Velocity Profile

This simulation shows the behavior of three immiscible, incompressible fluids in laminar flow. The fluids are layered vertically and subjected to steady-state Couette flow; the top plate moves to the right at a constant velocity, and the bottom plate is stationary. At each height, the black arrow is proportional to the velocity of the fluid at that height. You can change the viscosities and heights of fluids 2 and 3 with sliders. The velocity distribution shows the impact of fluid viscosity on the fluid's velocity gradient. The velocity versus height plot indicates the relative slopes (or velocity gradients) in each fluid. All numbers are dimensionless.

Couette flow is fluid flow through parallel plates where one plate is stationary and the other is moving at a constant velocity. In this Demonstration, the flow is at steady-state and fully developed. The Navier-Stokes equations in the $x$ and $y$ directions are used:

$$ \rho g_x - \frac{\partial P}{\partial x} + \mu \left(\frac{\partial^{2} u_x}{\partial x^{2}} + \frac{\partial^{2} u_x}{\partial y^{2}} + \frac{\partial^{2} u_x}{\partial z^{2}}\right) = \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y} + u_z \frac{\partial u_x}{\partial z}\right), $$ $$ \rho g_y - \frac{\partial P}{\partial y} + \mu \left(\frac{\partial^{2} u_y}{\partial x^{2}} + \frac{\partial^{2} u_y}{\partial y^{2}} + \frac{\partial^{2} u_y}{\partial z^{2}}\right) = \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y} + u_z \frac{\partial u_y}{\partial z}\right). $$

Since flow is steady-state and there is no flow in the $z$ direction, these equations simplify to:

$$ \rho g_x - \frac{\partial P}{\partial x} + \mu \left(\frac{\partial^{2} u_x}{\partial x^{2}} + \frac{\partial^{2} u_x}{\partial y^{2}}\right) = \rho \left(u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y}\right), $$ $$ \rho g_y - \frac{\partial P}{\partial y} + \mu \left(\frac{\partial^{2} u_y}{\partial x^{2}} + \frac{\partial^{2} u_y}{\partial y^{2}}\right) = \rho \left(u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}\right). $$

Flow is in the $x$ direction, so $u_y$ drops out:

$$ \rho g_x - \frac{\partial P}{\partial x} + \mu \left(\frac{\partial^{2} u_x}{\partial x^{2}} + \frac{\partial^{2} u_x}{\partial y^{2}}\right) = \rho u_x \frac{\partial u_x}{\partial x}, $$ $$ \rho g_y - \frac{\partial P}{\partial y} = 0. $$

Since the flow is fully developed the velocity in the direction of flow does not change. It is assumed that the pressure is hydrostatic so pressure does not vary in the $x$ direction:

$$ \mu \frac{\partial^2 u_x}{\partial y^2} = 0, $$ $$ \rho g_y = \frac{\partial P}{\partial y}. $$

For the boundary conditions $ u_x = 0 $ at $ y = 0 $ and $u_x = U$ at $ y = H $, the simplified Navier-Stokes eqation is:

$$ u_x = \frac{U}{H} y, $$

where $\rho$ is density, $g$ is gravity, $P$ is pressure, $\mu$ is viscosity, $u$ is fluid velocity, $U$ is plate velocity, $t$ is time, and $H$ is the distance between plates.

Reference

B. R. Munson, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2010.