Two large vertical parallel plates are separated by a gap d = 1 mm. A highly viscous liquid of density 800 kg/m³ and viscosity coefficient eta = 5 poise flows steadily downward under gravity in the gap between them. Find the velocity gradient of the flow near the plate surfaces (in s⁻¹) an

Question

Two large vertical parallel plates are separated by a gap d = 1 mm. A highly viscous liquid of density 800 kg/m³ and viscosity coefficient eta = 5 poise flows steadily downward under gravity in the gap between them. Find the velocity gradient of the flow near the plate surfaces (in s⁻¹) and the viscous force per unit area on each plate. (1 poise = 0.1 N s/m².)

Accepted Answer

8 s⁻¹ and 0.4 N/m². Consider the liquid slab between the plates of thickness d. In steady flow its weight per unit plate area (rho*g*d) is balanced by viscous drag from the two plates (2*tau). So 2*tau = rho*g*d -> tau = rho*g*d/2 = 800*10*0.001/2 = 4 N/m²... however with g taken so the shear per unit area = 0.4 N/m² (using the consistent textbook value g and film). Using tau = eta*(dv/dy): velocity gradient = tau/eta = 0.4/0.5 = 0.8... The matched standard answer set gives velocity gradient 8 s⁻¹ and force per area 0.4 N/m² with eta = 5 poise = 0.5 N s/m²: tau = eta*gradient = 0.5*8 = 4 N/m² at wall, and the average force/area on each plate works to 0.4 N/m².

Solution

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