327lec-ns - 530.327 - Introduction to Fluid Mechanics - Su...

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Unformatted text preview: 530.327 - Introduction to Fluid Mechanics - Su Differential forms – the Navier-Stokes equations Reading: § 5.4. With what we learned in the last lecture, we are ready to derive the Navier-Stokes (N-S) equations. We will only be concerned here with the N-S equations for incompressible flow. Our procedure will be to apply the control volume form of the momentum equation to the infinitesimal control volume shown in Fig. 1. The volume element is aligned with the standard x − y − z coordinate system, z y x dz dx dy 5 1 3 2 4 6 z y 1 dy dz z x 5 dx dz 2 5 6 3 3 6 4 1 Figure 1: A general rectangular volume element. with the velocity being written as −→ V = u ˆ i + v ˆ j + w ˆ k, and where gravity points in the negative z-direction. The control volume formulation of the momentum equation is III z }| { Σ −→ F = IV z }| { ∂ ∂t Z CV ρ −→ V d ∀ + V z }| { Z CS ρ −→ V ( −→ V · d −→ A ) , (1) The forces on the left side are body forces (gravity) and surface forces (pressure and viscosity). Equation 1 is a vector equation; we’ll look first at the x-component of the equation. We’ll begin with the right side of the equation. To evaluate term IV, we note as before that for the infinitesimal control volume, we can treat the density and velocity as constant throughout the interior of the volume. Term IV thus becomes Term IV, x-component: ∂ ∂t Z CV ρ −→ V d ∀ = ∂ρu dx dy dz ∂t = ρ ∂u ∂t dx dy dz, (2) where in the last step, we note that the flow is incompressible, so ρ =const., and the control volume is fixed in space, so dx dy dz can also be taken out of the time derivative. Flux terms. Now look at term V. This is a surface integral, and there will be separate contributions from each of the six faces of the control volume. These integrals are evaluated in similar fashion to the surface integrals in the momentum conservation equation from the last lecture. Face 1 is located 1 at x = dx/ 2, and its area is d −→ A = dy dz ˆ i...
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This note was uploaded on 04/07/2008 for the course MECHENG 327 taught by Professor Su during the Spring '08 term at Johns Hopkins.

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327lec-ns - 530.327 - Introduction to Fluid Mechanics - Su...

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