Book cover for Fluid Mechanics

Fluid Mechanics

Frank M. White

ISBN #9789385965494

8th Edition

1,418 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter shifts the focus from control volume (integral) methods to a detailed point-by-point (differential) analysis of fluid flow. By deriving the differential forms of the conservation laws—continuity, momentum, and energy—we obtain the mathematical foundation for describing complex fluid behaviors. These equations, under appropriate simplifying assumptions (steady, incompressible, frictionless), lead to models such as the Bernoulli equation and form the basis for computational simulation techniques. Understanding and correctly applying boundary conditions are essential to solving these equations and capturing the nuances of real fluid flows.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

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Example Problems

Example 1

Discuss Newton's second law (the linear momentum relation) in these three forms: $$\begin{array}{cc} \sum \mathbf{F}=m \mathbf{a} & \sum \mathbf{F}=\frac{d}{d t}(m \mathbf{V}) \\ \sum \mathbf{F}=\frac{d}{d t}\left(\int_{\mathrm{system}} \mathbf{V} \rho d^{\gamma}\right) \end{array}$$ Are they all equally valid? Are they equivalent? Are some forms better for fluid mechanics as opposed to solid mechanics?

Example 2

Consider the angular momentum relation in the form $$\sum \mathbf{M}_{o}=\frac{d}{d t}\left[\int_{\mathrm{syscm}}(\mathbf{r} \times \mathbf{V}) \rho d^{v}\right]$$ What does $\mathbf{r}$ mean in this relation? Is this relation valid in both solid and fluid mechanics? Is it related to the linear momentum equation (Prob. 3.1)? In what manner?

Example 3

For steady low-Reynolds-number (laminar) flow through a long tube (see Prob. 1.12 ), the axial velocity distribution is given by $u=C\left(R^{2}-r^{2}\right),$ where $R$ is the tube radius and $r \leq R .$ Integrate $u(r)$ to find the total volume flow $Q$ through the tube.

Example 4

Water at $20^{\circ} \mathrm{C}$ flows through a long elliptical duct $30 \mathrm{cm}$ wide and $22 \mathrm{cm}$ high. What average velocity, in $\mathrm{m} / \mathrm{s}$, would cause the weight flow to be 500 lbf/s?

Example 5

Water at $20^{\circ} \mathrm{C}$ flows through a 5 -in-diameter smooth pipe at a high Reynolds number, for which the velocity profile is approximated by $u \approx U_{\mathrm{o}}(y / R)^{1 / 8},$ where $U_{\mathrm{o}}$ is the centerline velocity, $R$ is the pipe radius, and $y$ is the distance measured from the wall toward the centerline. If the centerline velocity is $25 \mathrm{ft} / \mathrm{s}$, estimate the volume flow rate in gallons per minute.
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Step-by-Step Explanations

QUESTION

How do we derive the differential form of mass conservation for an incompressible fluid?\nStep-by-step Answer:\nStep 1: Start with the integral form of mass conservation over a fixed control volume; the net mass flow leaving the volume must equal the rate of decrease of mass inside.\nStep 2: Shrink the control volume to an infinitesimal element so that the integral converts to a differential expression in terms of spatial gradients.\nStep 3: Express the net mass flux as the divergence of \u03c1V. For an incompressible fluid (constant \u03c1), this simplifies to the divergence of the velocity field being zero.\nStep 4: Write the resulting differential continuity equation: \u2207\u00b7V = 0.\n\n- Topic: Derivation of the Differential Momentum Equation \nQuestion: How is the momentum equation in differential form derived for an inviscid fluid?\nStep-by-step Answer:\nStep 1: Apply Newton\u2019s second law to an infinitesimal fluid element, considering the rate of change of momentum equals the sum of forces.\nStep 2: Express both unsteady (local) and convective acceleration components that act on the fluid element.\nStep 3: Incorporate body forces (like gravity) and pressure forces. For an inviscid, frictionless flow, neglect viscous stresses.\nStep 4: Arrive at the Euler equation in differential form: \u03c1[\u2202V/\u2202t + (V\u00b7\u2207)V] = \u2212\u2207p + \u03c1g.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Start with the integral form of mass conservation over a fixed control volume; the net mass flow leaving the volume must equal the rate of decrease of mass inside.\nStep 2: Shrink the control volume to an infinitesimal element so that the integral converts to a differential expression in terms of spatial gradients.\nStep 3: Express the net mass flux as the divergence of \u03c1V. For an incompressible fluid (constant \u03c1), this simplifies to the divergence of the velocity field being zero.\nStep 4: Write the resulting differential continuity equation: \u2207\u00b7V = 0.\n\n- Topic: Derivation of the Differential Momentum Equation \nQuestion: How is the momentum equation in differential form derived for an inviscid fluid?\nStep-by-step Answer:\nStep 1: Apply Newton\u2019s second law to an infinitesimal fluid element, considering the rate of change of momentum equals the sum of forces.\nStep 2: Express both unsteady (local) and convective acceleration components that act on the fluid element.\nStep 3: Incorporate body forces (like gravity) and pressure forces. For an inviscid, frictionless flow, neglect viscous stresses.\nStep 4: Arrive at the Euler equation in differential form: \u03c1[\u2202V/\u2202t + (V\u00b7\u2207)V] = \u2212\u2207p + \u03c1g.\n\n"
Final Answer:

"- Topic: Derivation of the Differential Continuity Equation \nQuestion: How do we derive the differential form of mass conservation for an incompressible fluid?\nStep-by-step Answer:\nStep 1: Start with the integral form of mass conservation over a fixed control volume; the net mass flow leaving the volume must equal the rate of decrease of mass inside.\nStep 2: Shrink the control volume to an infinitesimal element so that the integral converts to a differential expression in terms of spatial gradients.\nStep 3: Express the net mass flux as the divergence of \u03c1V. For an incompressible fluid (constant \u03c1), this simplifies to the divergence of the velocity field being zero.\nStep 4: Write the resulting differential continuity equation: \u2207\u00b7V = 0.\n\n- Topic: Derivation of the Differential Momentum Equation \nQuestion: How is the momentum equation in differential form derived for an inviscid fluid?\nStep-by-step Answer:\nStep 1: Apply Newton\u2019s second law to an infinitesimal fluid element, considering the rate of change of momentum equals the sum of forces.\nStep 2: Express both unsteady (local) and convective acceleration components that act on the fluid element.\nStep 3: Incorporate body forces (like gravity) and pressure forces. For an inviscid, frictionless flow, neglect viscous stresses.\nStep 4: Arrive at the Euler equation in differential form: \u03c1[\u2202V/\u2202t + (V\u00b7\u2207)V] = \u2212\u2207p + \u03c1g.\n\n"

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