In a parallel one-dimensional flow in the positive x...

In a parallel one-dimensional flow in the positive $x$ direction, the velocity varies linearly from zero at $y=0$ to $30 \mathrm{m} / \mathrm{s}$ at $y=1.5 \mathrm{m} .$ Determine an expression for the stream function, $\psi$. Also determine the $y$ coordinate above which the volume flow rate is half the total between $y=0$ and $y=1.5 \mathrm{m}$.

In a parallel one-dimensional flow in the positive $x$ direction, the velocity varies linearly from zero at $y=0$ to $30 \mathrm{m} / \mathrm{s}$ at $y=1.5 \mathrm{m} .$ Determine an expression for the stream function,
$\psi$. Also determine the $y$ coordinate above which the volume flow rate is half the total between $y=0$ and $y=1.5 \mathrm{m}$.

Key Concepts

Stream Function in Incompressible Flow

The stream function is a scalar function used to describe two-dimensional incompressible flow. Its key property is that its partial derivatives relate directly to the velocity components (with u equal to the partial derivative of ? with respect to y, and v equal to the negative partial derivative of ? with respect to x). This automatically satisfies the continuity equation, making it a powerful tool in fluid mechanics for visualizing flow patterns and streamlines.

Parallel Flow

Parallel flow refers to a situation where the flow is primarily in one direction while the velocity may vary in the perpendicular direction. In such flows, the velocity field simplifies considerably because one component is dominant, and the velocity profile depends only on the coordinate perpendicular to the direction of flow. This simplification streamlines the analysis and derivation of the stream function.

Velocity Profile

A velocity profile represents how the speed of the fluid changes across the flow’s cross-section. In many practical problems, such as the one with a linear variation from one boundary to another, the profile can be described with a simple mathematical function. Understanding the velocity profile is crucial to computing other flow-related quantities like the stream function and the volume flow rate.

Volume Flow Rate

The volume flow rate is the measure of the volume of fluid passing through a cross-sectional area per unit time. It is computed by integrating the velocity profile over that area. In problems where the flow rate is partitioned, such as identifying where a given fraction of the total flow occurs, integration is used to set up and solve for the relevant spatial position within the flow.

Integration in Fluid Mechanics

Integration is a fundamental tool in fluid mechanics used to calculate the net effect of continuously varying quantities, like velocity. It allows one to sum contributions over a given area or volume. In the context of calculating flow rate, integration of the velocity function over the flow domain yields the total or partial fluid discharge, making it indispensable for analyzing flow distributions.

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