Fluid Mechanics 2 Chapter 6 Quiz

Cauchy's equation of motion is a fundamental equation in fluid dynamics that describes the motion of a fluid element. It expresses the balance of forces acting on a fluid element, including inertial forces, pressure forces, and viscous forces.

The Navier-Stokes equation is a set of partial differential equations that describe the motion of viscous, incompressible fluids. It is derived from Cauchy's equation of motion by introducing viscous forces.

The Navier-Stokes equation is a powerful tool for understanding and predicting fluid flow behavior in various engineering applications. It is used to model and analyze fluid flow in pipes, around objects, and in many other scenarios.

The continuity equation expresses the conservation of mass. In fluid dynamics, it states that the rate of change of mass within a control volume is equal to the net mass flow rate into the volume.

Fluid flow in a pipe is a common example of how the Navier-Stokes equation is used to analyze fluid behavior. The equation helps predict the velocity profile and pressure distribution along the pipe.

The Navier-Stokes equation can be used to model and predict fluid flow around different shapes and objects. It is essential for understanding aerodynamics and other fluid-related phenomena.

Fluid flow in a bend is a complex problem that can be analyzed using the Navier-Stokes equation. The equation helps predict the velocity and pressure changes as the fluid navigates the bend.

Isothermal flow is a type of fluid flow where the temperature remains constant throughout the system. This type of flow can be analyzed using the Navier-Stokes equation by considering the heat transfer involved.

A fundamental equation in fluid dynamics that describes the conservation of linear momentum in a fluid. It's essentially a force balance equation, where the forces acting on a fluid element (pressure, viscous forces, and gravity) are equated to the rate of change of momentum of that element.

The force per unit area exerted by a fluid on a surface, acting perpendicular to the surface.

The force per unit area acting parallel to the surface, arising due to the internal friction between fluid layers in motion.

The acceleration due to gravity, approximately 9.81 m/s^2, acting on a fluid element.

The density of the fluid, representing its mass per unit volume.

The velocity vector of the fluid element, representing its speed and direction of motion.

The rate of change of the fluid velocity vector with respect to time. It indicates how the speed and direction of the fluid element are changing over time.

The partial derivatives of the velocity vector with respect to the spatial coordinates (x, y, z). It represents the spatial variability of the velocity field, describing how velocity changes across the fluid.

The alternative form of Cauchy's equation of motion expresses the balance of forces acting on a fluid element. It involves pressure, stress tensor, gravity, and acceleration.

The degree of freedom in solving Cauchy's equations depends on the number of unknowns and equations. It represents the difference between the number of unknowns and the number of independent equations.

Pressure (P) is a scalar quantity representing the force per unit area acting perpendicular to a surface. It is an important term in Cauchy's equation, affecting the fluid's motion.

The stress tensor (𝝉) is a tensor quantity representing the forces acting on a fluid element. It captures how these forces are distributed in different directions.

The gravitational acceleration (g) is a vector quantity that points downwards and influences the motion of the fluid due to gravity.

The velocity (v) of the fluid is a vector quantity that represents the speed and direction of the fluid's movement.

The density (ρ) of the fluid is a scalar quantity representing the mass per unit volume of the fluid. It affects the fluid's response to forces in Cauchy's equation.

A type of fluid where the viscosity remains constant, regardless of the applied shear stress.

A fluid whose density does not change with pressure, meaning its volume remains constant.

The mathematical expression that describes the motion of viscous fluids by relating forces, pressure, and viscosity.

A fluid whose viscosity changes with the applied shear stress, meaning it doesn't flow consistently.

A steady flow means the velocity at any point in the fluid doesn't change over time.

A flow where the velocity varies only in two dimensions, meaning the motion can be described within a plane.

The force exerted by the fluid's weight per unit volume. It acts downwards due to gravity.

A measure of how much a material stretches or compresses in a given direction due to an applied force.

A measure of how much a material changes shape due to an applied force.

A mathematical tool used to represent the rate of deformation of a material in all directions.

A relationship that links the components of the strain rate tensor to the velocity components of the fluid.

Relating the stress tensors to the velocity components helps understand the forces acting within a deforming fluid.

The strain rate tensor is a measure of how the material is deforming, while the stress tensor represents the forces acting within the material.

A simplified form of the Navier-Stokes equation, obtained by assuming a constant viscosity and neglecting certain terms.

The Navier-Stokes equation is derived by substituting the stress tensors in terms of velocity into Cauchy's equation.

The Navier-Stokes equation can be expressed in different coordinate systems, such as Cartesian coordinates, cylindrical coordinates, and spherical coordinates.

The Navier-Stokes equation for the x-direction describes how the velocity of the fluid in the x-direction changes with time and space due to various factors.

The Navier-Stokes equation can be written for the y-direction and z-direction to describe the complete motion of the fluid in three dimensions.

The Navier-Stokes equation is used to simulate and predict various fluid phenomena, like weather patterns, ocean currents, and blood flow.

Simplifying the Navier-Stokes equation can make it easier to solve in specific scenarios. The specific terms to neglect or simplify will depend on the particular problem and the underlying assumptions.

The Navier-Stokes equation is a set of equations that describe the motion of viscous fluids. It is derived from Newton's second law and accounts for forces like pressure, viscosity, and gravity.

The equation is simplified based on the assumptions made about the flow. For example, if the flow is steady, the partial derivative of velocity with respect to time is zero.

The relationship between velocity components indicates how the fluid moves in different directions. For example, if the flow is one-dimensional, only one velocity component depends on its corresponding coordinate.

The simplified Navier-Stokes equation can be solved to obtain the velocity profile and pressure distribution of the fluid. The solution involves finding the velocity components and pressure terms.

The pressure gradient in the x-direction depends on the velocity components based on the simplified Navier-Stokes equation. It reflects the change of pressure along the x-axis.

The pressure gradient in the y-direction is also determined by the simplified Navier-Stokes equation. It represents the change of pressure in the y-direction.

Integration of the pressure gradient equations with respect to x or y yields the pressure distribution in two dimensions. This represents the pressure variation across the entire flow region.

The pressure distribution is found by integrating the pressure gradient equations with respect to x, y, or both. The integration involves finding the integral of the pressure gradient terms.

The pressure distribution is influenced by the velocity components and the integration constants. The integration constants are determined by boundary conditions of the flow.

The pressure distribution is a function of both x and y coordinates. It represents the pressure variation at every point in the flow domain.

The pressure gradients in the x and y directions are given by 𝜕𝑃/𝜕𝑥 and 𝜕𝑃/𝜕𝑦, respectively. These represent the rate of change of pressure along the corresponding directions.

The integration constants in the pressure distribution equation are unknown values that can be determined by applying suitable boundary conditions, such as specifying the pressure at certain locations in the flow domain.

The pressure distribution is a crucial parameter in fluid flow analysis. It provides information about the pressure forces acting on the fluid and helps predict the flow behavior.

The simplified Navier-Stokes equation provides a framework for understanding the pressure distribution in viscous flow. It helps analyze fluid behavior and predict the pressure field in various applications.