GATE ME 2023 Set 1 — Question 63
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Fluid Mechanics → Fluid Kinematics & Dynamics → Continuity & Momentum Equations
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Question
Consider a unidirectional fluid flow with the velocity field given bywhere . If the spatially homogeneous density field varies with time asthe value of is ______________. (Rounded off to two decimal places)Assume all quantities to be dimensionless.
Correct answer
1.1 to 1.2
Solution
For a compressible fluid flow, the continuity equation is:Given that the density field is spatially homogeneous, , so . The equation simplifies to:Rearranging for the velocity gradient:Given , we find the time derivative:Substituting these into the gradient expression:Integrating with respect to :Using the boundary condition :Thus, the velocity field is:To find , substitute and :Rounding to two decimal places, .
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