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Chapter 1: Problem 16

Algebraic equations such as Bernoulli's relation, Eq. of Example \(1.3,\) are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer \(x\) -momentum equation, first derived by Ludwig Prandtl in 1904: $$\rho u \frac{\partial u}{\partial x}+\rho v \frac{\partial u}{\partial y}=-\frac{\partial p}{\partial x}+\rho g_{x}+\frac{\partial \tau}{\partial y}$$ where \(\tau\) is the boundary-layer shear stress and \(g_{x}\) is the component of gravity in the \(x\) direction. Is this equation dimensionally consistent? Can you draw a general conclusion?

Short Answer

Expert verified
The equation is dimensionally consistent; all terms have the same dimensions.

Step by step solution

01

Understand the Terms

First, identify the dimensional terms in the equation. The terms are: \( \rho \), which is density (\( ML^{-3}\)); \( u \) and \( v \), which are velocities (\( LT^{-1}\)); \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \), which are velocity gradients (\( T^{-1} \)); \( p \) is pressure (\( ML^{-1}T^{-2} \)); \( g_x \) is the gravitational acceleration component (\( LT^{-2} \)); and \( \tau \), the shear stress (\( ML^{-1}T^{-2} \)).
02

Analyze Each Term's Dimensions

Evaluate each term's dimensions:- \( \rho u \frac{\partial u}{\partial x} = (ML^{-3})(LT^{-1})(T^{-1}) = ML^{-2}T^{-2} \)- \( \rho v \frac{\partial u}{\partial y} = (ML^{-3})(LT^{-1})(T^{-1}) = ML^{-2}T^{-2} \)- \( \frac{\partial p}{\partial x} = (ML^{-1}T^{-2})(L^{-1}) = ML^{-2}T^{-2} \)- \( \rho g_x = (ML^{-3})(LT^{-2}) = ML^{-2}T^{-2} \)- \( \frac{\partial \tau}{\partial y} = (ML^{-1}T^{-2})(L^{-1}) = ML^{-2}T^{-2} \)
03

Compare Dimensions of the Equation

Compare the dimensions of all terms on both sides of the equation: Each term has the dimension \( ML^{-2}T^{-2} \), indicating that the equation is dimensionally consistent.
04

Draw General Conclusion

Since every term on each side of the differential equation has the same dimensional formula (\( ML^{-2}T^{-2} \)), the equation is dimensionally consistent. Thus, properly formulated differential equations maintain dimensional consistency as a rule.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boundary-Layer Theory
Boundary-layer theory is a fascinating area of fluid dynamics first introduced by Ludwig Prandtl in 1904. It primarily focuses on the behavior of fluid flow near solid boundaries. When a fluid flows over a surface, such as an airfoil, it experiences drag forces due to the viscosity of the fluid. This thin region, where the effects of viscosity are significant, is known as the boundary layer.

In this layer, viscous forces greatly affect the velocity profile of the fluid. Inside the boundary layer, there's a noticeable velocity gradient moving from the fluid interface to the surface. That means, while the fluid's speed is high far from the boundary, it decreases to zero at the boundary because of no-slip conditions.

Understanding the boundary-layer phenomenon allows engineers to optimize designs to reduce drag or enhance heat transfer, making it crucial in industries like aerospace and automotive engineering.
Momentum Equation
The momentum equation is an essential component of Newtonian physics applied in fluid dynamics. It expresses the balance of forces acting on a fluid element within the framework of Newton's Second Law of Motion. Fundamentally, it states that any change in the momentum of a fluid is equal to the sum of external forces acting on it, accounting for pressure, viscous, and body forces.

For boundary-layer flows, the momentum equation highlights the interaction between pressure gradients and viscous forces. Specifically, the equation detailed here involves terms like pressure gradient ( abla p ep e), gravitational components, and shear stress ( abla au ep e).

By analyzing each term's dimensionality, as done in the original solution, consistency can be verified. This consistency is crucial for ensuring that the physical predictions made by the equation are correct and dependable.
Differential Equations
Differential equations are mathematical tools used to describe the relationship between functions and their derivatives. They are incredibly powerful in modeling numerous physical phenomena, including fluid flows, like those described in boundary-layer theory.

In essence, differential equations, such as the boundary-layer momentum equation, are used to capture changes across infinitesimally small regions of space and time.

The equation in the exercise is a partial differential equation because it involves derivatives with respect to more than one independent variable. This type of differential equation is employed when dealing with fields like temperature, velocity, and pressure, which depend on several variables.

Proper dimensional analysis, as illustrated by ensuring each term has the same dimensions, guarantees that these equations will give meaningful and accurate solutions applicable in real-world scenarios. Understanding how each part of a differential equation interacts allows for a robust analysis and solution of complex physical problems.

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Most popular questions from this chapter

Oil, with a vapor pressure of \(20 \mathrm{kPa}\), is delivered through a pipeline by equally spaced pumps, each of which increases the oil pressure by 1.3 MPa. Friction losses in the pipe are 150 Pa per meter of pipe. What is the maximum possible pump spacing to avoid cavitation of the oil?

A dimensionless parameter, important in natural convection heat transfer of fluids, is the Grashof number: \\[\mathrm{Gr}=\frac{g \beta \rho^{2} L^{3} \Delta T}{\mu^{2}}\\] where \(g\) is the acceleration of gravity, \(\beta\) is the thermal expansion coefficient, \(\rho\) the density, \(L\) a characteristic length, \(\Delta T\) a temperature difference, and \(\mu\) the viscosity. If the uncertainty of each of these variables is 卤2 percent, determine the overall uncertainty of the Grashof number.

An amazing number of commercial and laboratory devices have been developed to measure fluid viscosity, as described in Refs. 29 and \(49 .\) Consider a concentric shaft, fixed axially and rotated inside the sleeve. Let the inner and outer cylinders have radii \(r_{i}\) and \(r_{o},\) respectively, with total sleeve length \(L\). Let the rotational rate be \(\Omega(\mathrm{rad} / \mathrm{s})\) and the applied torque be \(M .\) Using these parameters, derive a theoretical relation for the viscosity \(\mu\) of the fluid between the cylinders.

A two-dimensional unsteady velocity field is given by \(u=\) \(x(1+2 t), v=y .\) Find the equation of the time-varying streamlines that all pass through the point \(\left(x_{0}, y_{0}\right)\) at some time \(t .\) Sketch a few of these.

Books on porous media and atomization claim that the viscosity \(\mu\) and surface tension \(Y\) of a fluid can be combined with a characteristic velocity \(U\) to form an important dimensionless parameter. (a) Verify that this is so. (b) Evaluate this parameter for water at \(20^{\circ} \mathrm{C}\) and a velocity of \(3.5 \mathrm{cm} / \mathrm{s} .\) Note: You get extra credit if you know the name of this parameter.
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