The concept of friction is easily understood for a rigid, solid object. For a solid object to move, it must overcome the resistance from another object in contact with it. A similar situation can be envisioned in the case of a flowing fluid. Consider the velocity profile for laminar flow shown in Figure 5.2. The molecules are flowing in layers stacked upon each other. The molecules in contact with the wall can be considered to be stationary because of friction with the wall—a state called the no-slip condition. The layer of molecules adjoining this layer has to slide against this stationary layer. Similarly, there is relative motion between all adjacent layers, causing frictional losses. The resistance to motion between the layers is highest at the wall, decreasing toward the center of the conduit.
This variation in axial velocity with the position in the direction normal to the flow direction causes a shear stress in the fluid. This shear stress multiplied by the area over which it enacts yields the shear force—the frictional resistance to the flow. The shear stress for a fluid depends on the velocity gradient—how rapidly the velocity changes with distance. The following shows this mathematically [4]:
where τ is the shear stress, and vx the velocity in the x-direction, which depends on the position in the y-direction. Equation 5.3, known as Newton’s law of viscosity, indicates that the shear stress is directly proportional to the velocity gradient, with the constant of proportionality being μ, which is the dynamic viscosity of the fluid. Fluids that follow this simple relationship are termed Newtonian fluids. Many other fluids not conforming to this relationship are termed non-Newtonian. The negative sign in the equation arises from the directional considerations, with velocity and position being vector quantities, viscosity being a scalar, and the shear stress a tensor. The directional considerations are disregarded in the calculations in this book, and only absolute values are considered, reducing the equation to the following form:
The SI unit of viscosity is N·s/m2 (equivalently, Pa·s or kg/m·s), the centimeter-gram-second (CGS) system unit being poise (P or g/cm·s). Viscosity is a function of temperature, the viscosity of water being 1 cP or 1 mPa·s at ~21°C. In contrast, viscosities of honey and glycerin are between 1500 and 2000 cP at ambient temperatures. The difficulty with which these fluids flow can be explained on the basis of their viscosities. Viscosities of substances can be predicted from theory or experimentally determined.
The force needed for getting the liquid to flow can be obtained by multiplying the shear stress with the area over which it acts, which is the contact surface area between the layers. The pressure differential needed to make the fluid flow can then be obtained by dividing the force by the cross-sectional flow area, which is the area in the direction perpendicular to the flow direction.
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