NS_PracticeProblems

Practice Problems on the Navier-Stokes Equations ns_02

A viscous, incompressible, Newtonian liquid flows in steady, laminar, planar flow down a vertical wall. The thickness, , of the liquid film remains constant. Since the liquid free surface is exposed 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. 2. 3.

Determine the velocity distribution for this gravity driven flow. Clearly state all assumptions and boundary conditions. Determine the shear stress acting on the wall by the fluid. Determine the maximum velocity of the fluid. y g

x 

wall

liquid

air

Answer(s): ---

C. Wassgren, Purdue University

Page 1 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations 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, U1 and U2. The pressure gradient in the x direction is zero and the only body force is due to gravity which acts in the y-direction. 1. Derive an expression for the velocity distribution between the plates assuming laminar flow. 2. Determine the volumetric flow rate of the fluid between the plates. 3. Determine the magnitude of the shear stress the fluid exerts on the upper plate (at y=b) and clearly indicate its direction on a sketch. U1

b

fluid with density, , and dynamic viscosity, 

g

y x U2

Answer(s): ---

C. Wassgren, Purdue University

Page 2 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_05

In cylindrical coordinates, the momentum equations for an inviscid fluid (Euler’s equations) become:  Du u 2  p   r        fr Dt r r   1 p  Du u ur      f r  r   Dt Du p  z     fz Dt z where ur, u, and uz are the velocities in the r,  and z directions, p is the pressure,  is the fluid density, and fr, f, and fz are the body force components. The Lagrangian derivative is: D   u     ur   uz Dt t r r  z A cylinder is rotated at a constant angular velocity denoted by . The cylinder contains a compressible fluid which rotates with the cylinder so that the fluid velocity at any point is u=r (ur=uz=0). If the density of the fluid, , is related to the pressure, p, by the polytropic relation: p  A k where A and k are known constants, find the pressure distribution p(r) assuming that the pressure, p0, at the center (r=0) is known. Neglect all body forces.



Answer(s):  p  r    p0k  k2k1 A k  2 r 2    k 1

1

k k 1

C. Wassgren, Purdue University

Page 3 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_08

Two immiscible viscous liquids are introduced into a Couette flow device so that they form two layers of equal height as shown:

liquid A (A=)

H/2

y

x

liquid B (B=4)

H/2 stationary wall

The dynamic viscosity, , of liquid A is one quarter that of liquid B. The upper plate is moved at a constant velocity, U, while the bottom plate remains stationary. a. b.

Determine the velocity of the interface between the two liquids. Determine the “apparent viscosity” of the mixture as seen by an experimenter who believes that only one liquid is in the device.

Answer(s): Vi  15 U

 *  85 

C. Wassgren, Purdue University

Page 4 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_09

Consider the fully-developed, steady, laminar circular pipe flow of an incompressible, non-Newtonian fluid due to a constant pressure gradient dp/dz < 0. Gravitational effects may 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 velocity gradient by: 2  du   rz  C   z   dr  where C is a known constant. Find: 1. the velocity profile, uz(r), and 2. the friction factor, f, (i.e. the wall shear stress made dimensionless using the dynamic pressure based on the average velocity in the pipe) for this pipe flow in terms of C,  (the fluid density), dp/dz, r, and R (the radius of the pipe), or a subset of these parameters.

Answer(s): 2  1 dp  uz    3  2C dz  49 C f  2  R2

1

2

R

3

2

r

3

2



C. Wassgren, Purdue University

Page 5 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_13

An incompressible fluid flows between two porous, parallel flat plates as shown: porous plate flow direction

h

y

V

x

porous plate

An identical fluid is injected at a constant speed V through the bottom plate and simultaneously extracted from the upper plate at the same velocity. Assume the flow to be steady, fully-developed, the pressure gradient in the xdirection is a constant, and neglect body forces. Determine appropriate expressions for the x and y velocity components.

Answer(s): uy  V    Vy    1  exp     h  p        y ux    V  x     Vh   h  1  exp             

C. Wassgren, Purdue University

Page 6 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_14

Determine the torque (in lbf-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 2 in. and the shaft is turning at 200 rpm. The viscosity of the lubricant is 200 cp and the fluid density is 50 lbm/ft3.

2.00” 0.002”

Answer(s): 

u  r 

R



1

r R

2   r R

 R r   R 2  r



C. Wassgren, Purdue University

r R

  1  Rr 

Page 7 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_19

Consider two concentric cylinders with a Newtonian liquid of constant density, , and constant dynamic viscosity, , contained between them. The outer pipe, with radius, Ro, is fixed while the inner pipe, with radius, Ri, and mass per unit length, m, falls under the action of gravity at a constant speed. There is no pressure gradient within flow and no swirl velocity component. Determine the vertical speed, V, of the inner cylinder as a function of the following (subset of) parameters: g, Ro, Ri, m, , and .

Ro Ri

annular region filled with liquid of density, , and dynamic viscosity, 

g movable inner cylinder with radius, Ri, and mass per unit length, m

fixed outer cylinder with radius, Ro

V

Answer(s):  R   g mg   g 2 V   Ri ln i  Ri  Ri  Ro2  R R     2 2 4 o  i  



C. Wassgren, Purdue University



Page 8 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_20

A wide flat belt moves vertically upward at constant speed, U, through a large bath of viscous liquid as shown in the figure. The belt carries with it a layer of liquid of constant thickness, h. The motion is steady and fully-developed after a small distance above the liquid surface level. The external pressure is atmospheric (constant) everywhere. U gravity h belt

atmosphere

liquid a. b. c. d.

Simplify the governing equations to a form applicable for this particular problem. State the appropriate boundary conditions Determine the velocity profile in the liquid. Determine the volumetric flow rate per unit depth.

Answer(s): 1 g 2 gh uy  x  x U 2  Q  

gh3  Uh 3

C. Wassgren, Purdue University

Page 9 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_24

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 fully developed region is reached where fluid shear balances gravity and the film thickness remains constant. Assuming incompressible laminar flow and negligible shear interaction with the atmosphere, find an expression for uz(r) and a relation for the volumetric flow rate Q.

film

film r

atmosphere

a

gravity

b

z

Answer(s):  g 2 2  gb 2  r  ln   uz   r a  4 2 a



Q  



  g  3b 4  4a 2b 2  a 4  b 4 ln b   a 2  4 

C. Wassgren, Purdue University

 

Page 10 of 13

Last Updated: 2010 Oct 13

Practice Problems on the Navier-Stokes Equations ns_27

Consider steady flow at horizontal velocity U (at y → ∞) past an infinitely long and wide plate. The plate is porous and there is uniform flow normal to the surface at a constant velocity, V. Assume there are no pressure gradients and that gravity is negligible.

horizontal velocity is U as y → ∞ incompressible, constant viscosity Newtonian fluid

y x V

a. b. c. d.

Determine the y-velocity at all points in the flow field. Determine the x-velocity at all points in the flow field. What restriction is there on the velocity V? Quantify how far into the flow the wall effects are felt. Clearly indicate what criterion you are using.

Answer(s): uy  V   V  u x  U 1  exp  y       V