Learning Objectives
After completing this module, you should be able to:
1. Develop a better understanding of and distinguish between dimensions and units.
2. Understand and be able to identify dimensional homogeneity in equations.
3. Identify dimensionless parameters in fluid mechanics and understand what it means for a parameter to be dimensionless.
4. Understand vector math and the function of the del operator, particularly using the expanded acceleration equation.
5. Apply Taylor series to solve equations in fluid mechanics.
6. Identify and understand the relationship between shear stress and viscosity for Newtonian Fluids.
Videos
Title: M8V1 – Dimensionless Parameters
Summary: This video explores the concept of dimensionless parameters in fluid mechanics, building on foundational knowledge of primary dimensions such as mass, length, and time. It emphasizes dimensional homogeneity and the importance of ensuring equations are dimensionally consistent. Through step-by-step dimensional analysis, the video examines common fluid mechanics parameters and demonstrates how to confirm that parameters like the Darcy friction factor, Reynolds number, and drag coefficient are dimensionless.
Learning Objectives: After watching this video, students will be able to perform dimensional analysis, identify the primary dimensions of physical quantities, and verify the dimensionless nature of key fluid mechanics parameters such as the Reynolds number, Darcy friction factor, and drag coefficient.
Transcript: Read the transcript
Slides With Annotations: See the slides
Title: M8V2 – Taylor Series
Summary: This lecture defines the Taylor series and explains how it can be used to estimate the value of a function at a location different from where it is known. The video walks through the expansion process, demonstrating how each term in the series progressively brings the estimate closer to the true value of the function. The importance of the Taylor series in numerical modeling is highlighted, especially in reducing model prediction errors in complex functions, such as nonlinear equations. The lecture explains the rule for using Taylor series, including requirement for continuous derivatives of the function in the domain. It also demonstrates how Taylor series can be applied in fluid mechanics specifically for estimating mass flux in a flow domain.
Learning Objectives: After watching this lecture, you will be able to understand the definition and structure of a Taylor series expansion, apply the Taylor series to approximate the value of a function, recognize the role of the Taylor series in minimizing errors in numerical modeling, and use the Taylor series to estimate values of fluid properties in a flow domain.
Transcript: Read the transcript
Slides With Annotations: See the slides
Title: M8V3 – Linear Momentum
Summary: This video reviews explains how linear momentum is accounted for through two primary terms: the time rate of change of momentum within the control volume and the net flux of momentum across the control surface. Unlike conservation laws like mass or energy where the sum of these terms is zero, here they are balanced by the net external force acting on the control volume. The video also discusses dimensional consistency, showing how momentum flux has the same dimensions as force. In the case of steady flows, the time derivative term disappears, simplifying the momentum equation. The video also introduces the momentum flux correction factor (β) to adjust for non-uniform velocity profiles.
Learning Objectives: After watching this video, students will be able to interpret the general form of the linear momentum equation for a control volume, differentiate between the terms representing time rate of change of momentum and net momentum flux, demonstrate dimensional consistency by showing that momentum flux has the same units as force, and simplify the momentum equation for steady flows by eliminating the time derivative term.
Transcript: Read the transcript