F11AE1 1. C = ρν r r. r u z r

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1 F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions at r = a, for all z, i.e. on the inside surface of the cylinder. Using the incompressible Navier Stokes equations in cylindrical polar coordinates see the formulae sheet at the back of the exam paper), look for a stationary solution to the fluid flow in the pipe of the following form. Assume there is no radial flow, u r = 0, and no swirl, u θ = 0. Further assume there is a constant pressure gradient down the pipe, i.e. that p = Cz for some constant C, and there is no body force. Lastly, suppose that the flow down the pipe, i.e. the velocity component u z, has the form u z = u z r) it is a function of r only). a) Using the Navier Stokes equations, show that 1 C = ρν r r b) Integrate the equation above to show that r u z r )). u z = C 4ρν r2 + A log r + B, where A and B are constants. We naturally require that the solution be bounded. Explain why this implies A = 0. Now use the no-slip boundary condition to determine B. Hence show that u z = C 4ρν a2 r 2 ). c) Explain why the mass-flow rate across any cross section of the pipe is given by known as the fourth power law) ρu z ds and then evaluate this quantity for the case above.

2 F11AE1 2 a z Figure 1: Poiseuille flow: a viscous fluid flows along an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis. A constant pressure gradient is assumed, as well as axisymmetry, no radial flow and no swirl.

3 F11AE1 3 Question 2 20 Marks) Consider the problem of an ideal, steady, incompressible flow in a channel over a gently undulating bed see Figure 2. Assume that the flow is shallow and uniform in crosssection. Upstream the flow is characterized by flow velocity U and depth H. The flow then impinges on a gently undulating bed of height y = yx) as shown in Figure 2, where x measures distance downstream. The depth of the flow is given by h = hx). The fluid velocity at that point is u = ux) and independent of the depth throughout. Also assume that the fluid depth h = hx) varies slowly as a function of position x. a) Explain why for all x we must have uh = UH. b) By applying Bernoulli s theorem to the surface streamline, for which the pressure is constant and equal to atmospheric pressure P 0, show that for all x: 1 2 U2 + gh = 1 2 u2 + gy + h). c) Using part a), show that the relation in part b) is equivalent to y = U2 2g + H h UH)2 2gh 2. For this relation, sketch y as a function of h, and show that this function has a unique global maximum at where Here F is known as the Froude number. y 0 := H F2 3 2 F2/3), h 0 = HF 2/3 F := U/ gh. d) Assume that the actual given undulation y = yx) attains an actual maximum value y max < y 0. Using part a) show that ) dy 1 dh = u2, gh and hence explain why the sign of 1 u 2 /gh is the same throughout the flow and determined by whether the Froude number F < 1 or F > 1. In each case, assuming the undulation y = yx) is a simple bump with a single maximum y max, sketch how you expect the depth of the fluid h = hx) to vary as a function of x.

4 F11AE1 4 P 0 hx) u H U yx) x Figure 2: Channel flow.

5 F11AE1 5 Question 3 20 Marks) Consider a uniform incompressible ideal flow of velocity U, into which we place a spherical obstacle, radius a. The set up is shown in Figure 3. Assume that the flow around the sphere is steady, incompressible and irrotational. Suppose we use spherical polar coordinates r,θ,ϕ) to represent the flow with the south-north pole axis passing through the centre of the sphere and aligned with the uniform flow U at infinity. Assume further that the flow is axisymmetric, i.e. independent of the azimuthal angle ϕ, and there is no swirl so that u ϕ = 0. a) Explain why there exists a potential function φ satisfying 2 φ = 0 for this flow. Show that this is equivalent to 1 r 2φ ) + 1 sin θ φ )) = 0. r 2 r r sin θ θ θ b) The general solution to Laplace s equation in part a) is well known and it is given by φr,θ) = A n r n + B ) n P r n+1 n cos θ), n=0 where P n are the Legendre polynomials, with P 1 x) = x. The coefficients A n and B n are constants. State the uniform flow conditions as r and the boundary conditions at r = a. Hence show that the potential for this flow around the sphere is φ = Ur + a 3 /2r 2 ) cos θ. c) Show that the velocity field u = φ has components u r,u θ ) = U1 a 3 /r 3 ) cos θ, U1 + a 3 /2r 3 ) sin θ ). d) Now use Bernoulli s theorem to explain why 1 2 u 2 + P/ρ = 1 2 U2 + P /ρ, where P and u are the pressure and velocity fields, respectively, at any point r,θ,ϕ) with r a. Then using part c) show that on the sphere r = a the relation P P ρ = 1 2 U sin2 θ ), holds. Hence explain why the fluid exerts no net force on the sphere known as D Alembert s paradox).

6 F11AE1 6 U U r θ U U Figure 3: D Alembert s paradox: there is no net force on a solid sphere obstacle placed in a uniform flow for an ideal fluid.

7 F11AE1 7 Question 4 20 Marks) An incompressible homogeneous fluid occupies the region between two horizontal rigid parallel planes, which are a distance h apart, and outside a rigid cylinder of diameter a) which intersects the planes normally; see Figure 4 for the set up. Suppose that for this question x and y are horizontal Cartesian coordinates and z is the vertical coordinate. Let u,v,w) be the fluid velocity components in the three coordinate directions x, y and z, respectively. Assume throughout that a typical horizontal velocity scale for u,v) is U and a typical vertical velocity scale for w is W. a) Explain very briefly why a is a typical horizontal scale for x,y) and h a typical vertical scale for z. b) Hereafter further assume that h a and further that ρuh 2 aµ where µ = ρν is the first coefficient of viscosity. Using these assumptions, show that the Navier Stokes equations for an incompressible homogeneous fluid are well approximated by µ 2 u z 2 = p x, v µ2 z = p 2 y and p z = 0 for a steady flow, where p = px,y,z) is the pressure. c) We define the vertically averaged velocity components u,v) = ux,y),vx,y) ) for the flow in part b) by ux,y),vx,y) ) := 1 h h 0 ux,y),vx,y) ) dz. Use the incompressibility of u, v, w) to show that the vertically averaged velocity field u,v) is incompressible. Use that p/z = 0 to show that u,v) is irrotational.

8 F11AE1 8 fluid a cylinder h cylinder Side view Above view fluid Figure 4: Hele Shaw cell: an incompressible homogeneous fluid occupies the region between two parallel planes a distance h apart) and outside the cylinder of diameter a normal to the planes).

9 F11AE1 9 Question 5 20 Marks) Hill s spherical vortex is an exact solution of Euler s equation of motion for an incompressible fluid. The vorticity is confined to the interior of a uniformly translating sphere, of radius a, translating with velocity U. Outside the sphere the flow is irrotational. If we use spherical polar coordinates r, θ, ϕ) centred at the middle of the sphere, then without loss of generality, we can assume that the sphere is translating along the south to north pole axis. Further, we can assume the flow is axisymmetric with no swirl. a) Use the properties of axisymmetry and no swirl to show that in spherical polar coordinates the only non-zero component of vorticity ω = ω r,ω θ,ω ϕ ) inside the sphere is ω ϕ. b) Explain why there is a stream function ψ = ψr,θ,t) Stokes stream function) which is a solution of the differential equations u r = 1 ψ r 2 sin θ θ and u θ = 1 ψ r sin θ r in this case, where u r and u θ are the radial and latitudinal components of velocity. Show that the azimuthal vorticity ω ϕ is given in terms of Stokes stream function by ω ϕ = 1 2 ψ r sin θ r + sin θ 2 r 2 1 θ sin θ )) ψ. θ c) Use the two identities given in part IV) of the formulae sheet at the end of the exam paper, to prove that the incompressible Euler equations of motion imply that the vorticity field ω = u satisfies ω t + u ω = ω u. d) Now assume that the flow is steady so that ω/t = 0. Using the assumption of axisymmetry, no swirl, part a) above, and that in these circumstances ω ϕ ω u = 0, 0, ur sin θ + u θ cos θ )), r sin θ which you may assume without proof), show that the equations for the vorticity in part c) above reduce to the equation ω ϕ u r r + u θ ω ϕ r θ = ω ϕ ur sin θ + u θ cos θ ). r sin θ in spherical polar coordinates. Use this last result to prove that ) D ωϕ = 0. Dt r sin θ Question 5 continues overleaf...)

10 F11AE1 10 Question 5 continued e) You may take as given that the flow outside the sphere is given by an irrotational flow around a solid sphere refer to Question 3 earlier in this exam paper) and described by the stream function ψ = ψ out where ψ = 1 2 U r 2 a 3 /r ) sin 2 θ for r > a. If ψ = ψ in is the stream function in the interior of the sphere, explain why at the boundary the sphere surface r = a) we require that ψ in ψ in a,θ) = ψ out a,θ) and r = ψ out r=a r. r=a Hence deduce that the boundary conditions for ψ = ψ in required are ψ in ψ in a,θ) = 0 and r = 3Ua 2 sin2 θ. r=a f) To finish, let us combine parts b), d) and e) above. From part d) we know that ω ϕ = Ar sin θ for some constant A. Thus inside the sphere Stokes stream function ψ = ψ in must satisfy the boundary value problem 2 ψ r + sin θ 2 r 2 θ 1 sin θ ψ θ ) = Ar 2 sin 2 θ, with the boundary conditions given at the end of part e) above. Show that ψ = A 10 r2 C 2 r 2 ) sin 2 θ is a general solution to the partial differential equation for ψ inside the sphere, where C is some constant. Use the boundary conditions to deduce specific values for C and A in terms of the given physical parameters.

11 F11AE1 11 Formulae I) The Navier Stokes equations for an incompressible homogeneous fluid are u t + u u = ν u 1 p + f, ρ u = 0, where u = ux,t) is the fluid velocity at position x and time t, ρ is the uniform constant density, p = px,t) is the pressure, ν is the constant kinematic viscosity and f is the body force per unit mass. II) The incompressible Navier Stokes equations in cylindrical polar coordinates r,θ,z) with the velocity field u = u r,u θ,u z ) are u r t + u )u r u2 θ r = 1 p ρ r + ν u r u r r 2 ) u θ + f 2 r 2 r, θ u θ t + u )u θ + u ru θ = 1 p r ρr θ + ν u θ + 2 u r r 2 θ u ) θ + f r 2 θ, u z t + u )u z = 1 p ρ z + ν u z + f z, where p = pr,θ,z,t) is the pressure, ρ is the mass density and f = f r,f θ,f z ) is the body force per unit mass. Here we also have and u = u r r + u θ r = 1 r θ + u z z r ) + 1r 2 r r 2 θ z 2 Further the gradient of a scalar function Φ and the divergence of a vector field u are given in cylindrical coordinates, respectively, by Φ Φ = r, 1 Φ r θ, Φ ) z and u = 1 r r ru r) + 1 u θ r θ + u z z. Lastly in cylindrical coordinates u is given by ω r u = ω θ = ω z 1 r 1 u z u θ r θ z u r uz z r r ru θ) 1 r III) The incompressible Navier Stokes equations in spherical polar coordinates r, θ, ϕ) with the velocity field u = u r,u θ,u ϕ ) are note θ is the angle to the south-north u r θ.

12 F11AE1 12 pole axis and ϕ is the azimuthal angle) u r t + u )u r u2 θ r u2 ϕ r = 1 p ρ r + ν u r 2 u r r 2 2 r 2 sin θ u θ t + u )u θ + u ru θ u2 ϕ cos θ r r sin θ = 1 ρr + ν u θ + 2 u r r 2 θ u ϕ t + u )u ϕ + u ru ϕ r + ν u ϕ + 2 r 2 sin θ θ u θ sin θ) 2 r 2 sin θ p θ u θ r 2 sin 2 θ 2 cos θ r 2 sin 2 θ + u θu ϕ cosθ = 1 r sin θ u r ϕ + 2 cos θ r 2 sin 2 θ u θ ϕ ) u ϕ + f r, ϕ ) u ϕ + f θ, ϕ p ρr sin θ ϕ u ) ϕ r 2 sin 2 + f z, θ where p = pr,θ,ϕ,t) is the pressure, ρ is the mass density and f = f r,f θ,f ϕ ) is the body force per unit mass. Here we also have and u = u r r + u θ r = 1 r 2 ) + 1 r 2 r r r 2 sin θ θ + u ϕ r sin θ ϕ sin θ ) + θ θ 1 r 2 sin 2 θ 2 ϕ 2. Further the gradient of a scalar function Φ and the divergence of a vector field u are given in spherical coordinates, respectively, by Φ Φ = r, 1 ) Φ r θ, 1 Φ r sin θ ϕ and u = 1 r 2 r r2 u r ) + 1 r sin θ θ sin θu θ) + 1 u ϕ r sin θ ϕ. Lastly in spherical coordinates u is given by ω r u = ω θ = ω ϕ 1 r sin θ 1 r sin θ 1 r sin θu θ ϕ) u θ ϕ u r 1 ru ϕ r r ϕ). r ru θ) 1 r IV) The following two identities hold for any two vectors u and v: and 1 2 u 2) = u )u + u u) u v) = u v) v u) + v )u u )v. u r θ

F1.9AB2 1. r 2 θ2 + sin 2 α. and. p θ = mr 2 θ. p2 θ. (d) In light of the information in part (c) above, we can express the Hamiltonian in the form

F1.9AB2 1. r 2 θ2 + sin 2 α. and. p θ = mr 2 θ. p2 θ. (d) In light of the information in part (c) above, we can express the Hamiltonian in the form F1.9AB2 1 Question 1 (20 Marks) A cone of semi-angle α has its axis vertical and vertex downwards, as in Figure 1 (overleaf). A point mass m slides without friction on the inside of the cone under the