry, by

h(r, t) ≈ h0(t) + r2/2a .

Using the approximation of lubrication theory, show that the pressure p(r, t) in thegap satisfies

∇ · (h3∇p) ≡ 1

r

∂

∂r

(rh3 ∂p

∂r

)= 12µ

dh0

dt,

where µ is the viscosity of the fluid and ∇ is the horizontal gradient operator.Hence find p and show that the force on the sphere is

F = 2π

∞∫

0

p(r, t)r dr = −6πµa2

h0

dh0

dt.

Deduce that, if the sphere is settling under its own weight (corrected for buoyancy),h0(t) decreases exponentially. What is the exponential decay rate if the sphere hasdensity ρs and the fluid has density ρ?


Starting from the Navier–Stokes equations for incompressible flow of aviscous fluid, obtain the equation governing the evolution of the vorticity field ω.Show that, if the flow is two-dimensional, with u = (u, v, 0) in Cartesian coordinates(x, y, z), then this equation reduces to the equation

∂ω

∂t+ u · ∇ω = ν∇2ω ,

where ω = |ω| and ν is the kinematic viscosity of the fluid.

Consider the evolution of a flow which at time t = 0 corresponds to a pointvortex of circulation Γ at x = y = 0. Show that, on dimensional grounds,

ω(r, t) =Γ

νtf(η) ,

where η = 12r(νt)

−1/2 and r2 = x2 + y2, and find the function f(η).

Show that the rate of dissipation of energy D (per unit length in the z-direction) is proportional to ρΓ2/t, where ρ is the density of the fluid, and find theconstant of proportionality.[

You may assume that D = 2ρν∫ω2 dA .

]


Fluid having kinematic viscosity ν and density ρ is confined between a fixedplate at y = h and a plate at y = 0 whose velocity is (U cosωt, 0, 0), where U is aconstant. Explain the physical significance of the dimensionless number S = ωh2/ν.

Assuming that the flow remains time-periodic and unidirectional, findexpressions for the flow profile and the time-average rate of working Φ per unitarea by the plates on the fluid.

Sketch the profile and evaluate Φ in the limits S → 0 and S → ∞, andexplain why in these limits Φ becomes independent of ω and h respectively.

[TURN OVER


Starting from the steady vorticity equation

u · ∇ω = ω · ∇u + ν∇2ω ,

explain briefly how the two-dimensional boundary layer equation

ψyψxy − ψxψyy = UU ′ + ν ψyyy

may be derived, and give the meaning of U(x).

Consider steady flow, with uniform velocity (U, 0) far upstream, past a flatplate that occupies 0 6 x <∞. Verify that a self-similar solution of the boundarylayer equation may be found, in which the boundary layer thickness grows like x1/2.Find an ordinary differential equation for the flow profile, and specify the boundaryconditions that must be satisfied.

Suppose that for a plate of finite length L, the flow profile nearby remainsthe same as for an infinite plate. How does the total drag on the plate depend uponL?


(a) A steady pressure drop ∆p is applied along a cylindrical pipe of radius Rand length L containing liquid of viscosity µ. Calculate the resulting volume fluxQ of the liquid.[In cylindrical polar coordinates, with φ = φ(r),

∇2φ =1

r

d

dr

(rdφ

dr

).]

(b) The centre of a sphere of radius a falls under gravity g with speed V alongthe axis of a vertical fluid-filled cylindrical tube of radius a + h0, where h0 ≪ a.The density of the sphere exceeds that of the fluid by amount ∆ρ. The ends of thecylinder, which are many diameters away from the sphere, are sealed. In a frameof reference in which the sphere is stationary, what is the fluid flux Q through thegap? Deduce that at leading order in h0/a, the velocity difference between thesphere and the tube is negligible compared with the fluid velocity in the gap, anduse lubrication methods to calculate the difference in fluid pressure ∆p above andbelow the sphere.

Show, again at leading order in h0/a, that this pressure difference supportsthe buoyant weight of the sphere, and deduce V .[You may assume that

∞∫

0

dx

(1 + x2)3 =

3π

16.

]

(c) Suppose instead that the ends of the vertical cylinder are unsealed andjoined by a fluid-filled tube again having radius a+ h0, so that, taken as a whole,the fluid now occupies a tube having the form of a ‘D-shaped’ loop with the spherein the vertical section. Assuming that the result of Part (a) applies away from thesphere, and the total length of the tube is L ≫ a, determine how large L/a mustbe if the fall speed of the sphere in this new geometry is to be unchanged from theresult in Part (b).

[TURN OVER


What is meant by a Stokes flow?

Fluid of viscosity µ is contained in the annulus between two concentric spheres of radius a and b > a. The inner sphere rotates about an axis through its centrewith angular velocity Ω; the outer sphere is stationary. Assuming that inertia isnegligible, explain why the pressure in the fluid is uniform, find the fluid velocityand deduce the couple that must be applied to the inner sphere to maintain themotion. What couple is exerted by the fluid on the outer sphere? How small must|Ω| be for your analysis to be self-consistent?

[Hint: You may find it helpful to consider a velocity of the form

u = Ω ∧ xf(r)

for a suitably chosen function f(r). For such a velocity field you should show that

x ∧ (σ · x) = µx ∧ (Ω ∧ x)rf ′(r).]


An infinite rigid cylinder of radius a falls, with its axis horizontal, throughfluid of viscosity µ, toward a rigid horizontal plane. Use lubrication theory to showthat, if the distance of closest approach of the cylinder to the plane is h0(t) ≪ a,then

dh0

dt= − W

3√

2πµ

(h0

a

)3/2

,

where W is the weight of the cylinder per unit length adjusted for buoyancy.


Fluid having kinematic viscosity ν flows steadily in the annular regiona < r < b between two concentric porous circular cylinders r = a, r = b. Anaxisymmetric radial flow having total flux Q per unit length of each cylinder ismaintained across the annulus. Explain briefly the physical significance of thedimensionless number R = Q/2πν, and obtain the radial velocity u(r).

Now suppose that the inner cylinder rotates about its axis with angularvelocity Ω, and the outer cylinder remains stationary. Assuming that u(r) remainsunchanged, find the steady azimuthal velocity v(r) in the annulus for an arbitraryfixed value of R.

In the limits R → ±∞, find and sketch the forms of v(r) and comment onyour results.

[In cylindrical polar coordinates, the azimuthal component of the Navier–

Stokes equation takes the form

udv

dr+

uv

r= ν

d

dr

[1

r

d

dr(rv)

].

You may find it helpful to look for solutions of the form v = rα. ]


(a) An unbounded fluid, having kinematic viscosity ν, has velocity at timet = 0 given in Cartesian coordinates as

u =

(− 1

2U, 0, 0), y > 0(

12U, 0, 0

), y < 0 ,

where U is a constant, so that there is a vortex sheet at y = 0. Assuming the flowremains unidirectional, find its velocity u(y, t) for times t > 0 in the form of anintegral, and sketch the velocity profile.

(b) Now suppose that the fluid is inviscid. Show that the vortex sheet inPart (a) is linearly unstable to disturbances of all wavelengths k, and find thecorresponding growth rates. Which waves are expected to grow most rapidly?

(c) Without further detailed calculation, use the result from Part (a) toindicate qualitatively how your conclusions in Part (b) should be modified if thefluid has viscosity ν.

[TURN OVER


The velocity field u and stress tensor σ satisfy the Stokes equations in avolume V bounded by a surface S. Let u be another solenoidal velocity field. Showthat ∫∫

S

σijnj ui dS =

∫∫∫

V

2µeij eij dV ,

where e and e are the strain-rates corresponding to the velocity fields u and urespectively, and n is the outward normal vector. Hence, or otherwise, prove theminimum dissipation theorem for Stokes flow.

A particle slowly moves at velocity U through a highly viscous fluid ofviscosity µ contained within a stationary vessel. As the particle moves, the fluidexerts a drag force F on it. Show that

−F · U =

∫∫ ∫

fluid

2µeijeij dV .

Consider now the case when the particle is a small cube, with sides of length l,moving in a very large vessel. You may assume that

F = −kµlU

for some constant k. Use the minimum dissipation theorem, being careful to declarethe domain involved, to show that

3π 6 k 6 3√

3π .

[Hint: You may assume Stokes’ result for the drag on a sphere of radius a,F = −6πµaU.]


The steady two-dimensional boundary-layer equations for flow primarily inthe x-direction are

ρ

(u∂u

∂x+ v

∂u

∂y

)= −dP

dx+ µ

∂2u

∂y2,

∂u

∂x+∂v

∂y= 0.

A thin, steady, two-dimensional jet emerges from a point at the origin andflows along the x-axis in a fluid at rest far from the x-axis. Show that the momentumflux

F =

∞∫

−∞

ρu2 dy

is independent of the position x along the jet. Deduce that the jet thicknessscale δ(x) increases along the jet as x2/3 while the centre-line velocity U(x) decreasesas x−1/3.

A similarity solution for the jet is sought with a streamfunction ψ of theform

ψ(x, y) = U(x)δ(x)f(η) with η = y/δ(x).

Find the nonlinear third-order non-dimensional differential equation governing f ,and state the boundary and normalisation conditions that must be applied. Finda solution for f of the form

f(η) = α tanh(βη) ,

where α and β are constants to be determined.[You may quote the result

∞∫

−∞

sech4ζ dζ = 43 .

]

[TURN OVER


A thin axisymmetric drop of a fixed volume V of viscous liquid is spreadingunder the action of gravity on a horizontal table. Let the radius of the drop beR(t) and the thickness of the liquid layer be h(r, t) in 0 6 r 6 R(t). Use lubricationtheory to derive the equation for the pressure,

p = ρg(h(r) − z

),

where the table is z = 0, and thence the equation governing the spreading of thedrop,

∂h

∂t=

g

3ν

1

r

∂

∂r

(rh3 ∂h

∂r

).

Consider a similarity solution of the form

h(r, t) = H(t)f(η) with η = r/R(t) ,

where R(t) is proportional to the radius of the drop R(t). On the assumption thatthe only scale for time is t itself, show that the radius R(t) increases like t1/8 and theheight H(t) decreases like t−1/4. Hence obtain the equation governing the shape,

− 14f − 1

8ηf′ =

1

3η

(ηf3f ′

)′.

Solve this equation subject to the volume normalisation.


Incompressible fluid having kinematic viscosity ν is confined in a channelhaving rigid walls at y = ±h. A spatially uniform pressure gradient −G cos ωt isapplied in the x-direction. What is the physical significance of the dimensionlessgroup S = ωh2/ν?

Assuming that the flow is unidirectional and time harmonic obtainexpressions for the flow profile and the total flux.

In each of the limits S → 0 and S → ∞, sketch the flow profile, find anasymptotic expression for the total flux, and give a physical interpretation of theflow field.

Suppose instead that G = 0 and the channel walls each oscillate in their ownplane with velocity U cosωt. Without explicitly calculating the solution sketch theflow profile in each of the limits S → 0 and S → ∞.

[You may assume that as z → 0,

cosh z = 1 + 12z2 + . . . and tanh z = z − 1

3z3 + . . . . ]


Show that the streamfunction ψ for a planar Stokes flow satisfies

∇4ψ = 0 .

A flat paint-scraper touches, and is held perpendicular to, a stationary horizontalsurface. The scraper moves parallel to the surface with speed U . Viscous fluidfills the region ahead of the scraper. Assuming that inertia is negligible, find thestreamfunction ψ(r, t) describing the flow in the corner between the scraper andthe surface, in the form

ψ(r, θ) = rαf(θ) ,

where r measures distance from the corner and α is a constant which is to bedetermined. Estimate the size of the corner region within which inertia remainsnegligible.

Calculate the tangential stress exerted by the fluid on the horizontal surface,and comment on the resulting prediction of the total force exerted on the horizontalsurface.

[In plane polar coordinates (r, θ)

ur =1

r

∂ψ

∂θ, uθ = −∂ψ

∂r,

∇2ψ =1

r

∂

∂r

(r∂ψ

∂r

)+

1

r2∂2ψ

∂θ2. ]


Fluid of viscosity µ is placed in the gap between two concentric parallelcircular discs of radius a. The gap width h(t) is much smaller than a. Uselubrication theory to find the radial fluid velocity in the gap when the plates movewith relative velocity dh/dt. What are the pressure forces on the plates? Underwhat circumstances does the lubrication approximation hold for this flow.

At t = 0 the gap width is h0. For t > 0 a constant force F is applied to eachof the discs in order to press them together. Solve for h(t) and briefly comment onyour result.

[TURN OVER


The vorticity ω in a steady incompressible two-dimensional flow of a viscousliquid satisfies

u · ∇ω = ν∇2ω .

Give a physical interpretation of this result.

Fluid flows at high Reynolds number past a rigid body. Derive the boundary-layer approximation to the vorticity equation

uωx + vωy = νωyy ,

stating clearly the region where this equation is valid and any approximation thatyou make. Give the corresponding boundary-layer approximation to the Navier-Stokes equations, and state appropriate boundary conditions.

Viscous fluid occupies the region y > 0 and is bounded by a rigid wall alongy = 0. For y → ∞, the fluid velocity streamfunction ψ → αxy, where α is aconstant. Show that the boundary-layer equation for this flow is satisfied by

ψ = |α|δxf(yδ

), where δ =

(ν

|α|

)1/2

,

provided thatf ′2 − ff ′′ = 1 + f ′′′ .

What are the boundary conditions on f?

A numerical solution to this problem can be found for α > 0. Explainphysically why no solution exists if α < 0.

Consider steady high-Reynolds-number flow with constant velocity past acircular cylinder. Briefly indicate what features of this flow can be identified usingthe results above.


The energy equation for the motion of a viscous, incompressible fluid statesthat

d

dt

∫

V (t)

1

2ρu2 dV =

∫

S(t)

uiσijnj dS − 2µ

∫

V (t)

eijeij dV .

Interpret each term in this equation and explain the meaning of the symbols used.

For steady rectilinear flow in a (not necessarily circular) pipe having rigidstationary walls, deduce a relation between the viscous dissipation per unit lengthof the pipe, the pressure gradient G, and the volume flux Q.

Starting from the Navier-Stokes equations, calculate the velocity field forsteady rectilinear flow in a circular pipe of radius a. Using the relationship derivedabove, or otherwise, find in terms of G the viscous dissipation per unit length forthis flow.

[In cylindrical polar coordinates,

∇2w(r) =1

r

d

dr

(rdw

dr

). ]


Explain what is meant by a Stokes flow and show that, in such a flow, inthe absence of body forces, ∂σij/∂xj = 0, where σij is the stress tensor.

State and prove the reciprocal theorem for Stokes flow.

When a rigid sphere of radius a translates with velocity U throughunbounded fluid at rest at infinity, it may be shown that the traction per unitarea, σ · n, exerted by the sphere on the fluid, has the uniform value 3µU/2a overthe sphere surface. Find the drag on the sphere.

Suppose that the same sphere is free of external forces and is placed withits centre at the origin in an unbounded Stokes flow given in the absence of thesphere as us(x). By applying the reciprocal theorem to the perturbation to theflow generated by the presence of the sphere, and assuming this to tend to zerosufficiently rapidly at infinity, show that the instantaneous velocity of the centre ofthe sphere is

V =1

4πa2

∫

r=a

us(x) dS .

[TURN OVER


Starting from the steady planar vorticity equation

u .∇ω = ν∇2ω ,

outline briefly the derivation of the boundary layer equation

uux + υuy = UdU/dx+ νuyy ,

explaining the significance of the symbols used.

Viscous fluid occupies the region y > 0 with rigid stationary walls alongy = 0 for x > 0 and x < 0. There is a line sink at the origin of strength πQ, Q > 0,with Q/ν ≫ 1. Assuming that vorticity is confined to boundary layers along therigid walls:

(a) Find the flow outside the boundary layers.

(b) Explain why the boundary layer thickness δ along the wall x > 0 is proportionalto x, and deduce that

δ =

(ν

Q

) 1

2

x .

(c) Show that the boundary layer equation admits a solution having streamfunction

ψ = (νQ)1/2f(η) with η = y/δ .

Find the equation and boundary conditions satisfied by f .

(d) Verify that a solution is

f ′ =6

1 + cosh(η√

2 + c)− 1 ,

provided that c has one of two values to be determined. Should the positive ornegative value be chosen?


State the minimum dissipation theorem for Stokes flow in a bounded domain.

Fluid of density ρ and viscosity µ fills an infinite cylindrical annulus a ≤r ≤ b between a fixed cylinder r = a and a cylinder r = b which rotates about itsaxis with constant angular velocity Ω . In cylindrical polar coordinates (r, θ, z) , thefluid velocity is u = (0, v(r), 0) . The Reynolds number ρΩb2/µ is not necessarilysmall.

Show that v (r) = Ar+B/r , where A and B are constants to be determined.

[You may assume that ∇2u = (0, ∇2v − v/r2, 0) and (u · ∇)u =(−v2/r, 0, 0) .]

Show that the outer cylinder exerts a couple G0 per unit length on the fluid,where

G0 =4πµΩ a2b2

b2 − a2.

[You may assume that, in standard notation, e rθ =r

2

d

dr

(vr

).]

Suppose now that b ≥√

2a and that the cylinder r = a is replaced by afixed cylinder whose cross-section is a square of side 2a centred on r = 0 , all otherconditions being unchanged. The flow may still be assumed steady. Explainingyour argument carefully, show that the couple G now required to maintain themotion of the outer cylinder is greater than G0 .

[TURN OVER


A thin layer of liquid of kinematic viscosity ν flows under the influence ofgravity down a plane inclined at an angle α to the horizontal (0 ≤ α ≤ π/2). Withorigin O on the plane, and axes Ox down the line of steepest slope and Oy normalto the plane, the free surface is given by y = h(x, t), where |∂h/∂x| ≪ 1. Thepressure distribution in the liquid may be assumed to be hydrostatic. Using theapproximations of lubrication theory, show that

∂h

∂t=

g

3ν

∂

∂x

h3(cosα

∂h

∂x− sinα)

.

Now suppose that

h = h0 + η (x, t) ,

whereη (x, 0) = η0 e

−x2/a2

and h0 , η0 and a are constants with η0 ≪ a, h0 . Show that, to leading order,

η (x, t) =a η0

(a2 + 4Dt)1/2exp

− (x− Ut)2

a2 + 4Dt

,

where U and D are constants to be determined.

Explain in physical terms the meaning of this solution.


(i) Suppose that, with spherical polar coordinates, the Stokes streamfunction

Ψλ(r, θ) = rλ sin2 θ cos θ

represents a Stokes flow and thus satisfies the equation D2(D2Ψλ) = 0 , where

D2 =∂2

∂r2+

sin θ

r2∂

∂θ

1

sin θ

∂

∂θ.

Show that the possible values of λ are 5, 3, 0 and −2 . For which of these valuesis the corresponding flow irrotational ? Sketch the streamlines of the flow for thecase λ = 3 .

(ii) A spherical drop of liquid of viscosity µ1 , radius a and centre at r = 0 ,is suspended in another liquid of viscosity µ2 which flows with streamfunction

Ψ ∼ Ψ∞(r, θ) = αr3 sin2 θ cos θ

far from the drop. The two liquids are of equal densities, surface tension issufficiently strong to keep the drop spherical, and inertia is negligible. Show that

Ψ =

(Ar5 +Br3) sin2 θ cos θ (r < a) ,(αr3 + C +D/r2) sin2 θ cos θ (r > a)

and obtain four equations determining the constants A , B , C , and D . (You neednot solve these equations.)

[You may assume, with standard notation, that

ur =1

r2 sin θ

∂Ψ

∂θ, uθ = − 1

r sin θ

∂Ψ

∂r, e rθ =

1

2

r∂

∂r

(uθ

r

)+

1

r

∂ur

∂θ

.]


Consider a two-dimensional horizontal vortex sheet of strength U at heighth above a horizontal rigid boundary at y = 0, so that the inviscid fluid velocity is

u =

(U, 0) 0 < y < h(0, 0) y > h

.

Examine the temporal linear instability of the sheet and determine the relevantdispersion relationship.

For what wavelengths is the sheet unstable?

Evaluate the temporal growth rate and the wave propagation speed in thelimit of both short and long waves. Comment briefly on the significance of yourresults.

[TURN OVER


A plate is drawn vertically out of a bath and the resultant liquid drains offthe plate as a thin film. Using lubrication theory, show that the governing equationfor the thickness of the film, h(x, t), is

∂h

∂t+

(gh2

ν

)∂h

∂x= 0 , (∗)

where t is time and x is distance down the plate measured from the top.

Verify that

h(x, t) = F (x− gh2

νt)

satisfies (∗) and identify the function F (x). Using this relationship or otherwise,determine an explicit expression for the thickness of the film assuming that it wasinitially of uniform thickness h0.


A steady two-dimensional jet is generated in an infinite, incompressiblefluid of density ρ and kinematic viscosity ν by a point source of momentum withmomentum flux in the x direction F per unit length located at the origin.

Using boundary layer theory, analyse the motion in the jet and show thatthe x-component of the velocity is given by

u = U(x)f ′(η) ,

where

η = y/δ(x) , δ(x) = (ρν2x2/F )1/3 and U(x) = (F 2/ρ2νx)1/3 .

Show that f satisfies the differential equation

f ′′′ +1

3(ff ′′ + f ′2) = 0 .

Write down the appropriate boundary conditions for the equation. [You

need not solve the equation.]


Consider a uniform stream of inviscid incompressible fluid incident onto atwo-dimensional body (such as a circular cylinder). Sketch the flow in the regionclose to the stagnation point, S, at the front of the body.

Let the fluid now have a small but non-zero viscosity. Using local co-ordinates x along the boundary and y normal to it, with the stagnation pointas origin and y > 0 in the fluid, explain why the local outer, inviscid flow isapproximately of the form

u = (Ex,−Ey)

for some positive constant E.

Use scaling arguments to find the thickness δ of the boundary layer on thebody near S. Hence show that there is a solution of the boundary layer equationsof the form

u(x, y) = Exf ′(η),

where η is a suitable similarity variable and f satisfies

f ′′′ + ff ′′ − f ′2

= −1. (∗)

What are the appropriate boundary conditions for (∗) and why? Explain briefly howyou would obtain a numerical solution to (∗) subject to the appropriate boundaryconditions.

Explain why it is neither possible nor appropriate to perform a similaranalysis near the rear stagnation point of the inviscid flow.


An incompressible fluid with density ρ and viscosity µ is forced by a pressuredifference ∆p through the narrow gap between two parallel circular cylinders ofradius a with axes 2a+ b apart. Explaining any approximations made, show that,provided b≪ a and ρb3∆p≪ µ2a, the volume flux (per unit length of cylinder) is

2b5/2∆p

9πa1/2µ

when the cylinders are stationary.

Show also that when the two cylinders rotate with angular velocities Ω and−Ω respectively, the change in the volume flux is

4

3baΩ.

For the case ∆p = 0, find and sketch the function f(x) = u0(x)/(aΩ), where u0 isthe centreline velocity at position x along the gap in the direction of flow. Commenton the values taken by f .

[TURN OVER


Write an essay on the Kelvin-Helmholtz instability of a vortex sheet. Youressay should include a detailed linearised analysis, a physical interpretation of theinstability, and an informal discussion of nonlinear effects and of the effects ofviscosity.


Consider a unidirectional flow with dynamic viscosity µ along a straightrigid-walled channel of uniform cross-sectional shape D driven by a uniform appliedpressure gradient G. Write down the differential equation and boundary conditionsgoverning the velocity w along the channel.

Consider the situation when the boundary includes a sharp corner of angle2α. Explain why one might expect that, sufficiently close to the corner, the solutionshould be of the form

w = (G/µ)r2f(θ) ,

where r and θ are polar co-ordinates with origin at the vertex and θ = ±α describingthe two planes emanating from the corner. Determine f(θ).

If D is the sector bounded by the lines θ = ±α and the circular arc r = a,show that the flow is given by

w = (G/µ)r2f(θ) +

∞∑

n=0

Anrλn cos λnθ,

where λn and An are to be determined.

[Note that∫

cos(ax) cos(bx) dx = a sin(ax) cos(bx) − b sin(bx) cos(ax)/(a2 − b2).]

Considering the values of λ0 and λ1, comment briefly on the cases: (i)2α < 1

2π; (ii) 12π < 2α < 3

2π; and (iii) 32π < 2α < 2π.


A volume V of very viscous fluid of density ρ and dynamic viscosity µis released at the origin on a rigid horizontal boundary at time t = 0. Usinglubrication theory, determine the velocity profile in the gravity current once it hasspread sufficiently that the axisymmetric thickness h(r, t) of the current is muchless than the radius R(t) of the front.

Derive the differential equation

∂h

∂t=β

r

∂

∂r

(rh3 ∂h

∂r

),

where β is to be determined.

Write down the other equations that are needed to determine the appropriatesimilarity solution for this problem.

Determine the similarity solution and calculate R(t).


Write down the Navier–Stokes equations for an incompressible fluid.

Explain the concepts of the Euler and Prandtl limits applied to the Navier–Stokes equations near a rigid boundary.

A steady two-dimensional flow given by (U, 0) far upstream flows past asemi-infinite flat plate, held at y = 0, x > 0. Derive the boundary layer equation

∂ψ

∂y

∂2ψ

∂x∂y− ∂ψ

∂x

∂2ψ

∂y2= ν

∂3ψ

∂y3

for the stream-function ψ(x, y) near the plate, explaining any approximations made.

Show that the appropriate solution must be of the form

ψ(x, y) = (νUx)1/2f(η),

and determine the dimensionless variable η.

Derive the equation and boundary conditions satisfied by f(η). [You need

not solve them.]

Suppose now that the plate has a finite length L in the direction of the flow.Show that the force F on the plate (per unit width perpendicular to the flow) isgiven by

F =4ρU2L

(UL/ν)1/2

f ′′(0)

[f ′(∞)]2.

[TURN OVER


Consider flow of an incompressible fluid of uniform density ρ and dynamicviscosity µ. Show that the rate of viscous dissipation per unit volume is given by

Φ = 2µeijeij ,

where eij is the strain rate.

Determine expressions for eij and Φ when the flow is irrotational withvelocity potential φ. Hence determine the rate of viscous dissipation, averagedover a wave period 2π/ω, for an irrotational two-dimensional surface wave ofwavenumber k and small amplitude a ≪ k−1 in a fluid of very small viscosityµ≪ ρω/k2 and great depth H ≫ 1/k.

[You may use without derivation that in deep water a linearised wave with surface

displacement η = a cos (kx−wt) has velocity potential φ = −(ωa/k)e−kz sin (kx−ωt).]

Calculate the depth-integrated time-averaged kinetic energy per wavelength.Assuming that the average potential energy is equal to the average kinetic energy,show that the total wave energy decreases to leading order like e−γt, where

γ = 4µk2/ρ .

2007/2 Fluid Dynamics II (MODIFIED)

Viscous fluid is extracted through a small hole in the tip of the conegiven by θ = α in spherical polar coordinates (R, θ, φ). The total volume fluxthrough the hole takes the constant value Q. It is given that there is a steadyaxisymmetric solution of the Navier-Stokes equations for the fluid velocity u. Forsmall enough R, the velocity u is well approximated by u ∼ (−A/R2, 0, 0), whereA = Q/[2π (1 − cosα)] except in thin boundary layers near θ = α.

(i) Verify that the volume flux through the hole is approximately Q.

(ii) Construct a Reynolds number (depending on R) in terms of Q and thekinematic viscosity ν, and thus give an estimate of the value of R below whichsolutions of this type appear.

(iii) Assume that there is a boundary layer near θ = α described by theapproximate equations

1

x

∂

∂x(xu) +

∂v

∂y= 0 , u

∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ ν

∂2u

∂y2,

∂p

∂y= 0

in terms of orthogonal non-cartesian coordinates (x, y, φ) where x is distancealong the cone from the tip and y is distance perpendicular to the cone, withcorresponding velocity components (u(x, y), v(x, y), 0) and (modified) pressure p.Write down the appropriate boundary conditions, and explain why the localboundary-layer thickness has order-of-magnitude

δ(x) =

(νx3

A

)1/2

.

By writing the boundary-layer problem in terms of a (Stokes) streamfunction, orotherwise, show that it admits a similarity solution with

u = − A

x2F ′ (η) , where η =

y

δ(x),

and give the equation and boundary conditions satisfied by F . Obtain anexpression, in terms of F , for the volume flux deficit through the boundary layer,and deduce (or otherwise determine) the R-dependence of the first correction tothe flow outside the boundary layer.

1996/1 Fluid Dynamics II - University of · PDF file1996/1 Fluid Dynamics II Viscous...

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