# 0.7 Laminar flows with dependence on one dimension  (Page 3/6)

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$\begin{array}{c}-\nabla p+\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{f}=-\nabla P\hfill \\ \mathrm{where}\hfill \\ P=p+\rho \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}h\hfill \end{array}$

The product $gh$ is the gravitational potential, where $g$ is the acceleration of gravity and $h$ is distance upward relative to some datum. The pressure, $p$ , is also relative to a datum, which may be the datum for $h$ .

The primary spatial dependence is in the direction normal to the plane of the plates. If there is no dependence on one spatial direction, then the flow is truly one-dimensional. However, if the velocity and pressure gradients have components in two directions in the plane of the plates, the flow is not strictly 1-D and nonlinear, inertial terms will be present in the equations of motion. The significance of these terms is quantified by the Reynolds number. If the flow is steady, and the Reynolds number negligible, the equations of motion are as follows.

$\begin{array}{c}0=-\frac{\partial P}{\partial {x}_{j}}-\frac{\partial {\tau }_{j3}}{\partial {x}_{3}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2\hfill \\ 0=-\frac{\partial P}{\partial {x}_{3}}-0\hfill \\ 0=-\frac{\partial P}{\partial {x}_{j}}+\mu \frac{{\partial }^{2}{v}_{j}}{\partial {x}_{3}^{2}},\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{1.em}{0ex}}\mathrm{Newtonian}\phantom{\rule{0.277778em}{0ex}}\mathrm{fluid}\hfill \end{array}$

Let $h$ be the spacing between the plates and the velocity is zero at each surface.

${v}_{j}=0,\phantom{\rule{1.em}{0ex}}{x}_{3}=0,\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2$

The velocity profile for a Newtonian fluid in plane-Poiseuille flow is

${v}_{j}=\frac{{h}^{2}}{2\mu }\frac{\partial P}{\partial {x}_{j}}\left[{\left(\frac{{x}_{3}}{h}\right)}^{2}-\frac{{x}_{3}}{h}\right],\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{1.em}{0ex}}0\le {x}_{3}\le h$

The average velocity over the thickness of the plate can be determined by integrating the profile.

${\overline{v}}_{j}=-\frac{{h}^{2}}{12\mu }\phantom{\rule{0.166667em}{0ex}}\frac{\partial P}{\partial {x}_{j}},\phantom{\rule{1.em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2$

This equation for the average velocity can be written as a vector equation if it is recognized that the vectors have components only in the $\left(1,2\right)$ directions.

$\overline{\mathbf{v}}=-\frac{{h}^{2}}{12\mu }\phantom{\rule{0.166667em}{0ex}}\nabla P,\phantom{\rule{1.em}{0ex}}\overline{\mathbf{v}}=\overline{\mathbf{v}}\left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\nabla P=\nabla P\left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2}\right)$

If the flow is incompressible, the divergence of velocity is zero and the potential, $P$ , is a solution of the Laplace equation except where sources are present. If the strength of the sources or the flux at boundaries are known, the potential, $P$ , can be determined from the methods for the solution of the Laplace equation.

We now have the result that the average velocity vector is proportional to a potential gradient. Thus the average velocity field in a Hele-Shaw flow is irrotational. If the fluid is incompressible, the average velocity field is also solenoidal can can be expressed as the curl of a vector potential or the stream function. The average velocity field of Hele-Shaw flow is an physical analog for the irrotational, solenoidal, 2-D flow described by the complex potential. It is also a physical analog for 2-D flow of incompressible fluids through porous media by Darcy's law and was used for that purpose before numerical reservoir simulators were developed.

## Poiseuille flow

Poiseuille law describes laminar flow of a Newtonian fluid in a round tube (case 1). We will derive Poiseuille law for a Newtonian fluid and leave the flow of a power-law fluid as an assignment. The equation of motion for the steady, developed (from end effects) flow of a fluid in a round tube of uniform radius is as follows.

$\begin{array}{c}0=-\frac{\partial P}{\partial r}\hfill \\ 0=-\frac{\partial P}{\partial z}-\frac{1}{r}\frac{\partial }{\partial r}\left(r{\tau }_{rz}\right),\phantom{\rule{1.em}{0ex}}0

The boundary conditions are symmetry at $r=0$ and no slip at $r=R$ .

$\begin{array}{c}{\left({\tau }_{rz}|}_{r=0}=-\mu {\left(\frac{\partial {v}_{z}}{\partial r}|}_{r=0}=0\hfill \\ {v}_{z}=0,\phantom{\rule{1.em}{0ex}}r=R\hfill \end{array}$

From the radial component of the equations of motion, $P$ does not depend on radial position. Since the flow is steady and fully developed, the gradient of $P$ is a constant. The $z$ component of the equations of motion can be integrated once to derive the stress profile and wall shear stress .

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