Fluid Statics: Defining a Fluid at Rest

The concept of a fluid being "at rest" is fundamental in fluid mechanics and is referred to as fluid statics. A fluid is a substance that continually deforms under an applied shear stress. However, when a fluid is at rest, it means there is no relative motion between adjacent fluid particles or layers. This state has specific implications for the stresses acting within the fluid.

Understanding Stress in Static Fluids

Stresses are forces acting over an area. In fluids, we consider two main types:

• Normal Stress: This stress acts perpendicular to a surface. In a fluid, the normal stress is related to pressure. For a fluid at rest, the normal stress acting on any imaginary plane within the fluid is non-zero (unless the pressure itself is zero, which is rare). This normal stress is solely due to the weight of the fluid above and is equal in all directions. It's essentially the pressure ($p$) exerted by the fluid. Mathematically, the normal stress $\sigma_{nn}$ is equal to the pressure $p$.
• Shear Stress: This stress acts parallel (tangential) to a surface. Shear stress in fluids arises from the fluid's viscosity and the relative motion (velocity gradient) between adjacent layers. Since a fluid at rest has no relative motion between its layers, the velocity gradient is zero. Consequently, the shear stress ($\tau$) resulting from viscosity must also be zero.

Analyzing the Options

Based on the principles of fluid statics:

• A static fluid experiences pressure (normal stress) due to the fluid's weight.
• A static fluid does not experience shear stress because there is no motion or velocity gradient.

Let's examine the given options:

• Option 1: states zero normal stress and non-zero shear stress. This is incorrect because a static fluid has non-zero normal stress (pressure) and zero shear stress.
• Option 2: states non-zero normal stress and zero shear stress. This aligns perfectly with the definition of a static fluid. The fluid exerts pressure on its container and itself, but there are no tangential forces due to motion.
• Option 3: states non-zero normal stress and shear stress. This is incorrect as shear stress must be zero for a fluid at rest.
• Option 4: states zero normal stress and zero shear stress. This is incorrect because the normal stress (pressure) is generally non-zero due to the weight of the fluid.

Conclusion on Fluid State

Therefore, the statement "a fluid is at rest" accurately means that it exhibits non-zero normal stress (pressure) and zero shear stress.