The velocity distribution for laminar flow between parallel...

The velocity distribution for laminar flow between parallel plates is given by \[\frac{u}{u_{\max }}=1-\left(\frac{2 y}{h}\right)^{2}\] where $h$ is the distance separating the plates and the origin is placed midway between the plates, Consider a flow of water at $15^{\circ} \mathrm{C}$ with maximum speed of $0.05 \mathrm{m} / \mathrm{s}$ and $h=0.1 \mathrm{mm}$. Calculate the force on a $1 \mathrm{m}^{2}$ section of the lower plate and give its direction.

The velocity distribution for laminar flow between parallel plates is given by
\[\frac{u}{u_{\max }}=1-\left(\frac{2 y}{h}\right)^{2}\]
where $h$ is the distance separating the plates and the origin is placed midway between the plates, Consider a flow of water at $15^{\circ} \mathrm{C}$ with maximum speed of $0.05 \mathrm{m} / \mathrm{s}$ and $h=0.1 \mathrm{mm}$. Calculate the force on a $1 \mathrm{m}^{2}$ section of the lower plate and give its direction.

Key Concepts

Dynamic Viscosity

Dynamic viscosity quantifies a fluid's resistance to shear or flow. In calculations involving shear stress, the fluid's dynamic viscosity is a key parameter that, together with the velocity gradient, determines the magnitude of the shear force applied to solid surfaces. Accurate values of viscosity are necessary for precise force calculations in any viscous flow scenario.

Shear Stress

Shear stress in a fluid is the tangential force per unit area arising from the fluid's viscosity and the velocity gradient at the wall. In the context of the problem, calculating the shear stress on the plate involves evaluating the rate at which the fluid velocity changes in the direction normal to the plate. This concept is fundamental in fluid mechanics, especially for predicting frictional forces in laminar flow.

Velocity Gradient

The velocity gradient is the rate at which the fluid velocity changes with respect to the distance normal to the surface. It is a crucial factor in determining shear stress because, according to Newton’s law of viscosity, shear stress is the product of the dynamic viscosity and the velocity gradient at the wall. Understanding how this gradient develops, especially in a parabolic velocity profile, is essential in the analysis of wall shear forces.

Laminar Flow

This is a flow regime characterized by smooth, orderly motion in parallel layers with little to no mixing between them. In such flows, the fluid particles move along well-defined streamlines. The laminar nature of the flow ensures that the velocity profile is predictable and parabolic between the plates, simplifying related calculations such as shear stress on boundaries.

Velocity Profile

The velocity profile describes how the speed of the fluid varies across the space between the plates. For laminar flow between parallel plates, this profile is parabolic, with the maximum velocity occurring at the center of the channel and decreasing toward the plates. Understanding this profile is essential for determining the velocity gradients at the walls which directly affect the force exerted by the fluid.

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