Is there one or two forces acting on a plate in a fluid? In my Fluids Dynamics course, we have the following two examples


What I don't understand is: Why is it that in the first example we only calculated one shear stress force while in the second example we have two shear stress forces?
 A: From the comments, it seems like the confusion might arise from the difference between forces and stresses; the difference between a stress state at a point, a stress existing at a surface, and the corresponding force applied to that surface; and the number of forces that produce a stress state and how to count and distinguish them. Is that roughly the case? If so, it might be helpful to review some points to check where the confusion arises.

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*A stress can broadly be thought of as a force per unit area.


*If the force points toward the area, the stress is a normal stress; otherwise, it is a shear stress.


*Thus, for a surface (the surface of a horizontal plate, for example), we can have three stresses, composing a general stress state: a normal stress from a force pointing up or down, a shear stress from a force pointing in and out of the page, and a shear stress from a force pointing left or right. (For this problem, only the last is relevant.)


*For this surface, there's a one-to-one relationship between the stress and a single force. If we know that the surface of a plate is subjected to a shear stress (from fluid flow, for example), then it's subjected to a single parallel—or traction—force, and vice versa.


*Points in a region are different. We can imagine a point as the infinitesimal limit of a very small 3D cube. At equilibrium, we need equal opposing forces on that cube, or it would start to accelerate. As a simple example, any upward force must be paired with a downward force to produce a normal—or axial—stress state. One (normal) stress state, but two forces that together impart that stress.


*For a shear stress state at a point, we need fully four forces to prevent translational and rotational acceleration of the infinitesimal cube. One (shear) stress state, four forces:




*In the two questions you show above, the first is asking about a stress state at a point, and the second is asking about the total force on a plate. That's why we obtain a single answer in the first case even though pairs of forces are acting at that point, as you note. In the second case, there are spatially distinct shear stresses acting at the top and bottom of the plate. Each shear stress is associated with balanced pairs of forces as discussed in point 6, but only one force for each is applied to the plate surface, as discussed in point 3. We take these two forces and add them to obtain the total associated load on the plate.


*Note some fundamental differences between the two cases: In the first case, the single shear stress (if decomposed into forces acting on a fluid element) consists of forces pointing in opposite directions as shown in the animation above. Nothing is pulling the water. In the second case, the forces on the plate are pointing in the same direction. One force comes from a shear stress on the top, and one comes from a different shear stress on the bottom. (Only in the symmetric case are they equal.) These two forces are balanced by a third force pulling the plate. Finally, the first problem asks for a shear stress magnitude, whereas the second problem asks for a net drag force.


*Moving forward, it would be good to avoid terms such as "shear stress force" to avoid confusion. A stress state can exist at a point; an associated force can be applied to that point. It's best not to conflate the two, or you might conclude that a shear stress = two "shear stress forces" = two shear stresses that should be added together. This would be incorrect.


*As an aside, note that the plate is not under simple shear. It is under a complex stress state (that varies across the plate) arising from the two traction forces on the top and bottom and the axial load pulling it forward. Analysis of the stress state in the plate is beyond the scope of these questions.
Do these points make sense?
A: Examples of sheer stress on a surface with flow on only one side:

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*Fluid flowing inside a pipe (and if you want a flat surface then make the pipe rectangular). The surface is the wall of the pipe.


*Force on the bed of a river owing to the water flow (and if you want a flat bed then make it a man-made canal with a flat bottom). The surface is the river bed.


*etc.
In these examples, if the fluid is flowing to the right then the sheer stress on the boundary owing to viscosity of the fluid is in the direction to the right. Assuming the boundary (e.g. the pipe or river bed) is not moving then there is another force acting on it, directed to the left, from whatever other objects it is connected to. In this example the other force would typically be stress forces in the solid materials which are ultimately connected to the ground of planet Earth.
