Practice Problems on the Navier-Stokes Equations

Transcription

1 ns_0 A viscous, incompressible, Newtonian liquid flows in stead, laminar, planar flow down a vertical wall. The thickness,, of the liquid film remains constant. Since the liquid free surface is eposed to atmospheric pressure, there is no pressure gradient in the liquid film. Furthermore, the air provides a negligible resistance to the motion of the fluid. 1. Determine the velocit distribution for this gravit driven flow. Clearl state all assumptions and boundar conditions.. Determine the shear stress acting on the wall b the fluid. 3. Determine the maimum velocit of the fluid. g wall liquid air --- C. Wassgren, Purdue Universit Page 1 of 13 Last Updated: 010 Oct 13

2 ns_03 An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in the figure. The two plates move in opposite directions with constant velocities, U 1 and U. The pressure gradient in the direction is zero and the onl bod force is due to gravit which acts in the -direction. 1. Derive an epression for the velocit distribution between the plates assuming laminar flow.. Determine the volumetric flow rate of the fluid between the plates. 3. Determine the magnitude of the shear stress the fluid eerts on the upper plate (at =b) and clearl indicate its direction on a sketch. U 1 b fluid with densit,, and dnamic viscosit, g U --- C. Wassgren, Purdue Universit Page of 13 Last Updated: 010 Oct 13

3 ns_05 In clindrical coordinates, the momentum equations for an inviscid fluid (Euler s equations) become: Dur u p fr Dt r r Du uur 1 p f Dt r r Duz p fz Dt z where u r, u, and u z are the velocities in the r, and z directions, p is the pressure, is the fluid densit, and f r, f, and f z are the bod force components. The Lagrangian derivative is: D u ur uz Dt t r r z A clinder is rotated at a constant angular velocit denoted b. The clinder contains a compressible fluid which rotates with the clinder so that the fluid velocit at an point is u =r (u r =u z =0). If the densit of the fluid,, is related to the pressure, p, b the poltropic relation: k p A where A and k are known constants, find the pressure distribution p(r) assuming that the pressure, p 0, at the center (r=0) is known. Neglect all bod forces. k k1 1 k1 k k 1 k 0 k p r p A r C. Wassgren, Purdue Universit Page 3 of 13 Last Updated: 010 Oct 13

4 ns_08 Two immiscible viscous liquids are introduced into a Couette flow device so that the form two laers of equal height as shown: H/ H/ liquid A ( A =) liquid B ( B =4) stationar wall The dnamic viscosit,, of liquid A is one quarter that of liquid B. The upper plate is moved at a constant velocit, U, while the bottom plate remains stationar. a. Determine the velocit of the interface between the two liquids. b. Determine the apparent viscosit of the miture as seen b an eperimenter who believes that onl one liquid is in the device. 1 V U i * C. Wassgren, Purdue Universit Page 4 of 13 Last Updated: 010 Oct 13

5 ns_09 Consider the full-developed, stead, laminar circular pipe flow of an incompressible, non-newtonian fluid due to a constant pressure gradient dp/dz < 0. Gravitational effects ma be neglected. The normal stress in this fluid in the z-direction, i.e. zz, is equal to p where p is the pressure. The shear stress, rz, is related to the velocit gradient b: duz rz C dr where C is a known constant. Find: 1. the velocit profile, u z (r), and. the friction factor, f, (i.e. the wall shear stress made dimensionless using the dnamic pressure based on the average velocit in the pipe) for this pipe flow in terms of C, (the fluid densit), dp/dz, r, and R (the radius of the pipe), or a subset of these parameters dp uz R r 3Cdz f 49 C R C. Wassgren, Purdue Universit Page 5 of 13 Last Updated: 010 Oct 13

6 ns_13 An incompressible fluid flows between two porous, parallel flat plates as shown: porous plate h porous plate V flow direction An identical fluid is injected at a constant speed V through the bottom plate and simultaneousl etracted from the upper plate at the same velocit. Assume the flow to be stead, full-developed, the pressure gradient in the - direction is a constant, and neglect bod forces. Determine appropriate epressions for the and velocit components. u V u V 1 ep h p V Vh h 1 ep C. Wassgren, Purdue Universit Page 6 of 13 Last Updated: 010 Oct 13

7 ns_14 Determine the torque (in lb f -ft) and power consumption (in hp) required to turn the shaft in the friction bearing shown in the figure. The length of the bearing into the page is in. and the shaft is turning at 00 rpm. The viscosit of the lubricant is 00 cp and the fluid densit is 50 lb m /ft u r 1 R r r r 1 r R R R r r R R R C. Wassgren, Purdue Universit Page 7 of 13 Last Updated: 010 Oct 13

8 ns_19 Consider two concentric clinders with a Newtonian liquid of constant densit,, and constant dnamic viscosit,, contained between them. The outer pipe, with radius, R o, is fied while the inner pipe, with radius, R i, and mass per unit length, m, falls under the action of gravit at a constant speed. There is no pressure gradient within flow and no swirl velocit component. Determine the vertical speed, V, of the inner clinder as a function of the following (subset of) parameters: g, R o, R i, m,, and. R o R i annular region filled with liquid of densit,, and dnamic viscosit, g fied outer clinder with radius, R o movable inner clinder with radius, R i, and mass per unit length, m V R g mg g V R R R R i i ln i i o Ro Ri 4 C. Wassgren, Purdue Universit Page 8 of 13 Last Updated: 010 Oct 13

9 ns_0 A wide flat belt moves verticall upward at constant speed, U, through a large bath of viscous liquid as shown in the figure. The belt carries with it a laer of liquid of constant thickness, h. The motion is stead and full-developed after a small distance above the liquid surface level. The eternal pressure is atmospheric (constant) everwhere. U gravit belt h atmosphere liquid a. Simplif the governing equations to a form applicable for this particular problem. b. State the appropriate boundar conditions c. Determine the velocit profile in the liquid. d. Determine the volumetric flow rate per unit depth. 1 g gh u U gh Q Uh 3 3 C. Wassgren, Purdue Universit Page 9 of 13 Last Updated: 010 Oct 13

10 ns_4 Consider a film of Newtonian liquid draining at volume flow rate Q down the outside of a vertical rod of radius, a, as shown in the figure. Some distance down the rod, a full developed region is reached where fluid shear balances gravit and the film thickness remains constant. Assuming incompressible laminar flow and negligible shear interaction with the atmosphere, find an epression for u z (r) and a relation for the volumetric flow rate Q. film film r atmosphere gravit a b z g gb r uz r a ln 4 a Q b 4 g b 4 a b a 4 b ln a C. Wassgren, Purdue Universit Page 10 of 13 Last Updated: 010 Oct 13

11 ns_7 Consider stead flow at horizontal velocit U (at ) past an infinitel long and wide plate. The plate is porous and there is uniform flow normal to the surface at a constant velocit, V. Assume there are no pressure gradients and that gravit is negligible. horizontal velocit is U as incompressible, constant viscosit Newtonian fluid V a. Determine the -velocit at all points in the flow field. b. Determine the -velocit at all points in the flow field. c. What restriction is there on the velocit V? d. Quantif how far into the flow the wall effects are felt. Clearl indicate what criterion ou are using. u V V u U 1ep V < 0 ln 0.01 V C. Wassgren, Purdue Universit Page 11 of 13 Last Updated: 010 Oct 13

12 ns_30 An incompressible, Newtonian liquid of densit and dnamic viscosit is sheared between concentric clinders as shown in the sketch below. The inner clinder radius is R i and the outer clinder radius is R o. R i R o a. Determine the velocit profile for the liquid in the gap assuming that the inner clinder rotates with constant angular speed,. Do not assume that (R o R i ) << R o. b. Determine the torque (per unit depth into the page) acting on the outer wall of the clinder. RR i o R0 r u r r Ro R i r R 0 RR i o T 4 Ro Ri C. Wassgren, Purdue Universit Page 1 of 13 Last Updated: 010 Oct 13

13 ns_3 A viscous, incompressible fluid flows between the two infinite, vertical, parallel plates shown in the figure. Determine, b use of the Navier-Stokes equations, an epression for the pressure gradient in the direction of flow. Epress our answer in terms of the mean velocit. Assume that the flow is laminar, stead, and full developed. g h u 1 8 d h 1 1 dp g h dp 1u g d h C. Wassgren, Purdue Universit Page 13 of 13 Last Updated: 010 Oct 13