Explanation Given information: The flow is steady, incompressible, parallel, two dimensional and laminar. The velocity components are u s and u n and angle of wall is α . Write the expression for the continuity equation. ∂ u s ∂ s + ∂ u n ∂ n = 0 ...... (I) Here, The velocity components are u s , and u n time is t and position coordinates of flow field are s and n . Write the expression for the momentum conservation in n-direction. ( u s ∂ u n ∂ s + u n ∂ u n ∂ n ) = − 1 ρ ∂ P ∂ n + υ ( ∂ 2 u n ∂ s 2 + ∂ 2 u n ∂ n 2 ) + g n ...... (II) Here, pressure of the fluid is P , kinematic viscosity of fluid is υ , gravity in n-direction is g n . Write the expression for the momentum conservation in s-direction. ( u s ∂ u s ∂ s + u n ∂ u s ∂ n ) = − 1 ρ ∂ P ∂ s + υ ( ∂ 2 u s ∂ s 2 + ∂ 2 u s ∂ n 2 ) + g s ...... (III) Here, gravity in s-direction is g s . Write the expression for the kinematic viscosity. υ = μ ρ Here, density is ρ and dynamic viscosity is μ . Calculation: Substitute 0 for u n in Equation (I). ∂ u s ∂ s + ∂ ( 0 ) ∂ n = 0 ∂ u s ∂ s = 0 Substitute 0 for u n , 0 for ∂ u s ∂ s and − g cos α for g n in Equation (II). ( u s ( 0 ) + ( 0 ) ( 0 ) ) = − 1 ρ ∂ P ∂ n + υ ( 0 + 0 ) + ( − g cos α ) 0 = − 1 ρ ∂ P ∂ n − g cos α ∂ P ∂ n = − ρ g cos α ...... (IV) Integrate Equation (III) with respect to n . P = − ρ g n cos α + f ( s ) ....... (V) Substitute h for n and P atm for P in Equation (IV). P atm = − ρ g h cos α + f ( s ) f ( s ) = P atm + ρ g h cos α ....... (VI) Substitute P atm + ρ g h cos α for f ( s ) in Equation (V). P = − ρ g n cos α + P atm + ρ g h cos α = P atm + ρ g ( h − n ) cos α ....... (VII) Differentiate Equation (VII) with respect to s . ∂ P ∂ s = 0 Substitute 0 for u n , 0 for ∂ u s ∂ s , 0 for ∂ P ∂ s and g sin α for g s in Equation (III). ( u s ( 0 ) + ( 0 ) ∂ u s ∂ n ) = − 1 ρ ( 0 ) + υ ( ( 0 ) + ∂ 2 u s ∂ n 2 ) + ( g sin α ) 0 = υ ∂ 2 u s ∂ n 2 + g sin α ∂ 2 u s ∂ n 2 = − g sin α υ ...... (VIII) Integrate Equation (VIII) with respect to n . ∂ u s ∂ n = − g sin α υ n + C 1 ....... (IX) Here, arbitrary constant is C 1 . Again, Integrate Equation (IX) with respect to n . u s = − g sin α 2 υ n 2 + C 1 n + C 2

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ISBN: 2810022150991

Author: CENGEL

Publisher: MCG

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Chapter 9, Problem 90P

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The velocity field.

The plot of velocity profile.

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Chapter 9, Problem 89P

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The velocity field. The plot of velocity profile.

Solution Summary: The author explains the velocity field and plot of velocity profile. The flow is steady, incompressible, parallel, two dimensional, and laminar.

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The velocity field. The plot of velocity profile.

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Chapter 9 Solutions

The velocity field.

The plot of velocity profile.