*7-84. A horizontal velocity is defined by u = 2(x2 – y?) ft/s and u = (-4xy) ft/s. Show that these expressions satisfy the continuity equation. Using the Navier-Stokes equations, show that the pressure distribution is defined by p = C -pV/2 – pgz.
*7-84. A horizontal velocity is defined by u = 2(x2 – y?) ft/s and u = (-4xy) ft/s. Show that these expressions satisfy the continuity equation. Using the Navier-Stokes equations, show that the pressure distribution is defined by p = C -pV/2 – pgz.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Problem 7-84**
A horizontal velocity is defined by the equations:
\[ u = 2(x^2 - y^2) \, \text{ft/s} \]
\[ v = (-4xy) \, \text{ft/s} \]
**Tasks:**
1. Show that these expressions satisfy the continuity equation.
2. Using the Navier-Stokes equations, show that the pressure distribution is defined by:
\[ p = C - \frac{\rho V^2}{2} - \rho gz \]
**Explanation:**
- The continuity equation in fluid dynamics ensures mass conservation in a flow field. For an incompressible flow, it is represented as:
\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]
- The Navier-Stokes equations describe the motion of viscous fluid substances. They represent conservation of momentum and are used to model the behavior of fluids.
- The pressure distribution formula combines kinetic and potential energy terms to describe how pressure varies within the fluid. Here, \( C \) is a constant, \(\rho\) is the fluid density, \( V \) is the velocity magnitude, and \( gz \) represents gravitational potential energy per unit mass.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17a6ed38-66c3-4809-bb99-3d4867aba49e%2F9e4ecf3d-64a2-40d5-aa24-a4c681966ec7%2Fsr0dzge_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 7-84**
A horizontal velocity is defined by the equations:
\[ u = 2(x^2 - y^2) \, \text{ft/s} \]
\[ v = (-4xy) \, \text{ft/s} \]
**Tasks:**
1. Show that these expressions satisfy the continuity equation.
2. Using the Navier-Stokes equations, show that the pressure distribution is defined by:
\[ p = C - \frac{\rho V^2}{2} - \rho gz \]
**Explanation:**
- The continuity equation in fluid dynamics ensures mass conservation in a flow field. For an incompressible flow, it is represented as:
\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]
- The Navier-Stokes equations describe the motion of viscous fluid substances. They represent conservation of momentum and are used to model the behavior of fluids.
- The pressure distribution formula combines kinetic and potential energy terms to describe how pressure varies within the fluid. Here, \( C \) is a constant, \(\rho\) is the fluid density, \( V \) is the velocity magnitude, and \( gz \) represents gravitational potential energy per unit mass.
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