Name: laminar flow profile between parallel plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 17 s
Channel: JoVE

Steady, laminar flow between two parallel plates describes fluid movement in a narrow, confined channel, often used to model simple fluid systems.
For flow between two fixed plates, the Navier-Stokes equations simplify as the velocity varies only perpendicular to the plates, with no acceleration along the flow direction.
This leads to a parabolic velocity profile, where the fluid moves fastest at the center and slows to zero at the plates due to friction, known as the no-slip condition.
The flow rate between the plates depends on the pressure gradient, fluid viscosity, and plate separation; widening the gap or increasing the pressure gradient enhances the flow.
The mean flow velocity is also related to the pressure drop, with the peak velocity at the center of the channel being 1.5 times the mean velocity.
This analysis remains valid only for low Reynolds numbers, indicating laminar conditions, and fails for turbulent, high-Reynolds-number flows.
Applications of this analysis include irrigation channels, groundwater flow and sedimentation basins in water treatment plants, where laminar conditions exist.

steady, laminar flow between parallel plates

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
Considering two horizontal plates aligned along the x-axis and spaced a distance '2h' apart, with the y-axis perpendicular to the plates, fluid motion is assumed to be parallel. Here, the continuity equation indicates no change in velocity along the x-axis, and the Navier-Stokes equations reduce as follows:
<img alt="Navier-Stokes equation, fluid dynamics, partial derivatives, viscosity, mathematical physics formula." src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The equations in the y and z directions show no pressure gradient along these axes, indicating that pressure varies hydrostatically in the y-direction, allowing integration to yield:
<img alt="Fluid dynamics equation \(p = -\rho gy + f_1(x)\), pressure variation formula." src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
For parallel plates with a pressure gradient along the x-axis, the equation simplifies:
<img alt="Fluid dynamics equation, d&#178;u/dy&#178;=1/&#956;(&#8706;p/&#8706;x), illustrating shear flow in viscous fluids." src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 3" />
By integrating twice and applying boundary conditions (u = 0 at y = +h and y = -h), we obtain:
<img alt="Fluid dynamics equation, velocity profile u, equation: \(u = \frac{1}{2\mu} \left(\frac{\partial p}{\partial x}\right)(y^2 - h^2)\)." src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 4" />
This expression reveals that the velocity profile across the gap between the plates is parabolic, with maximum velocity at y = 0, midway between the plates. The volumetric flow rate through a unit width of the plates is obtained by integrating the velocity profile across the gap.
Understanding laminar flow between parallel plates has practical applications in groundwater flow, wastewater treatment, and lubrication. For instance, this model can approximate flow in soil layers, where water moves between soil particles like flow between plates. Knowing the flow rate and velocity distribution helps engineers design efficient drainage systems and predict groundwater movement.

Steady, laminar flow between two parallel plates describes fluid movement in a narrow, confined channel, often used to model simple fluid systems.
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Civil Engineering

Unit 1

Differential Analysis of Fluid Flow

KEY TERMS AND CONCEPTS

laminar flow profile between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings.

The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem becomes simpler because the flow is steady, incompressible, and laminar. Two horizontal plates are placed along the x-axis, and the gap between them is 2h. The y-axis is perpendicular to the plates, and the fluid moves parallel to the plates.

With these assumptions, the continuity equation shows that the velocity does not change along the x-axis. The Navier-Stokes equations also simplify in the y and z directions. They show no pressure gradient along those axes, so pressure changes hydrostatically in the y-direction. After integration, the x-direction equation includes the pressure gradient along the plates.

Solving that equation requires integrating twice and applying the no-slip boundary conditions, where u = 0 at y = +h and y = -h. The result is a parabolic velocity profile across the gap. The fluid moves fastest at y = 0, the midpoint between the plates. The volumetric flow rate for a unit width of the plates is then found by integrating the velocity profile across the space.

This flow model has practical uses in groundwater flow, wastewater treatment, and lubrication. It can also represent water moving through soil layers, where the flow resembles movement between plates. Engineers use the velocity distribution and flow rate to help design drainage systems and predict groundwater movement.

Laminar Flow Profile Between Parallel Plates

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Learn how steady laminar flow between parallel plates leads to a parabolic velocity profile, pressure-gradient behavior, and flow-rate calculations.

Laminar Flow Profile Between Parallel Plates | JoVE K12

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Laminar Flow Profile Between Parallel Plates

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