The channel for a liquid is formed by two fixed plates. If...

The channel for a liquid is formed by two fixed plates. If laminar flow occurs between the plates, show that the Navier-Stokes and continuity equations reduce to $\partial^{2} u / \partial y^{2}=(1 / \mu) \partial p / \partial x$ and $\partial p / \partial y=0 .$ Integrate these equations to show that the velocity profile for the flow is $u=(1 /(2 \mu))(d p / d x)\left[y^{2}-(d / 2)^{2}\right] .$ Neglect the effect of gravity.

Key Concepts

Navier?Stokes Equations

These are the fundamental equations governing fluid motion, derived from the conservation of momentum. They include terms that account for the inertial forces, pressure forces, viscous forces, and any external forces such as gravity. Under certain simplifying assumptions, such as steady, incompressible, and laminar flow, these equations can be reduced to a simpler form that is easier to analyze and solve.

Continuity Equation

This equation represents the conservation of mass in a fluid flow. For an incompressible fluid, it simplifies to a statement that the divergence of the velocity field is zero. In many flow problems, such as channel flows, the continuity equation helps in reducing the complexity by linking different velocity components and ensuring that mass is neither created nor destroyed.

Laminar Flow

Laminar flow refers to a flow regime characterized by smooth, orderly motion of fluid particles, with layers sliding past each other without mixing. In such flows, the viscous effects dominate over inertial effects, which permits the neglect of convective acceleration terms in the Navier?Stokes equations, leading to simpler forms that are amenable to analytical solutions.

Pressure Gradient

In fluid dynamics, a pressure gradient refers to the spatial rate of change of pressure, which acts as a driving force for fluid flow. In many practical situations like Poiseuille flow, the constant pressure gradient along the flow direction is the primary factor that drives the motion, and its role becomes central in the derivation of the velocity profile for the fluid.

Integration of Differential Equations with Boundary Conditions

Once the governing differential equations are simplified, they are typically ordinary differential equations that can be integrated to obtain the velocity profile. The integration process involves determining constants of integration, which are physically interpreted and computed using boundary conditions such as the no?slip condition on solid surfaces.

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