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SOLUTIONS MANUAL: Fundamentals of Fluid Mechanics, 7th Edition by
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Is it two- or three-dimensional? One obvious one is k. The streamlines are in the x-y plane and are found from the velocities: The streamlines converge and the velocity increases to the right.
According to the theory of Chap. Note that x is negative along line AB. Why does the fluid acceleration become negative after condition b? This is a one-dimensional unsteady flow. Simply substitute the given velocity components into the incompressible continuity equation: Make a sketch of the velocity components. This is harder than it looks. Make a sketch of each separate cartesian component: The new version, in polar coordinates, requires some effort.
The problem is only meant to acquaint students with spherical coordinates.
SOLUTIONS MANUAL: Fundamentals of Fluid Mechanics, 7th Edition by
Substitute into continuity, Eq. What might this flow field simulate? Does this flow satisfy continuity? What might it represent physically? We check the incompressible continuity relation in cylindrical coordinates: What are the proper dimensions for constants K and a?
We can find the appropriate velocity v from two-dimensional continuity: If we apply one-dimensional Fig. If the tank temperature is assumed constant and the gas is ideal, find an expression for introdduction variation of density within the tank.
A special ediion solution iswhere A and B are wolution. For consistency, what should the dimensions of constants K and b be? Substitute into plane polar coordinate continuity: The pattern represents inviscid flow of 7rh uniform stream past a circular cylinder Chap.
This is a laborious derivation, really, the problem is only meant to acquaint the student with streamline coordinates. The second part is not too hard, though. Multiply the streamwise momentum equation by ds and integrate: Find an expression for the pressure gradient in the x direction. Neglect gravity and assume constant viscosity.
The flow is rotational. Evaluate and check the incompressible continuity equation: Compute the value and position of the maximum viscous normal stress along this streamline.
Is this also the position Fig. Do not reveal this to your friends if they are still working on Prob. Show that this flow field is an exact solution to the Navier- Stokes equations 4. Interpret these two cases physically. Neglect gravity and friction and assume purely radial r Q inflow. The laborious results are: The preliminary work for this! Find the volume flux Q per unit width in terms of these parameters. The x-momentum equation can easily be evaluated from the known velocity profile: There are no pressure gradients, only gravity.
Set up and solve the Navier-Stokes equation for the velocity profile w x. Only the z-component of Navier-Stokes is relevant: What might these two cases represent? Can you identify what type of partial differential equation it is? What is the physics represented by this equation, and the role played by the parameter v?
Is it dissipative or conservative? The N-S equation reduces to a momentum diffusion equation, where the kinematic viscosity plays the role of momentum diffusivity. Momentum or energy is dissipated because of viscosity. Determine a relation between the body couple and shear stress for equilibrium. What are the proper dimensions for Cz? Body couples are important in continuous media with microstructure, such as granular materials. The couple Cz has to be per unit volume to make physical sense in Eq.
The concentrated couple allows the stress tensor to have unsymmetrical shear stress terms. In the spirit of Ex. There are no variations with x or z, so the energy equation 4. Now integrate once more: The final solution is: What are the appropriate boundary and initial conditions for this problem? The initial condition is: The boundary conditions are Along the side walls: The physically realistic conditions at the upper and lower surfaces are: What are the proper boundary conditions to handle this problem?
First, at all walls, one would impose the no-slip condition: Finally, the pressure must be specified at either the inlet or the outlet section of the flow, usually at the upstream section: There are four different kinds of boundary conditions needed, as labeled.
The pipe walls are wound with an electric-resistance coil which delivers heat to the fluid at a rate qw energy per unit wall area. If we wish to analyze this problem by using the full continuity, Navier-Stokes, and energy equations, what are the proper boundary conditions for the analysis? On a surface about which the flow is symmetrical, the normal velocity, the tangential velocity gradient and hence the shear stress are equal to zero.
In cartesian coordinates the stream function is quite easy: Plot the streamlines in the full xy plane, find any stagnation points, and interpret what the flow could represent. Two velocity components and two continuity terms. The width into the paper is b, and the volume flow rate is Q. At any given distance r from the slot, the flow is radial inward, with constant velocity. Find an expression for the polar-coordinate stream function of this flow.
We can find velocity from continuity: Where in this chapter are the streamlines of this flow plotted? Use this stream function to find the volume flow Q passing through the Fig.
Show the direction of Q.
The streamlines are plotted in Fig. Here are the results: We are showing only the upper half plane, which is the mirror image of the lower half. Sketch the potential lines in the full xy plane, find any stagnation points, and sketch in by eye the orthogonal streamlines. What could the flow represent? The pattern represents plane stagnation flow Prob.
If they exist, find the stream function and velocity potential.