Fluid Mechanics 2 Chapter 6 Quiz

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Questions and Answers

What is the principal focus of the Navier-Stokes equation derivation?

Reduction steps from the law of thermodynamics

The impact of gravitational forces on fluid flow

Reduction steps from Cauchy's equation of motion (correct)

The study of isothermal and non-isothermal flows

Which equation is essential for estimating flow properties such as velocity and pressure?

Euler's equation

Cauchy’s equation of motion

Continuity equation (correct)

Bernoulli's equation

In which scenario do velocity variations occur according to the content?

Flow in a pipe bend (correct)

Static fluid at rest

Uniform flow through a straight pipe

Laminar flow in open channels

What is indicated by the operational form of Cauchy’s equation of motion?

Complexity due to 2nd order tensors Signup and view all the answers

What does the linear momentum balance relate to in fluid dynamics?

Navier-Stokes equation Signup and view all the answers

Which of the following best describes energy dissipation in fluid flow?

Loss of energy due to friction and turbulence Signup and view all the answers

How can the velocity field in a pipe be characterized?

It varies at different locations based on flow dynamics Signup and view all the answers

Which of the following is not part of the foundation for modeling a flow system?

Speed of sound in fluids Signup and view all the answers

What characterizes a Newtonian fluid?

Constant kinematic viscosity Signup and view all the answers

In the context of incompressible fluids, what remains constant?

Density, 𝜌 Signup and view all the answers

For a non-Newtonian fluid, how are the viscous friction terms modified?

They are modified to include the fluid's rheology. Signup and view all the answers

In the given velocity field, what does the variable 𝑏 represent?

A constant related to the velocity in the y-direction Signup and view all the answers

Which assumption is NOT applicable in the provided scenario of fluid flow?

The flow is three-dimensional. Signup and view all the answers

What is the primary purpose of the Navier-Stokes equation in fluid dynamics?

To describe the motion of fluid substances Signup and view all the answers

What does the term 'steady flow' imply in fluid dynamics?

Fluid velocity at a point remains constant over time. Signup and view all the answers

In the given velocity field, what does the constant 𝑐 represent?

The constant velocity in the z-direction Signup and view all the answers

What is the primary goal of modifying Cauchy’s equation of motion into an alternative form?

To eliminate the second order stress tensor Signup and view all the answers

In the context of Cauchy's equations, what is the significance of having zero degrees of freedom?

It indicates the system has a unique solution Signup and view all the answers

Which component of the Cauchy’s equation corresponds to the gravitational force acting in the x-direction?

𝜌𝑔𝑥 Signup and view all the answers

What does the symbol 𝜏 represent in the alternative form of Cauchy’s equation?

Stress tensor Signup and view all the answers

How many unknowns are associated with pressure, P, in the context of Cauchy’s equations?

10 Signup and view all the answers

Which of the following represents the relationship between known and unknown variables in Cauchy’s equation?

3 equations with 10 unknowns Signup and view all the answers

In the y-direction form of Cauchy’s equation, what does the term 𝜏𝑥𝑦 represent?

Stress in the x-direction due to y-direction strain Signup and view all the answers

Which of the following is included in the modifications made to Cauchy’s equation for the x-direction?

$ ho + v_x abla v$ Signup and view all the answers

What is the general form of the equation representing forces in the x-direction?

$ ho g_x - abla P + ( au_{xx}) + ( au_{yx}) + ( au_{zx}) = ho v_x + ho v_x v_x + ho v_y v_x + ho v_z v_x$ Signup and view all the answers

Which term represents the effects of fluid momentum in the x-direction?

$ ho v_x + abla ho vv$ Signup and view all the answers

What does the symbol $ abla P$ represent in these equations?

The gradient of pressure Signup and view all the answers

How are the directions of tensors represented in these equations?

By using second order components for shear stresses Signup and view all the answers

In the equation for the z-direction, which components are explicitly mentioned?

Pressure, shear stresses, and inertial forces Signup and view all the answers

What does the term $ ho v_x v_y$ signify in the context of the equations?

The interaction of velocities Signup and view all the answers

Cauchy’s equation was introduced primarily to address what?

Limitations in understanding fluid dynamics Signup and view all the answers

Which of the following is NOT a term found in the equations for the y-direction?

$ ho v_z$ Signup and view all the answers

What is the primary assumption regarding the function of velocity components in the x-direction?

Velocity components depend only on the x coordinate. Signup and view all the answers

Which differential term is specifically omitted in the equation for the y-direction?

Gravity term. Signup and view all the answers

What is the relationship of $v_y$ as expressed in the simplified Navier-Stokes equations?

$v_y = -ay + cx$ Signup and view all the answers

In the Navier-Stokes equations, what does the term $\mu$ represent?

Viscosity of the fluid. Signup and view all the answers

How does the pressure gradient relate to the x-coordinate based on the Navier-Stokes equation?

$\frac{\partial P}{\partial x} = \rho - a^2 x - ab$ Signup and view all the answers

What happens when you integrate the expression for $\partial P / \partial y$?

You find the pressure $P(x, y)$ as a function of coordinates. Signup and view all the answers

In the Navier-Stokes equations, what is the significance of steady-state conditions?

The flow velocity varies only in space but not in time. Signup and view all the answers

Which of the following describes how $v_x$ is affected by the x-coordinate based on the equation provided?

$v_x$ is linearly dependent on $x$. Signup and view all the answers

What effect does the term $\rho g_y$ have in the y-direction Navier-Stokes equation?

It simplifies to zero under no gravity conditions. Signup and view all the answers

Which of the following variables plays a key role in determining the final expression for pressure $P(x,y)$?

Integration constants $C$. Signup and view all the answers

What does the term $a^2$ signify in the equations for velocity components?

It is a parameter influencing the flow characteristics. Signup and view all the answers

What does the integration of the expression $\partial P / \partial y$ ultimately yield in terms of function?

It produces the pressure as a function of both $x$ and $y$. Signup and view all the answers

What significance does the term $\partial^2 v_y / \partial y^2$ hold in the context of fluid dynamics?

It reflects the rate of shear deformation in the fluid. Signup and view all the answers

When integrating the expression for $h_x$, what final form does it take in relation to pressure?

$h_x = \rho - \frac{a^2}{2} x^2 - abx + D$ Signup and view all the answers

What is referred to as the rate of increase in length per unit length in linear strain?

Linear strain rate Signup and view all the answers

What is the formula for shear strain rate between two perpendicular lines initially intersecting at a point?

𝜀𝑥𝑦 = 1/2(𝜕𝑣𝑦/𝜕𝑥 + 𝜕𝑣𝑥/𝜕𝑦) Signup and view all the answers

Which equation represents the linear strain rate tensor for the x-component?

𝜀𝑥𝑥 = 𝜕𝑣𝑥/𝜕𝑥 Signup and view all the answers

In shear strain rates, what expression defines 𝜀𝑦𝑥?

𝜀𝑦𝑥 = 1/2(𝜕𝑣𝑦/𝜕𝑥 + 𝜕𝑣𝑥/𝜕𝑦) Signup and view all the answers

What does the stress tensor relate to in the context of velocity?

Rate of deformation of fluid elements Signup and view all the answers

In the Navier-Stokes equation, which term represents viscous stress?

2𝜇𝜀𝑖𝑗 Signup and view all the answers

How is the stress tensor for the x-direction expressed in the context of the Navier-Stokes equations?

𝜏𝑥𝑥 = 2𝜇𝜕𝑣𝑥/𝜕𝑥 Signup and view all the answers

Which of the following terms appears in the Cauchy’s equations for fluid motion?

Body force term Signup and view all the answers

Which of the following represents the relationship between stress and strain rate in a Newtonian fluid?

𝜏𝑖𝑗 = 𝜇𝜀𝑖𝑗 Signup and view all the answers

What is considered an assumption in the reduction to the Navier-Stokes equations?

Incompressible flow Signup and view all the answers

In the context of pressure, which term in the Navier-Stokes equations denotes pressure gradient force?

𝜕𝑃/𝜕𝑥 Signup and view all the answers

Which set of terms contributes to the nonlinear forces in the Navier-Stokes equations?

Inertial terms Signup and view all the answers

What does the symbol 𝜇 represent in fluid mechanics?

Dynamic viscosity Signup and view all the answers

What does modifying the Navier-Stokes equations aim to achieve?

Simplify the computation process Signup and view all the answers

Study Notes

Fluid Mechanics 2 - KIL 3002

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

Chapter 6: Derivation of Navier-Stokes Equations

Derivation steps: Reduction steps from Cauchy's equation of motion
Cauchy's equation of motion: An equation of motion which includes stress tensor terms
Navier-Stokes equation: Results from the reduction of Cauchy's equation of motion

Derivation of Navier-Stokes Equations

Reduction of equation from Cauchy's equation of motion, with the simplification of viscosity terms as a function of velocity
Navier-Stokes equation's components can be expressed as a function of velocity components
Constitutive equation relates viscous stress tensors with strain rates
Formula: τ_ij = 2µε_ij

Determination of Flow Properties

Modeling a flow system estimates flow velocity and pressure at different locations
Methods are derived from the law of physics
Conservation of mass (Continuity equation)
Linear momentum balance (Navier-Stokes equation)
Flow patterns and energy dissipations are factors to consider for modeling

Cauchy's Equation of Motion

Original form: complex, uses 2nd-order tensors
Alternative form: introduced to overcome complexity and limitations of the original form
Variables: Pressure (P), Velocities (vx, vy, vz), Stress tensors (Txx, Tyx, Tzx, Tyy, Tzy, Tzz)
Degree of freedom relates to the number of equations and unknowns.

Reduction to Navier-Stokes Equations

Simplifying the stress tensor components in terms of velocity components
Constitutive equation relates viscous stress tensors with strain rates, valid for incompressible and Newtonian fluids.
Formula: τ_ij = 2µε_ij
Strain rate tensor relation to velocity components using different relations.

Reduction to Navier-Stokes Eqn for X-Direction

Substituting stress tensors in Cauchy's equations yields the Navier-Stokes equation for the x-direction.
Includes density, pressure, viscosity, and velocity components.
Relevant for Newtonian fluids.

Reduction to Navier-Stokes Eqn - Simplifications

Simplifying the Navier-Stokes equations for incompressible fluids, setting ∇•v = 0.
Resultant equations have reduced complexity, making calculations more manageable for incompressible fluids.
Compact notation for easier handling of variables.

Navier-Stokes Equation - Compact Form

Combining the x, y, and z-components of the Navier-Stokes equation to form a general form for any direction.
Simplified notation to represent the three equations of motion using compact notation.
General form of the Navier-Stokes equation allows for a general solution for any direction (x,y,z).

Navier-Stokes Equation Applications

Assumptions made—Newtonian fluid, constant kinematic viscosity (µ), constant density (ρ), and incompressible fluids, flow is steady and 2-dimensional, gravity is negligible.

Application of Navier-Stokes Equation – Example

Example of a steady, two-dimensional, incompressible velocity field.
Determining the pressure field as a function of x and y
Simplification of the Navier-Stokes to solve for x and y components of the equations
Pressure field functions are determined by using integration techniques and by substituting equations to solve for the differential variables for x and y components.

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Description

Test your understanding of the Navier-Stokes equations in Fluid Mechanics 2, specifically focused on Chapter 6. This quiz covers the derivation steps from Cauchy's equation of motion and the components of the Navier-Stokes equation related to flow properties. Assess your grasp of essential concepts and formulas in fluid dynamics.