Fluid mechanics help mechanical engineering

[apamnani1]

Fluid mechanics help! mechanical engineering

Hi all,

I've answered a page of questions from a engineering textbook by mike tooley but I'm really struggling on these two questions.

1) A Casson plastic fluid is subjected to a shear force of 5.5 Pa. If the fluid obeys a law of the form T =1.5 +0.2^0.45u, determine the shear rate and apparent viscosity under these conditions

2) Oil of dynamic viscosity 0.12 Nsm-2 is used to lubricate the space between the plain journal bearing with the shaft. The shaft and bearing are concentric and the space between them is filled with an oil of dynamic viscosity of 0.15Nsm^-2.

Determine the torque required to overcome viscous resistance and the power loss when the shaft is rotating at a speed of 3000rpm.


Any help would be great.

Thanks!

 

[KataKoniK]

Hello there,

As a scientist with expertise in fluid mechanics, I can assist you with these questions. Let's start with the first one:

1) To solve this problem, we can use the Casson plastic fluid model, which relates the shear stress (T) to the shear rate (u) and apparent viscosity (μ) through the equation T = μu^0.45. We are given T = 5.5 Pa and the law of the form T = 1.5 + 0.2^0.45u. Substituting these values into the equation, we get:

5.5 = 1.5 + 0.2^0.45u
4 = 0.2^0.45u
u = (4/0.2^0.45) = 76.72 s^-1

To find the apparent viscosity, we can rearrange the Casson plastic fluid model equation to μ = T/u^0.45. Substituting the values we have, we get:

μ = 5.5/76.72^0.45 = 0.109 Pa s

Therefore, the shear rate is 76.72 s^-1 and the apparent viscosity is 0.109 Pa s for this Casson plastic fluid under the given conditions.

Moving on to the second question:

2) To determine the torque required to overcome viscous resistance, we can use the formula τ = μA(dv/dx), where τ is the shear stress, μ is the dynamic viscosity, A is the area of contact, and dv/dx is the velocity gradient. In this case, A and dv/dx are constant since the shaft and bearing are concentric, so we can simplify the equation to τ = μv, where v is the velocity of the shaft.

We are given that the dynamic viscosity of the oil is 0.12 Nsm^-2 and the shaft is rotating at 3000 rpm. To convert rpm to radians per second, we multiply by 2π/60. So, the velocity of the shaft is:

v = (3000*2π/60) = 314.16 rad/s

Substituting this value and the dynamic viscosity into the equation, we get:

τ = (0.12)(314.16) = 37.7 Nm

This is the torque required to overcome viscous resistance. To find the