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ma3102soln2-11

# ma3102soln2-11 - MATH3102 — Assignment 2 2011 Asterisked...

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Unformatted text preview: MATH3102 — Assignment 2, 2011 Asterisked questions form this assignment. Total marks: 25 Due Date: Hand in during tutorial on 8 September, 07‘ deposit in the MATH3102 assignment collection box on Level 4 of Building 67 before that date 1. The velocity ﬁeld in a 2—dimensional unsteady ﬂow is given by q(a:, y, t) = yeti + 2xe'tj. Show that (a) the streamline at t = O which passes through the point (1, 2) is (one sheet of) the hyperbola y2 — 232:2 = 2. (b) the pathline for the ﬂuid particle which is at the point (1, 2) at t = 0 is the parabola y2 = 4m. ' Sketch the two curves, indicating direction of motion on each, and note that they have that same slope. where they meet at (1, 2). 2. Show that (I) = K (x — t) (y —t) represents the velocity potential of an incompressible two-dimensional ﬂow. Show that the streamlines at time t are the curves (as — t)2 — (y — t)2 = constant, and that the paths of the ﬂuid particles have the equations . K -1nlw-yl=3{(x+y)— m y} + B, where A, B are consts. 3. Compressible inviscid fluid ﬂows along a very narrow tube of cross—sectional area {3(5) where s is the distance measured along the tube. Show that the equation; of continuity is ‘ 8p (9 . _-_ __ = 0 [3’ at + a8(pﬂv) . Where p(s, t) and 11(5, t) are the density and the speed of the ﬂuid. Show that under the steady conditions the density at any. section is inversely pro- portional to the volume ﬂux at the section. ' ‘ 4. * (6 marks.) A two—dimensional point source of homogeneous, incompressible, irro— tational ﬂuid is located at the point (a, b) with a, 6 positive constants. The source has a constant strength m, and the ﬂuid density is p. There is a solid wall along y = 0. Use the method of images to ﬁnd the velocity potential and hence velocity field q = ui + 'uj in the region y 2 0. Verify that o = 0 on the wall and show that it is the only stagnation point. The pressure P at inﬁnity is Poo; ﬁnd the maximal AND minimal values of P on the wall. 5. A point source of homogeneous, incompressible, irrotational ﬂuid is situated at the point with Cartesian coordinates (a, b, c), where a, b and c are positive. A solid wall occupies the region :1: g 0. The source has a constant strength m, the ﬂuid has density p, and there are no external forces acting. (a) Use the method of image to ﬁnd the velocity potential and hence the velocity ﬁeld q = ui + vj + wk in the region :r 2 O. (b) Show that the pressure on the wall is a minimum on the circle (y—b)2+(z—c)2 = éa2, and that it has there the value p m2 54m2 a4 where PS is the pressure on the wall at the stagnation point. 6. * (7 marks.) P=P5— (a) Find the velocity potential (say gbl) of a uniform velocity ﬁeld q = U i (U const) such that q = ngl. - (b) Consider a2D doublet of strength ,u at O with doublet axis along 03:, obtained from the limiting conﬁguration of a source—sink pair as described in lectures. Given that the velocity potential of this 2D doublet is (ﬁg = —/.b cos 6/27rr. Show that qb = ¢i+¢2 is the velocity potential for “uniform ﬂow past a cylinder” with axis along Oz and radius 0., provided [.11 = —27rUa2. (c) Find the pressure distribution on the cylinder. 7. Two inﬁnitely long concentric circular cylinders have their common axis along the z— axis. The inner and outer cylinders have radii r0 and r1 respectively. A non—viscous, incompressible ﬂuid ﬂows between the cylinders with velocity ﬁeld q = q(7‘, (9)89, where (r, 6, z) are cylindrical polar coordinates. Show that if the ﬂow is irrotational, then q = %, where a is constant. Find a corresponding velocity potential (I) and stream function W, noting that q /= curl(l\I/k). Working from Euler’s equation of motion, show that if there are no external forces acting, and the ﬂuid is homogeneous, then the ﬂuid. pressure at the outer cylinder is greater than that at the inner cylinder by an amount 1 2 1 1 —a ———. 2p 7% if (a) Show that the result div(D) =1 p implies f f S n - D d8 is the total charge within the closed surface S, where D is the electric displacement and p the charge density. 8. * (6 marks.) (b) The density of electric charge in a spherical region of radius 1) is given by p = b — 7" where 7" is the distance from the center of the sphere. There is no charge outside the sphere. Find the electric ﬁeld E and the electric potential gb at points inside and outside the sphere. 9. (a) Show that the electric potential for a dipole in two dimensions is . _ pcos0 9b 27mm" Show the equipotentials are circles. (962 + y2 = Am.) 2 (b) A conducting cylinder of inﬁnite length and radius a is placed with its axis normal to a uniform electric ﬁeld of strength E. Use the result in part (a) (or any other method) to ﬁnd the potential at any point outside the cylinder. 10. (a) A point source emits Q units of heat at the origin 0 at t = O in the plane Ozzy. For circular symmetry the heat diffusion equation has the form 8T 82T 18T a—k<w+;w>’ t>0 The solution is known to have the form A 2 T(7",t) = — exp (—1—) . Find A by considering the total amount of heat in the plane at any instant t > 0. (b) A point source of pollutant in three dimensions constantly emits Q units of mass per unit time in a uniform fluid velocity ﬁeld q = U k, where U is a const. Write the approximate form of the steady convection—diffusion equation that is appropriate for the downstream wake region. By referring to part (a) ﬁnd the concentration C(r, z), where (r, 0, z) are cylindrical polar coordinates. ll. * (6 marks.) 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