1 Fluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity components are 2 cos, sin, 0, where M is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the stream lines. 2. (Example1, p.79, Chorlton) Test whether the motion specified by, is a possible motion for an incompressible fluid. If so, determine the equations of the streamlines. Also, test whether the motion is of the potential kind and if so determine the velocity potential. 3. (Example2, p.81, Chorlton) For an incompressible fluid,,, 0 ( = constant). Discuss the nature of the flow. Find also the vortex lines. 4. (Example3, p.82, Chorlton) For a fluid moving in a fine tube of variable section A, prove from first principles that the equation of continuity is where v is the speed at a point P of the fluid and s the length of the tube up to P. What does this become for steady incompressible flow? 5. (Example4, p.83, Chorlton) Liquid flows through a pipe whose surface is the surface of revolution of the curve about the x-axis If the liquid enters at the end of the pipe with velocity V, show that the time taken by a liquid particle to traverse the entire length of the pipe from to is.
2 6. (Ex.1, p.90, Chorlton) Show that a fluid of constant density can have a velocity given by Find the vorticity vector.,,. 7. (Ex.2, p.90, Chorlton) Show that if, where m is a constant, the equation of continuity for an incompressible fluid is satisfied at all points other than the origin O. Further show that, if S is any closed surface not passing through O, the net volume of fluid flowing out of S per unit time is 4 if S encloses O and zero if O is outside S. 8. (Ex.3, p.90, Chorlton) If. 0 for all closed curves in a region R show that there exists a scalar function V such that. Show that the kinetic energy of uniform incompressible fluid moving irrotationally in the finite region between surfaces, is given by., where is a unit vector directed from the surface normally into the fluid and is the velocity potential. Hence or otherwise, show that, if, are at rest, the fluid cannot move irrotationally. 9. (Ex.4, p.90, Chorlton) Derive the equation of continuity, in Cartesians, for the flow of a compressible fluid. If the velocity is constant in magnitude everywhere in the irrotational, twodimensional flow of an incompressible fluid, prove that the flow must be uniform in direction. Prove also the converse of this statement. 10. (Ex.1, p.73, Kundu and Cohen) A two-dimensional steady flow has velocity components. Show that the stream lines are rectangular hyperbolas.
3 11. (Ex.6, p.74, Kundu and Cohen) The velocity components in an unsteady plane flow are given by and. Describe the path lines and the stream lines. 12. (Ex.13, p.75, Kundu and Cohen) Using the vector identity show that the acceleration of a fluid particle is given by, where q is the magnitude of velocity and is the vorticity. 13. (Example,p97, Chorltan) AB is a tube of small uniform bore forming a quadrantal arc of a circle of radius a and centre O, OA being horizontal and OB vertical with B below O. The tube is full of liquid of density, the end B being closed. If B is suddenly opened, show that initially, where is the velocity, and that the pressure at a point whose angular distance from A is immediately drops to above atmospheric pressure. Prove further that when the liquid remaining in the tube subtends an angle at the centre,. 14. (Example3,p102, Chorltan) A long pipe is of length l and has slowly tapering cross-section. It is inclined at angle to the horizontal and water flows steadily through it from the upper to the lower end. The section at the upper end has twice the radius of the lower end. At the lower end, the water is delivered at atmospheric pressure. If the pressure at the upper end is twice atmospheric find the velocity of delivery. 15. (Example,p126, Chorltan) Define the vorticity in the motion of a continuous medium with velocity,,,. Show that for a motion of an inviscid incompressible fluid of uniform density, under gravity, the vorticity satisfies the equation..,
4 A motion symmetric about the axis 0, is described in terms of cylindrical polar coordinates,,, the velocity having components,, 0,,. By evaluating the term., or otherwise show that if the fluid element has vorticity when at radius, its vorticity at radius r is given by. 16. (Ex.3, p.130, Chorlton) The particle velocity for a fluid motion referred to rectangular Cartesian axes,, with unit vectors,, is given by, where is a constant. Show that this is a possible motion of an incompressible fluid motion under no body forces in an infinite fixed rigid tube, 0 2 and that the pressure p is given by + constant. 17. (Ex.5, p.131, Chorlton) Prove that the equation of motion of a homogeneous inviscid liquid moving under forces arising from a potential V may be written in the form = Where is the vorticity. If the velocity referred to cylindrical polar coordinates,, is given by 0,, 0 0, 0,, 0, where w is a constant, prove that vorticity is given by =0,0,, 0, =0,0,0,.
5 Determine the corresponding pressure distribution if the motion takes place under no external forces, and find the smallest value of the pressure at infinity which will ensure that the pressure is everywhere positive. 18. (Ex.6, p.131, Chorlton) A tube AB of uniform fine bore is in the form of the arc of the cycloid 4, the ends A and B being given by 0,, respectively. It is fixed in a vertical plane so that A is its lowest point and the tangent there is horizontal. The tube, which is closed at A and open at B, is full of uniform liquid. If the end A is opened, show that the tube empties in time. 19. (Ex.10, p.132, Chorlton) The space between two spheres is filled with incompressible fluid. The spheres have radii, and move with constant speeds U,V respectively along the line of centres. Show that the instant when the spheres are concentric, the velocity potential is given by And that the equation of the streamlines takes the form,. where,, are spherical polar coordinates withorigin at the common centre of the spheres and axis 0 along their line of motion. 20. (Ex.11, p.132, Chorlton) Infinite inviscid fluid of constant density is attracted towards a fixed point O by a force per unit mass, where r is the distance of any point from O. Initially the liquid is at rest, and there is a cavity bounded by a sphere of radius a. If there is no pressure at infinity or in the cavity, prove that the radius R of the cavity at time t is such that 2 0. If, where k is a constant, and the cavity is filled after time T, show that 25 4.
6 19.(Ex.12, p.133, Chorlton) A spherical globule of gas of radius a and at pressure P expands in an infinite mass of liquid of density in which the pressure at infinity is zero. The gas is initially at rest and its pressure and volume are governed by the equation radius in time. Prove that the gas doubles its (Ex.13, p.133, Chorlton) A large mass of incompressible non-viscous fluid contains a spherical air bubble, the air inside the bubble obeying Boyle s law,. At a great distance from the bubble the pressure is zero. Neglecting the inertia of the system, show that the radius R of the bubble at time t satisfies an equation of the form where k is a constant., 21. (Ex.14, p.133, Chorlton) An infinite mass of ideal incompressible fluid is subjected to a force per unit mass directed towards the origin. If initially the fluid is at rest and there is a cavity in the form of the sphere in it, show that the cavity will be completely filled after an interval of time (Ex.17, p.132, Chorlton)The space between a uniform rigid sphere, of radius a and density and a fixed concentric spherical envelope of radius b is filled with inviscid liquid of constant density. An impulse is applied to the rigid sphere and the system is set in motion from rest. Show that, just after the impulse, the kinetic energy of the liquid and sphere is, where U is the velocity of the sphere. Hence or otherwise, determine the impulse.
7 23. (Ex.18, p.132, Chorlton)The space between two concentric spherical shells of radii, is filled with liquid of density. If the shells are set in motion, the inner one with velocity U in the x-direction and the outer with velocity V in the y-direction, show that the initial motion of the liquid is given by the velocity potential where, the coordinates being rectangular. Evaluete the velocity at any point of the liquid and hence, or atherwise, prove that the total momentum communicated has components,, (Ex.6, p.210, Chorlton) A two-dimensional motion of a liquid has the complex potential, where the constants U,k,a are real and positive. Show that (i) The velocity at infinity is U in the negative sense of the real axis; (ii) The circle is a streamline; (iii) There are, in general, two stagnation points; (iv) There is a circulation 2 about the circle; (v) Find the points on where the velocity is maximum and minimum. 25. (Ex.26, p.215, Chorlton) Find the resultant force on a fixed circular boundary C of radius a with centre at the origin if the fluid motion is caused entirely by a vortex of strength 2 at 2 0 and there is no circulation round C.
8 26. (Ex.27, p.215, Chorlton) Show that the force exerted on a circular cylinder in the irrotational flow produced by a line source m at 3 is 48, (Ex.25, p.215, Chorlton) Verify that is the complex potential of a steady flow of liquid about a circular cylinder, the plane being a rigid boundary. Find the force exerted by the liquid on unit length of the cylinder. 28. (Example1,p162, Chorlton) Uniform flow past a fixed infinite circular cylinder 29. (Example,p169, Chorlton) Discuss the flow for which. 30. (Example1,p175, Chorlton) Find the equations of the streamlines due to uniform line sources of strength m through the points, 0,, 0 and a uniform line sink of strength 2m through the origin. 31. (Example1,p183, Chorlton) Uniform flow past a stationary cylinder. 32. (Example2,p183, Chorlton) Uniform stream at incidence to OX. 33. (Example3,p184, Chorlton) Image of a line source in a circular cylinder. 34. (Example4,p184, Chorlton) Image of a line doublet parallel to the axis of a right circular cylinder. 35. (Example5,p185, Chorlton) A vortex of circulation 2 is at rest at the point 1, in the presence of a plane circular boundary, around which there is a circulation 2. Show that 1. 1 Show that there are two stagnation points on the circular boundary,symmetrically placid about the real axis in the quadrants nearest to the vortex, given by and prove that is real.
9 36. (Example,p188, Chorlton) A source and a sink of equal strength are placed at the points, 0 with in a fixed circular boundary. Show that the streamlines are given by (Example1,p191, Chorlton) A long infinite cylinder of radius a is placed in a uniform stream having velocity in the negative direction of the x-axis and in addition a circulation round the cylinder 2 is produced. Find the complex potential due to this flow and show that the cylinder experiences uplift force 2 and moment M=0. Determine velocity of the fluid at every point of the cylinder. 38. (Example2,p191, Chorlton) Verify that is the complex potential of a steady flow of liquid about a circular cylinder, the plane 0 being a rigid boundary. Find the force axerted by the liquid on unit length of the cylinder. 39. (Example,p145, Chorlton) A three-dimensional doublet of strength whose axis is in the direction is distant a from the rigid plane 0 Which is the sole boundary of liquid of density, infinite in extent. Find the pressure at a point on the boundary distant r from the doublet given that the pressure at infinity is. Show that the pressure on the plane is least at a distance 5 2 from the doublet. 40. (Example1,p149, Chorlton) Image of a source in a solid sphere. 41. (Example2,p149, Chorlton) Image of a doublet in a sphere when the axis of the doublet passes through the centre of the sphere. 42. (Ex.4, p.158, Chorlton) Three-dimensional doublets of strengths and are situated in an infinite expance of liquid of constant density at the points having Cartesian coordinates, 0,0 and, 0,0 respectively. Their axes are directed towards and away from the origin respectively. Find the velocity potential and hence show that no fluid crosses the sphere centre (0,0,0) and radius a.
10 43. (Ex.9, p.159, Chorlton) The three -dimensional motion of a liquid is symmetrical with respect to the axis 0 of spherical polar coordinates, and the velocity components are, in directions associated with the coordinates,. Obtain the equation of continuity and deduce that u and v can be expressed in terms of the stream function,. Show that, if the motion is irrotational, then satisfies the equation Show that if., the motion is irrotational. 44. (Ex.8, p.159, Chorlton) A three dimensional doublet points in the direction of flow of a uniform strem in which it is situated. Show that one of the equipotential surfaces is a sphere and show that the stagnation points lie in this sphere. Taking the centre of the doublet as the origin and its axis as the line 0, show that the equations of the stream lines in spherical polar coordinates are of the form 2, where a is the radius of the sphere. 45. (Example1,p111, Chorltan) Stationary Sphere in a Uniform Stream A solid sphere of radius a is fixed in a uniform stream of incompressible fluid which is flowing with constant speed U in the negative direction of x- axis (in the absence of body forces). Show that (i) Velocity potential at any point,, is given by,, (ii) Stagnation points are given by, 0 or. (iii) The pressure at any point,, on the surface of a sphere is given by 1, where is pressure at infinity and is density of fluid. At what point minimum pressure occurs and to which velocity it corresponds.
11 (iv) Show that the total thrust on the hemisphere, 0 is given By. (v) Obtain stream lines of this flow. 46. (Example2,p115, Chorltan) Sphere moving with constant velocity in liquid which is otherwise at rest (at ). A solid sphere of radius a is moving with uniform velocity U in an incompressible fluid of infinite extent which is at rest at. Show that (i) velocity potential at,,, is given by, /2 (ii) Kinetic energy of the fluid is given by, where is the mass of the fluid displaced. Find total K.E. of the sphere and the fluid. What is virtual mass of the sphere. 47. (Example3,p116, Chorltan) Accelerating sphere moving in a fluid at rest at infinity A sphere of centre O and radius a moves through an infinite liquid of constant density at rest at infinity, O describing a straight line with velocity. If there are no body forces, show that the pressure p at points on the surface of the sphere in a plane perpendicular to the straight line at a distance x from O measured positively in the direction of v is given by, where is the pressure at infinity. Deduce that the thrust on the sphere is, where is the mass of the liquid having the volume of the sphere. 48. (Example,p121, Chorltan) A stationary infinite right circular solid cylinder of radius a is placed in a uniform stream of incompressible fluid which is flowing in the negative direction of x-axis( in the absence of body forces), its axis being perpendicular to the direction of flow. Determine (i) Velocity potential at any point,,
12 (ii) Stagnation points (iii) and for what value of U, occurs. (iv) The thrust on unit half cylinder. (v) Stream lines. 49. (Ex.18, p.134, Chorlton) The space between two concentric spherical shells of radii a,b (a<b) is filled with liquid of density. If the shells are set in motion, the inner one with velocity U in the x-direction and the outer with velocity V in the y-direction, show that the initial motion of the liquid is given by the velocity potential where,the coordinates being rectangular. Evaluate the velocity at any point of the liquid and hence, or otherwise prove that the total momentum communicated has components,, (Ex.13,p.128, Kundu and Cohen)Prove that the velocity field given by 0, can have only two possible values of the circulation. They are (a) Γ 0 for any path not enclosing the origin, and (b) Γ k for any path enclosing the origin. 51. (Ex.11,p.75, Kundu and Cohen) A flow field on the xy-plane has the velocity components 3, 2 3. Show that the circulation around the circle is 4.
13 52. (Ex.1, p.349, Chorlton) Show that th Navier-Stokes equation for steady, viscous incompressible flow under conservative body forces may be developed in the form Ω, where, the vorticity vector. Show further that for two dimensional flow in the (x,y) plane, on taking the curl of both sides of this equation,,, where is the stream function such that. 53. (Ex.8, p.159, Chorlton) A three dimensional doublet points in the direction of flow of a uniform stream in which it is situated. Show that one of the equipotential surfaces is a sphere and show that the stagnation points lie in this sphere. Taking the centre of the doublet as the origin and its axis as the line 0, show that the equations of the stream lines in spherical polar coordinates are of the form 2, where a is the radius of the sphere. 54. (Ex.9, p.159, Chorlton) Show that, if the motion is irrotational, then satisfies the equation Show that if., the motion is irrotational.