Question 9.1: Equation (9-19) may be applied to numerous flow systems to p......
Equation (9-19) may be applied to numerous flow systems to provide information regarding velocity variation, pressure gradients, and other information of the type achieved in Chapter 8. Many situations are of sufficient complexity to make the solution extremely difficult and are beyond the scope of this text. A situation for which a solution can be obtained is illustrated in Figure 9.4.
\rho{\frac{D\mathbf{v}}{D t}}=\rho\mathbf{g}-\nabla P+\mu\nabla^{2}\mathbf{v} (9-19)
Figure 9.4 shows the situation of an incompressible fluid confined between two parallel, vertical surfaces. One surface, shown to the left, is stationary, whereas the other is moving upward at a constant velocity v_{\mathrm{0}}. If we consider the fluid Newtonian and the flow laminar, the governing equation of motion is the Navier–Stokes equation in the form given by equation (9-19). The reduction of each term in the vector equation into its applicable form is shown below:
\rho\frac{{ D}{\bf v}}{{ D}t}=\rho\biggl\{\frac{\partial{\bf v}}{\partial t}+v_{x}\frac{\partial{\bf v}}{\partial x}+v_{y}\frac{\partial{\bf v}}{\partial y}+v_{z}\frac{\partial{\bf v}}{\partial z}\biggr\}=0
\rho\mathrm{g}=-\rho\,g\mathrm{e}_{y}
\nabla P={\frac{d P}{d y}}\mathbf{e}_{y}
where dP/dy is constant, and
\mu\nabla^{2}\mathbf{v}=\mu\frac{d^{2}v_{y}}{d x^{2}}\mathbf{e}_{y}
The resulting equation to be solved is
0=-\rho g-\frac{d P}{d y}+\mu\frac{d^{2}v_{y}}{d x^{2}}
This differential equation is separable. The first integration yields
\frac{d v_{y}}{d x}+\frac{x}{\mu}\left\{-\rho g-\frac{d P}{d y}\right\}=C_{1}
Integrating once more, we obtain
v_{y}+{\frac{x^{2}}{2\mu}}\Biggl\{-\rho g-{\frac{d P}{d y}}\Biggr\}=C_{1}x+C_{2}
The integration constants may be evaluated, using the boundary conditions that
v_{\mathrm{y}}=0\,\mathrm{at}\ x=0,\;\mathrm{and}\;v_{\mathrm{y}}=v_{0}\,\mathrm{at}\ x=L. The constants thus become
C_{1}={\frac{v_{0}}{L}}+{\frac{L}{2\mu}}\left\{-\rho g-{\frac{d P}{d y}}\right\}\quad{\mathrm{and}}\quad C_{2}=0
The velocity profile may now be expressed as
v_{y}=\underbrace{{\frac{1}{2\mu}}\left\{-\,\rho g-{\frac{d P}{d y}}\right\}\{L x-x^{2}\}}_{ \underset{①}{} } \;+\,\underbrace{v_{0}{\frac{x}{L}}}_{\underset{②}{} } (9-21)
It is interesting to note, in equation (9-21), the effect of the terms labeled ① and ②, which are added. The first term is the equation for a symmetric parabola, the second for a straight line. Equation (9-21) is valid whether 0 is upward, downward, or zero. In each case, the terms may be added to yield the complete velocity profile. These results are indicated in Figure 9.5. The resulting velocity profile obtained by superposing the two parts is shown in each case.
Euler’s equation may also be solved to determine velocity profiles, as will be shown in Chapter 10. The vector properties of Euler’s equation are illustrated by the example below, in which the form of the velocity profile is given.
