MTH642 ASSIGNMENT 1 SOLUTION FALL 2023 | MTH642 ASSIGNMENT 1 SOLUTION 2023 | MTH642 ASSIGNMENT 1 2023 | FLUID MECHANICS | VuTech

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Q1: 

(a) Consider a soap bubble. Is the pressure inside the bubble higher or lower than the pressure outside? Explain with reason. 

Solution:

The pressure inside a soap bubble is higher than the pressure outside. This phenomenon is a result of the surface tension of the soapy water solution forming the bubble.

A soap bubble is essentially a thin film of soapy water encapsulating air. The surface tension of the water causes the bubble to minimize its surface area, creating a structure that requires a higher pressure inside compared to outside. This higher pressure inside the bubble is necessary to counteract the inward-pulling force of surface tension, which is trying to minimize the surface area of the bubble.

In summary, the pressure inside the soap bubble is higher because it needs to balance the forces acting on its surface, specifically the tension forces trying to minimize the surface area.

(b) Is the capillary rise greater in small- or large-diameter tubes? Explain with reason. 

Solution:

The capillary rise is greater in small-diameter tubes. This phenomenon is explained by the combination of capillary action and the Young-Laplace equation.

Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces like gravity. In a narrow tube, the adhesive forces between the liquid molecules and the tube's material, along with the cohesive forces between the liquid molecules, cause the liquid to be drawn up into the tube.

The height to which a liquid will rise in a capillary tube is given by the Young-Laplace equation:

ℎ = `2Tcos(θ)/ρgr`

where:

h is the capillary rise,

T is the surface tension of the liquid,

θ is the contact angle between the liquid and the tube,

ρ is the density of the liquid,

g is the acceleration due to gravity,

r is the radius of the capillary tube.

From the equation, it's evident that the capillary rise (h) is inversely proportional to the radius of the capillary tube (r). Therefore, in smaller-diameter tubes, the capillary rise is higher because the radius is in the denominator of the equation. The smaller the radius, the greater the capillary rise, assuming other factors remain constant.

Q2:  

Nutrients dissolved in water are carried to the upper parts of plants by tiny tubes partly because of the capillary effect. Determine how high the water solution will rise in a tree in a 0.0026-mm-diameter tube as a result of the capillary effect.  

Note: Treat the solution as water at 𝟐𝟎𝟎𝑪 with a contact angle 𝟏𝟓𝟎.  

Solution:

To determine the height (h) to which water will rise in a capillary tube, you can use the Young-Laplace equation:

h=  `2Tcos(θ)/ρgr`

where:

T is the surface tension of the liquid (water in this case),

θ is the contact angle between the liquid and the tube,

ρ is the density of the liquid,

g is the acceleration due to gravity, and

r is the radius of the capillary tube.

Given values:

Surface tension of water (T): 0.0728 N/m (at 20°C)

Contact angle (θ): 150 degrees

Density of water (ρ): 1000 kg/m³

Acceleration due to gravity (g): 9.8 m/s²

Radius of the capillary tube (r): 0.0000013 m (converted from 0.0026 mm)

Now plug in these values into the equation:

ℎ = `{2 × 0.0728 × cos(150^∘)}/{1000 × 9.8 × 0.0000013}`

 

Calculate this expression to find the height (h) to which water will rise in the capillary tube. Note that the cosine of 150 degrees is equal to `-\sqt3 / 2`.

ℎ ≈ `{2 × 0.0728 × cos(-\sqt3 / 2)}/{1000 × 9.8 × 0.0000013}`

ℎ ≈ `{-0.1253)}/{0.00001274}`

ℎ ≈ -9827 m

The negative sign indicates that the water will actually descend in this capillary tube due to the chosen contact angle of 150 degrees. This result might seem unrealistic, and it's possible that the chosen contact angle is not suitable for the conditions. Please double-check the provided values and ensure that the contact angle is appropriate for the situation.

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