A fluid flows steadily between two parallel plates. The flow is fully developed and laminar. The

step 1 Here the velocity profile, let's solve the part A first. The velocity profile for laminar flow b

A fluid flows steadily between two parallel plates. The flow...

Key Concepts

Momentum Balance Equation

The momentum balance equation is fundamental in fluid mechanics as it provides a relation between the forces acting on a fluid element, such as pressure and viscous forces. In deriving the shear stress distribution, the momentum balance helps relate the constant pressure gradient to the spatial variation of the shear stress across the flow domain.

Pressure Gradient

The pressure gradient is the spatial rate of change of pressure and serves as the primary driving force in many fluid flow problems, including flow between parallel plates. It directly influences the velocity profile and, through the momentum balance, the distribution of shear stress in the fluid.

Dynamic Viscosity

Dynamic viscosity is a measure of a fluid's resistance to shear or flow. It appears in constitutive equations relating the shear stress to the rate of strain (velocity gradient) in the fluid. Its value is critical in determining the magnitude of shear stress for a given velocity profile in viscous flows such as between parallel plates.

Shear Stress Distribution

Shear stress in a fluid arises from viscosity as a result of velocity gradients perpendicular to the direction of flow. In the context of flow between parallel plates, the shear stress can be derived by balancing the pressure forces and viscous forces in a differential control volume, leading to a linear variation across the gap between the plates.

Laminar Flow

Laminar flow is characterized by smooth, orderly fluid motion in distinct layers, with little to no mixing between them. In this regime, viscous forces dominate over inertial forces, which makes the mathematical treatment of flow simpler and allows for analytical solutions such as those found in plane Poiseuille flow between parallel plates.

Fully Developed Flow

In fluid mechanics, fully developed flow refers to a flow regime where the velocity profile does not change along the direction of flow. This means that the shape of the profile has reached a steady state and the derivatives with respect to the streamwise direction vanish, simplifying the analysis and derivation of other flow properties such as the shear stress distribution.

Transcript

00:01 Here the velocity profile, let's solve the part a first.

00:04 The velocity profile for laminar flow between stationary parallel clarity is given as u is equals to minus h squared divided by 8 mu multiplied by 1 minus 2 y divided by h whole square multiplied by del p divided by del x.

00:25 Let this equation v equation number 1.

00:27 Also we know that c r stress tau x can can be written as mu du by d y, mu d u by d y, d u divided by d y.

00:39 Let this equation be question number two.

00:42 Now we will put this equation now substitute the value of velocity from equation 1 into equation 2 we get t tau y x is equal to mu multiplied by d divided by d y put the value of velocity as minus h squared divided by 8 mu multiplied by del p divided by del x multiplied by 1 minus 2 i divided by h whole square bracket closed from here we get tau x x is equal to taw x is equals to minus mu multiplied by h squared by 2 mu h squared divided by 2 mu minus del p by del x x multiplied by d divided by d divided by d by d divided by d by d divided by d by d minus 4 y squared divided by h square so upon further solving we get tau x square tau x is equal to tau y x is equal to h squared divided by 2 del p divided of del x multiplied by 0 minus 0 minus 8 y divided by h square or we get this as or we will get this as tau y x is equal to tau x is equal to h square minus 4 h square y divided by h square or this will be equals to minus 4 y here we have a slight change here it will be 8 as here we have written the value of you is below so the denominator is 8 mu.

03:00 8 will be 8 mu.

03:02 Well, now this will cut each other and here we have only h squared by x squared y multiplied by del p divided by del x.

03:16 This is the final term.

03:18 And finally we will have a value of taw y x will be equal to it will be equal to y d y multiplied by del p divided by del x so this is the final value of tau y x hence the equation for the c r states as a function of y will be tau y x will be minus del p divided by del x multiplied by y.

04:06 So this is the shear stress function.

04:10 Now as the velocity increases, the change in pressure decreases.

04:15 So as velocity will increase, the change in pressure del p, del p divided by del x will decrease.

04:27 So, so, cr stress tau x is less than 0 when y is greater than 0.

04:37 Also, tau y x is greater than 0 when y is less than 0...

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