Fluid Mechanics 2: Navier-Stokes Solutions

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Questions and Answers

Which equation describes the relationship for the x-direction component in terms of other variables?

$ ext{direction component} = 7 - ext{variable} + ext{other variables}$

$ ext{direction component} = 7 - rac{9}{3} - rac{9}{2}$

$ ext{direction component} = 7 - rac{9}{4} - rac{9}{5}$ (correct)

$ ext{direction component} = 7 - ext{functions of variables}$

What does the new variable represent after integrating the differential equation?

A differential value

An error term

A function of another variable (correct)

A constant value

What is the general form of the equation obtained for # $, & after integration?

# $, & = ext{constant} - f(x,y)$

# $, & = ( * - ,- )$

# $, & = * - , - $ + ext{function of variables}$

# $, & = ( * - ,!$ - ,-$) + h(&)$ (correct)

How do you find the new variable when differentiating the integrated equation?

Differentiate with respect to another variable Signup and view all the answers

Which of the following components is part of the equation described in the document?

Direction component Signup and view all the answers

What is an important part of the process when integrating the differential equation?

Adding a variable with a function of other variables Signup and view all the answers

What differential equation format is presented for the integration step?

$# $, & = ( * - ,!$ - ,-$)$ Signup and view all the answers

Which function appears in the process of determining the new variable through differentiation?

Function of variable(s) Signup and view all the answers

What is the primary objective of the exercise described?

To determine the pressure field from known velocity Signup and view all the answers

Which equation is used to express the pressure field in relation to velocity components?

$p = -% + ,&$ Signup and view all the answers

In which direction is the velocity component not specified?

No direction mentioned Signup and view all the answers

What role does gravity play in the described pressure field determination?

It is not considered in this context Signup and view all the answers

How are velocity fields related to pressure fields in this document?

Pressure fields are derived from velocity fields Signup and view all the answers

Which component is included in the pressure field calculation?

Velocity magnitude Signup and view all the answers

What factor is important when substituting velocity fields into pressure fields?

The accuracy of velocity measurements Signup and view all the answers

What does 'steady' imply in the context of pressure fields?

No changes in pressure over time Signup and view all the answers

What is the first step in the problem setup process?

Sketch the geometry and identify dimensions Signup and view all the answers

Which equation represents the final pressure field?

<h1>$, & = * - (0.05)(1.5)$ + * - (1.5)(0.35)& + +5</h1> Signup and view all the answers

What is the purpose of specifying boundary conditions in problem-solving?

To solve for unknown parameters Signup and view all the answers

Which of the following is NOT a part of the problem setup process?

Verifying the results Signup and view all the answers

What is typically the first assumption made in a mechanical problem?

Neglecting air resistance Signup and view all the answers

What is a constant of integration used for in this context?

To express solutions uniquely in the problem Signup and view all the answers

What should be done after integrating the equations of motion?

Apply the boundary conditions Signup and view all the answers

Which term best describes the purpose of gathering relevant dimensions in the setup?

To formulate governing equations Signup and view all the answers

In the context of fluid flow, what does the term 'no-slip boundary condition' refer to?

Fluid velocity matches the solid boundary's velocity. Signup and view all the answers

What is one of the key assumptions made when determining pressure from a known velocity field in incompressible fluid flow?

Flow is steady. Signup and view all the answers

In a steady, two-dimensional, incompressible flow, which of the following holds true regarding the velocity field?

The pressure field can be determined directly from the velocity field. Signup and view all the answers

In the analysis of Couette flow, what type of flow is generally assumed?

Incompressible and laminar flow. Signup and view all the answers

When determining the pressure field using the Navier-Stokes equations, which assumption simplifies the equations?

No external forces such as gravity act on the fluid. Signup and view all the answers

How is the continuity equation linked to the determination of pressure in fluid flow?

It ensures mass conservation, which is vital for incompressible flows. Signup and view all the answers

Which of the following constants is NOT typically considered in the analysis of a steady two-dimensional incompressible velocity field?

Velocity of sound within the fluid. Signup and view all the answers

What does 'fully developed flow' in the context of fluid dynamics imply?

Velocity is consistent across all cross-sections of the flow. Signup and view all the answers

Study Notes

Fluid Mechanics 2 - KIL 3002

Course code: KIL 3002
Course name: Fluid Mechanics 2
Department: Chemical Engineering
University: Universiti Malaya

Chapter 7: Analytical Solutions to Navier-Stokes Equations 1

Chapter focus: Analytical solutions to Navier-Stokes equations, specifically exploring approximate solutions.
Key topics include:
- Flow applications, solving continuity and Navier-Stokes equations
- Analysis of fully developed Couette flow, a common flow case
- Numerical solutions to similar flow cases in Computational Fluid Dynamics (CFD) chapters
- Applications of no-slip boundary conditions (solid/wall)

Determination of Pressure Field from Known Velocity

Problem context: Determining the pressure field based on a provided velocity field (steady, two-dimensional, incompressible)
Velocity field representation:
- v = (ax + b)i + (-ay + cx)j, with a, b, and c as constants.
Assumptions:
- Steady flow
- Incompressible flow
- Laminar flow
- Constant fluid properties
- No gravity in x and y directions
Goal: Determines the pressure (as a function of x and y).

Key Procedures for Navier-Stokes

Procedures for analyzing common flow problems, including the Couette flow.
Key steps:
- Problem setup: Geometrical depiction, dimensions, and parameters.
- Assumptions & BCs: Specify relevant assumptions and boundary conditions.
- Equations of motion: Set up the governing equations (differential equations) using fundamentals, and Simplify where possible.
- Integration: Method of integration to find constants of integration from steps above
- Solve: Apply boundary conditions to find constants of integration. Verify.

Common Flow Case - Fully Developed Couette Flow

Flow description: Incompressible, laminar flow between two infinite parallel plates, one stationary and the other moving at a constant velocity S.
Assumptions:
- Incompressible and Newtonian fluid
- Steady and laminar flow
- One-dimensional flow in the x direction
- Constant pressure in the x direction
- Gravity acting only in the z direction
Boundary conditions:
- No-slip condition at both plates (y = 0 and y = h).

Common Flow Case - Falling Fluid Between Parallel Plates

Flow description: Viscous fluid falling between two infinite parallel vertical plates. Gravity in the negative y-direction is the driving force, no applied pressure.
Assumptions:
- Steady and incompressible
- Newtonian fluid with constant properties
- Laminar flow
- Parallel flow (Vx = 0)
- Two-dimensional problem (Vz = 0)
- Constant pressure
- Gravity in the y-direction only
Boundary conditions:
- No-slip condition at both plates (y = 0 and y = h).

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Description

This quiz covers Chapter 7 of Fluid Mechanics 2, focusing on analytical solutions to the Navier-Stokes equations. You'll explore flow applications, the determination of pressure fields from known velocity fields, and the analysis of fully developed Couette flow. Prepare to dive into the complexities of incompressible flow and boundary conditions.