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When we consider force balance on a static fluid element which is cut along some arbitrary plane as shown, we start by saying that the fluid is static and hence there is no shear stress along the plane. And from there we define the pressure as acting normal to the plane, and go on to derive the hydrostatic condition of pressures at a point. Fluid Element in Hydrostatic condition

My question is, where does the assumption that the shear stress is zero come from? Is it purely due to observation that a fluid flows continuously under any applied shear, or is there a fundamental basis?

Suppose, say we do not initially let the shear forces vanish along the plane of the element. If we take a non-zero value of the shear force and do a force balance on the element, we do not reach the hydrostatic condition since the shear values are not zero now. How can we proceed now to prove that the shear stress is zero?

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  • $\begingroup$ Your question is a bit unclear. A diagram would help. $\endgroup$
    – Lelouch
    Jul 6, 2016 at 10:57

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One definition of a fluid is a substance that continually deforms under shear stress. So if we find a fluid that is static, i.e. not continually deforming, then it cannot have any shear stress by definition.

Note that the condition "continually deforms under shear stress" depends on the time period of observation. For example, on a timescale of hundreds of years, glass is a fluid that deforms continuously under shear stress. On a timescale of seconds, glass resists shear stress.

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  • $\begingroup$ When we consider force balance on a static fluid element which is cut along some arbitrary plane, we start by saying that the fluid is static and hence there is no shear stress along that plane. And from there we define the pressure on the face of the plane as acting normal to the plane, and go on to derive the hydrostatic condition of pressures at a point. My question is, where does the assumption that the shear stress is zero come from? Is it purely due to observation that a fluid flows continuously under any applied shear, or is there a fundamental basis? $\endgroup$
    – Haridev V
    Jul 6, 2016 at 11:49
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    $\begingroup$ By empirical observation, some materials can support shear stress in their static state, whereas others cannot. We have chosen the term 'fluid' to describe those materials which cannot. $\endgroup$ Jul 6, 2016 at 11:54
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    $\begingroup$ @SergeiPatiakin I think your comment above is the correct answer! $\endgroup$
    – Diracology
    Jul 6, 2016 at 12:03
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    $\begingroup$ The fact that the fluid element is not deforming will conclude whether the shear force is zero or not. If the shear stress is T then T = M du/dy. For a static fluid, du/dy is zero and hence T is zero. $\endgroup$
    – Lelouch
    Jul 6, 2016 at 12:11
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    $\begingroup$ If there was a shear stress, the fluid would flow, would it not? And if it did not flow, then we would no longer call it a fluid, right? It seems definitional to me. Unless I'm missing something, it seems similar to asking for a proof why solids can sustain non-zero shear stresses. On the other hand, are you asking for an atomic model of a substance which guarantees fluid-like qualities, which therefore guarantee any shear stress would be eliminated through fluid-like accomodation (ie, molecules moving until shear forces are eliminated through equilibrium)? $\endgroup$ Jul 30, 2016 at 8:19

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