What Is a Fluid? — Question 3

Question 3

Separate a wall force into its meaningful parts

A small planar wall patch touches an ordinary fluid at rest. Neglect surface tension and other interfacial effects. In Cartesian coordinates, the outward unit normal from the fluid is $\hat{\mathbf n}=(3\hat{\mathbf i}+4\hat{\mathbf j})/5$. A proposed force exerted by the fluid on the patch is $\mathbf F=(10\hat{\mathbf i}+5\hat{\mathbf j})\,\mathrm{N}$. Let $\hat{\mathbf t}=(-4\hat{\mathbf i}+3\hat{\mathbf j})/5$ be a tangent.

1. Compute the signed normal and tangential components. Is this force compatible with the stated static-fluid model?

2. Find the force vector that retains the proposed normal component but removes the incompatibility. Explain why a nonzero normal force is allowed.

Question 3 - Solution

Separate a wall force into its meaningful parts

(a) Derivation and calculation

$$\hat{\mathbf n}\cdot\hat{\mathbf t}=0,\qquad |\hat{\mathbf n}|=|\hat{\mathbf t}|=1.$$$$\boxed{F_n=\mathbf F\cdot\hat{\mathbf n}=10(3/5)+5(4/5)=10\,\mathrm N}.$$$$\boxed{F_t=\mathbf F\cdot\hat{\mathbf t}=10(-4/5)+5(3/5)=-5\,\mathrm N}.$$The nonzero tangential force is incompatible with the stated static-fluid model.

(b) Result and interpretation

$$\boxed{\mathbf F_\perp=F_n\hat{\mathbf n}=(6\hat{\mathbf i}+8\hat{\mathbf j})\,\mathrm N}.$$$$\mathbf F-\mathbf F_\perp=(4\hat{\mathbf i}-3\hat{\mathbf j})\,\mathrm N=-5\hat{\mathbf t}\,\mathrm N.$$Normal compression is allowed; static shear resistance is excluded.

Physical check

The corrected vector has magnitude 10 N and zero projection on the tangent.