For the flow field ?⃗ = ? + Ω × ? , where ?⃗ and Ω...
For the flow field ?⃗ = ? + Ω × ? , where ?⃗ and Ω are constant
linear- and angular velocity vectors, use index notation to find
its strain rate tensor ??? and rotation tensor ???.
Solutions
Expert Solution
We have,
Now,
Thus,
Using relation for strain rate tensor and rotation tensor as
shown above
Consider an incompressible flow in cylindrical coordinates with
velocity field v = Cr^(a) θ^ where C is a constant (a) Show that
the flow satisfies the Navier-Stokes equations for only two values
of a. (b) Given that p(r; θ; z) = p(r) only, and neglecting
gravity, find the pressure distribution for each case, assuming
that the pressure at r = R is p0
Prove or disprove each of the
followings.
If f(n) = ω(g(n)), then
log2(f(n)) =
ω(log2g(n)), where
f(n) and g(n) are positive
functions.
ω(n) + ω(n2) =
theta(n).
f(n)g(n) =
ω(f(n)), where f(n) and
g(n) are positive functions.
If f(n) = theta(g(n)), then
f(n) = theta(20 g(n)), where
f(n) and g(n) are positive
functions.
If there are only finite number of points for which
f(n) > g(n), then
f(n) = O(g(n)), where
f(n) and g(n) are positive
functions.
The angular acceleration of a shaft is given by α = -0.20ω
rad/s2 , where ω represents the angular velocity of the shaft. If
at t = 0 the angular velocity ω of the shaft is 18 rad/s, determine
(a) the number of revolutions the shaft will execute before coming
to rest, (b) the time required for the shaft to come to rest, (c)
the time required for the angular velocity of the shaft to be
reduced to 1 rad/s
Assume that R1 = 44 Ω , R2 = 75 Ω ,
R3 = 19 Ω , R4 = 79 Ω , R5 = 20 Ω , and
R6 = 23 Ω .
fig 1
fig 2
fig 3
fig 4
Part A
Find the equivalent resistance of the combination shown in the
figure (Figure 1) .
Express your answer using two significant figures.
Req =______ Ω
Part B
Find the equivalent resistance of the combination shown in the
figure...
Evaluate ??? ?.???? where ?? ? is an Electric field which is
given as ?? ? = ?3????3??2???? ??? + 6??3??2?????? ??? ?
8??2??2??2???? ??? , for a surface bound by a cuboidal shape having
dimensions, ?2 ? ?? ? 0; 1 ? ?? ? 3; 0? ?? ? 4
Three resistors having resistances of 1.10 Ω , 2.40 Ω , and 5.00
Ω are connected in parallel to a 28.0 V battery that has negligible
internal resistance.
a) Find the equivalent resistance of the combination.
b) Find the current in each resistor.
c) Find the total current through the battery.
d) Find the voltage across each resistor.
e) Find the power dissipated in each resistor.
f) Which resistor dissipates the most power, the one with the
greatest resistance or...
1) Distinguish between blood flow rate and blood flow velocity.
When an expert in the field uses the term blood flow, does that
term usually mean rate or velocity?
2) Explain why a heart can keep beating after it has been
removed from a living body.
1. If an ideal solution of the flow field through a nozzle is
shown with a depiction of vectors distributed at points throughout
the nozzle that illustrate the magnitude and direction of steady
flow through those points, this depiction would be Eulerian or
Lagrangian (circle one)
2. If an ideal solution of the flow field through a nozzle is
shown with a depiction of vectors attached to particles moving
through the nozzle, changing magnitude and direction as they move
through...
The velocity field of a permanent, incompressible and isothermal flow is given by
u = a (x^2 - y^2) (1)
v = −2axy (2)
w = 0 (3)
Gravitational acceleration acts on the z axis only.
a) Write the Navier-Stokes equations for this flow, canceling the appropriate terms
and replacing the non-zero derivatives with their detailed expression. Be sure to indicate
clearly your assumptions. (4 points)
b) From the result obtained in (a), demonstrate that the pressure field is continuous...
The stream function ?? in a two-dimensional flow field is given
as ?? = 4?? − 3?? + 7???? (a) Prove that this flow field is
irrotational and that it satisfies the continuity equation. (b) Find the potential flow function Ф(??, ??) for this flow
field with boundary condition Ф = 0 at x = 2, y = 1.