Is the following velocity field a possible incompressible flow? ν(x,y,z,t)=ρg sin θ/μ(yh-y^2/2)i where ρ is the mass density of the fluid, g (=9.81 m/s^2) is the gravitational constant, μ is the viscosity of the fluid, θ is some fixed angle (number) relative to a horizontal datum, and h is some depth of a fluid film.

Question

Is the following velocity field a possible incompressible flow? [latex] u (x,y,z,t)=\frac{ ho g\sin heta }{\mu }\left(yh-\frac{y^{2}}{2} ight)\hat{i}[/latex] where ρ is the mass density of the fluid, [latex]g (=9.81 m/s^{2})[/latex] is the gravitational constant, [latex]\mu[/latex] is the viscosity of the fluid, [latex] heta[/latex] is some fixed angle (number) relative to a horizontal datum, and h is some depth of a fluid film.

Accepted Answer

[latex] riangledown \cdot u =\frac{\partial}{\partial x} \left[\frac{ ho g\sin heta }{\mu }\left(yh-\frac{y^{2}}{2} ight) ight]+\frac{\partial}{\partial y}(0)+\frac{\partial}{\partial z}(0)=0;[/latex]

so yes, this is a possible incompressible flow field. This velocity field will be determined formally in Example 9.3 via the solution of the equation of motion for a particular boundary value problem.

Question 8.1: Is the following velocity field a possible incompressible fl...

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