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MexChemE

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Greetings, PF! I have some questions regarding the problem I attached below. It is some sort of Couette flow variation. It's not homework, I'm just learning the basics of TP on my own. I fully solved the problem with the Navier-Stokes and continuity equations, using some assumptions in order to simplify the models. Here are the solutions

1. [itex]v_x = \left( \frac{U_1 - U_2}{b} \right)y + U_2[/itex]

2. [itex]Q = \int_0^W \int_0^b v_x \ dy \ dz = Wb \left( \left( \frac{U_1 - U_2}{2} \right) + U_2 \right)[/itex], where W is the width of the flow in the z direction

3. [itex]\tau_{yx} = - \mu \frac{d v_x}{dy} = -\mu\left( \frac{U_1 - U_2}{b} \right)[/itex]

Now, what I'm trying to understand is the meaning of the sign of the shear stress, and how it affects its direction. The velocity and stress profiles in the diagram were added by me. According to my understanding so far, and to the math also, if the velocity profile of the fluid is linear, then the shear stress is constant; now, [itex]\left( \frac{U_1 - U_2}{b} \right)[/itex] is always positive, so the shear stress component is always oriented in the negative x direction, even when the velocity of the fluid is oriented in the positive x direction. What does this result actually mean? How is this explained physically?

Thanks in advance for any input!

1. [itex]v_x = \left( \frac{U_1 - U_2}{b} \right)y + U_2[/itex]

2. [itex]Q = \int_0^W \int_0^b v_x \ dy \ dz = Wb \left( \left( \frac{U_1 - U_2}{2} \right) + U_2 \right)[/itex], where W is the width of the flow in the z direction

3. [itex]\tau_{yx} = - \mu \frac{d v_x}{dy} = -\mu\left( \frac{U_1 - U_2}{b} \right)[/itex]

Now, what I'm trying to understand is the meaning of the sign of the shear stress, and how it affects its direction. The velocity and stress profiles in the diagram were added by me. According to my understanding so far, and to the math also, if the velocity profile of the fluid is linear, then the shear stress is constant; now, [itex]\left( \frac{U_1 - U_2}{b} \right)[/itex] is always positive, so the shear stress component is always oriented in the negative x direction, even when the velocity of the fluid is oriented in the positive x direction. What does this result actually mean? How is this explained physically?

Thanks in advance for any input!

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