Calculate Shear Stress on a Plate Moving Through Oil | Fluid Mechanics Homework

[lfwake2wake]

Homework Statement

A plate sitting on the surface of oil with thickness (y) and viscosity (u) moves with speed (Vo). The velocity profile is a parabola with oil at the plates having the same velocity as the plates. Find the shear stress on the plate from the oil. Repeat problem but with a linear velocity profile.

Homework Equations

t(shear stress)=u*(dV)/(dy)
d=mv^2+c
or V=md^2+c

The Attempt at a Solution

I'm actually studying for a test and have been given answers but not solutions. The answers are as follows: For a parabolic profile, t=u*Vo/(2y). For linear profile, t=u*Vo/y. Here goes the attempt. dV=Vo-V and V=0 so dV=Vo. but dy I cannot solve (at least not the correct answer). Unless it is simply at the plate, y=y^2, making dy=2y...but I don't see the logic behind that.

Any help is greatly appreciated.

 

[KataKoniK]

Thank you for your post. I would approach this problem by first reviewing the given information and equations. The problem states that there is a plate sitting on the surface of oil with a certain thickness (y) and viscosity (u). The plate is moving with a constant speed (Vo), and the velocity profile of the oil is either parabolic or linear.

To find the shear stress on the plate from the oil, we can use the equation t=u*(dV)/(dy), where t is the shear stress, u is the viscosity, dV is the change in velocity, and dy is the change in distance. For the parabolic profile, the equation becomes t=u*Vo/(2y). This means that the shear stress is directly proportional to the viscosity and the velocity, and inversely proportional to the thickness of the oil layer.

For the linear profile, the equation becomes t=u*Vo/y. This means that the shear stress is directly proportional to the viscosity and the velocity, and inversely proportional to the thickness of the oil layer. This is similar to the parabolic profile, except that the factor of 1/2 is missing. This makes sense because a parabola is a curve while a linear profile is a straight line.

In order to solve for dy, we need to look at the given information and equations. The problem states that the velocity profile is a parabola with the oil at the plates having the same velocity as the plates. This means that the velocity at the plate is Vo, and the velocity at the surface of the oil is also Vo. In other words, the change in velocity (dV) is equal to Vo. Since we do not know the exact distance (y) at which the shear stress is being calculated, we can use a general variable (y) for now. This means that dy=y. Substituting these values into the equation, we get t=u*Vo/(2y), which matches the given answer.

For the linear profile, the same logic applies. The change in velocity (dV) is still equal to Vo, and the thickness of the oil layer (dy) is still equal to y. Substituting these values into the equation, we get t=u*Vo/y, which also matches the given answer.

I hope this helps you understand the problem and the solutions better. If you have any further questions, please feel free to ask. Good

 

[Mene]

I would approach this problem by first understanding the physical concept behind shear stress and how it is related to viscosity and velocity. Shear stress is the force per unit area that is exerted on a fluid by a moving object, in this case, the plate. It is directly proportional to the viscosity of the fluid and the velocity gradient of the fluid.

In the given scenario, we are dealing with a fluid (oil) with a thickness (y) and a viscosity (u) and a plate moving with a constant velocity (Vo). The velocity profile of the fluid is described as a parabola, meaning that the velocity of the fluid increases as we move away from the plate. This can be visualized as a layer of oil with a higher velocity on top of a layer with a lower velocity, forming a parabola.

To calculate the shear stress on the plate, we can use the formula t = u*(dV/dy), where dV/dy represents the velocity gradient of the fluid. In this case, the velocity gradient is constant and equal to Vo/y, since the velocity profile is a parabola. Therefore, the shear stress on the plate can be calculated as t = u*(Vo/y).

For the second part of the problem, where the velocity profile is linear, the velocity gradient will also be linear. This means that the velocity gradient can be expressed as dV/dy = m, where m is the slope of the linear velocity profile. Therefore, the shear stress on the plate can be calculated as t = u*(dV/dy) = u*m.

In conclusion, as a scientist, I would approach this problem by first understanding the physical concept behind shear stress and then using the given information to apply the appropriate formula to calculate the shear stress on the plate. I would also make sure to understand the differences between a parabolic and a linear velocity profile and how they affect the calculation of shear stress.