fluids final conceptual questions

Question 1

λ_1 (=10^-8m.) and λ_2 (=10m.) represent the mean free paths of gas molecules for two scenarios. Which of these scenarios will violate the continuum hypothesis, and why?

Answer 1

The scenario with 𝜆_2=10 m is likely to violate the continuum hypothesis because the mean free path is very large, leading to a high Knudsen number. This indicates that the gas cannot be approximated as a continuous medium, and molecular effects must be explicitly considered.

Question 2

There is a square fluid element experiencing various normal and shear forces. Find the magnitude of the net force in the x and y directions.

Answer 2

F_x = add up all the forces acting in the x-direction
F_y = add up all the forces acting in the y-direction

Question 3

At any point within a fluid, if the absolute pressure is 5 atm, what is the gauge pressure when the ambient pressure is 1 atm?

Answer 3

P_gage = P_abs - P_amb
5 - 1 = 4 atm

Question 4

You drop a lightweight ball, denoted as B, into the bucket shown below, and the ball starts to float. Assume that the ball is fully immersed in the water. Write an expression to equate the buoyancy force with the gravitational force acting on the ball. The ball has a mass m and volume V. The fluid density is ρ.

Answer 4

Fv = mg
ρvg = mg
ρv = m

Question 5

Write an expression for integral form of conservation of mass.

Answer 5

∂/∂t (∫_cv ρ∂V) + ∫_cs ρ(v*dA) = 0

Question 6

Apply the mass conservation equation from question 5 to the control volume (CV) below. Assuming steady, incompressible flow with uniform velocity at the boundaries, and no flow from other boundaries except at inlet 1 and outlet 2. The fluid density is ρ.

Answer 6

ρA_1 * V_1 = ρ A_2 * V_2
A_1 * V_1 = A_2 * V_2

Question 7

Write an expression for the integral form of the conservation of x-momentum.

Answer 7

F_Sx+F_Bx = ∂/∂t (∫_cv ρudV) + ∫_cs ρu(v*dA)

Question 8

Write an expression for the integral form of the conservation of y-momentum.

Answer 8

F_Sy+F_By = ∂/∂t (∫_cv ρvdV) + ∫_cs ρv(v *dA)

Question 9

Write down the mass conservation equation in differential equations form for an incompressible flow field.

Answer 9

∂u/∂x+∂v/∂y+∂w/∂z=0

Question 10

Write down the Navier-Stokes equation along the x-axis.

Answer 10

ρ(∂u/∂t+u∂u/∂x+v∂u/∂y)=-∂ρ/∂x+μ(∂^2/∂x^2+∂^2u/∂y^2)+pf_x

Question 11

Write down the Navier-Stokes equation along the y-axis.

Answer 11

ρ(∂v/∂t+u∂v/∂x+v∂v/∂y)=-∂ρ/∂y+μ(∂^2v/∂x^2+∂^v/∂y^2)+pf_y

Question 12

Deduce Euler’s equation (for the x-axis) by simplifying the equation mentioned in Question
2. Mention the assumptions you make.

Answer 12

Assume velocity (μ) = 0
ρ(∂u/∂t+u∂v/∂x+v∂v/∂y)=-∂ρ/∂x

Question 13

Write an expression for the stagnation pressure.

Answer 13

P_o=P+(1/2)ρv^2

Question 14

The velocity field is represented by V = U y/h i. Assume U to be constant. Calculate the rate of rotation (ω).

Answer 14

check work

Question 15

For an incompressible and irrotational flow field, if the given lines represent the stream function field (ψ), then draw the lines for the velocity potential field(ϕ).

Answer 15

make the lines perpendicular to given lines and ϕ go on the top horizontal and ψ goes on the vertical left side

Question 16

Find the shear stress distribution for a hypothetical flow field (u = y^2 + y) on a flat plate.
What is the value of wall-shear stress? The wall shear stress represents the value of shear stress right at the wall.

Answer 16

u=y^2+y
∂u/∂y=2y+1
τxy = μ ∂u/∂y= μ(2y+1)
any force at wall put y=0
τxy = μ

Question 17

Check if the following flow field is incompressible. u=2x^2+y^2-x^2y;v=x^3+x(y^2-4y)

Answer 17

find partial of u with respect to x, then find partial of v with respect to y and then add u and y together

Question 18

Check if this is irrotational: u=2x^2+y^2-x^2y;v=x^3+x(y^2-4y)

Answer 18

w_z=∂v/∂x-∂u/∂y and then if w_z is equal to zero then the flow field is irrotational

Question 19

Continuum hypothesis

Answer 19

Treat fluid as a continuous medium, rather than a collection of distinct particles

Question 20

Mean free path

Answer 20

The average distance between molecules

Question 21

Continuum hypothesis is only valid when it is a

Answer 21

short mean free path.

Question 22

The chances of fluid particles collision is … in a shortened mean free path. When there are … chances of collision, we can treat the fluid as continuous.

Answer 22

higher, higher

Question 23

Gauge pressure

Answer 23

A measurement of pressure relative to the surrounding atmospheric pressure.

Question 24

Ambient/Atmospheric pressure

Answer 24

The pressure exerted by the weight of the atmosphere, ~ 101,325 Pa, 14.6959 psi, 1 atm

Question 25

Equation for P_abs

Answer 25

P_abs=P_atm+P_gauge

Question 26

Buoyancy force

Answer 26

A force experienced by a body that is fully or partially immersed in a fluid and is equal to the weight of the displaced fluid that would occupy the same immersed volume.

Question 27

Equation for buoyancy force

Answer 27

F_buoyancy=(rho)gV

Question 28

Gravitational force

Answer 28

Force of gravity acting on an object

Question 29

Equation for gravitational force

Answer 29

F_g=mg

Question 30

Stagnation pressure

Answer 30

The pressure measured at a point in a fluid flow where the fluid comes to rest.

Question 31

Equation for stagnation pressure

Answer 31

P_O=P+(1/2)(rho)v^2

Question 32

rate of rotation equation

Answer 32

w=1/2(⛛ x v)
(x = cross product)