The phenomenon of downhill skiing or snowboarding, in its simplest form, refers to the motion of a human being sliding down an inclined plane on a porous medium (snow layer) (Fig. 1A). The lift mechanics of downhill skiing or snowboarding is intended to answer a fundamental question, that is, why a 70-kg human can glide through a soft snow layer without sinking into the base as would occur if the motion is arrested. This question was recently answered by Wu et al. (^{15,18}), who developed a simplified mathematic model to determine the pore pressure generation inside a snow layer as a planing surface glides over it. The theory, hereafter called the Wu-Weinbaum theory, was based on the idea of lift generation in a highly compressible porous media, a new concept in porous media flow. It was the first realistic model that incorporated the lift contributions from both the transiently trapped air inside a snow layer and that from the compressed solid phase (ice crystals). However, this theory has two major limitations. First, it is only applicable to the simple case where the planing surface is of constant width in the axial direction (Fig. 1B), whereas commercially available skis/snowboards have complex geometries and variable width to reduce weight and improve performance while maintaining the strength and rigidity of the planing surface (Fig. 1C). Second, for a certain gliding condition such as the board geometry, the speed of the skier or snowboarder, the frictional coefficient between the board surface and the snow layer, the position of the skier's center of gravity, and the properties of snow, the Wu-Weinbaum theory only predicts a single neutral stability position because of its limitations on the numerical simulation. However, an experienced skier or snowboarder would tell you that there are multiple equilibrium positions instead of just one. In the current study, we shall perform a systematic study to treat these limitations. The Wu-Weinbaum theory will be extended to more complex shapes where the width of the planing surface changes with the axial location. A modified mathematical model will be developed where a width factor, *f*(*x*), that characterizes the variation of width from the leading to the trailing edge is introduced. We shall thoroughly reexamine the validity of the Wu-Weinbaum theory by implementing significantly improved numerical simulations for a rectangular ski/snowboard, in which case, *f*(*x*) = 1. Furthermore, we shall evaluate the performance of a commercial snowboard of certain geometric characteristics on the basis of our new model, which provides a new viewpoint for the optimization of a ski or snowboard.

Lift generation in a soft porous medium under rapid compaction is a new concept for porous media flow. This concept was originally proposed by Feng and Weinbaum (^{7}) when they comparatively examined the motion of a red blood cell moving in a tightly fitting capillary and a human skiing or snowboarding on fresh powder snow. It is well accepted that the luminal surface of the endothelial cells that line our vasculature is coated with a thin (∼500 nm) negatively charged endothelial glycocalyx layer (EGL) of proteoglycans, glycoproteins, and glycosaminoglycans. From a structural standpoint, the EGL is a remarkably resilient fibrous porous material. It is a hydrated gel that in its fully distended state is >90% water (^{6}). One of the fundamental mysteries of microcirculation is how an 8-μm red cell is able to squeeze through a blood vessel of 5-6 μm in diameter, glide over the EGL with almost negligible friction, and survive 10^{5} passages without being damaged or undergoing hemolysis. This question, having puzzled biologists since the motion of red cells was discovered, was answered by Feng and Weinbaum (^{7}), who developed a new lubrication theory for highly compressible porous media and observed an intriguing hydrodynamic similarity between the motion of a red cell moving in a capillary and a human skiing or snowboarding, although they differ in mass by 10^{15}. The theory demonstrates that the excess pore pressure generated by a planing surface moving on a compressed porous layer scales as *α*^{2} = *h*_{2}^{2}/*K*, where *h*_{2} is the layer thickness at the leading edge of the planing surface (Fig. 1B) and *K* is the Darcy permeability, and that *α* is 10^{2} or larger for both red blood cells gliding over the EGL and a human skiing or snowboarding on fresh powder snow. Thus, the lift forces generated can be more than 4 orders of magnitude greater than the classical lubrication theory. The huge enhancement in the lift arises from the fact that as the porous medium (snow or glycocalyx) compresses, there is a dramatic increase in the lubrication pressure because of the marked increase in the hydrodynamic resistance that the fluid (air or blood plasma) encounters as it tries to escape from the confining boundary through the compressed porous layer. It explains the pop-out phenomenon for a red blood cell squeezing through a small capillary, as well as the question of why a 70-kg human can glide over a soft snow layer without sinking to the base as would occur if the motion is arrested.

Although the new lubrication theory for highly compressible porous media developed in Feng and Weinbaum (^{7}) illustrated the striking similarity between a red cell and a human skiing, direct experimental validation of this theory remained difficult because the dimensions of the red cells are obviously too small to obtain the detailed pressure measurements. In a recent study by Wu et al. (^{13,14,16}), the authors circumvented this difficulty by examining the dynamic behavior of snow under rapid compaction. The objective was to verify Feng and Weinbaum's (^{7}) lubrication theory for highly compressible porous media and deduce the behavior of red cells "skiing" on the EGL. At typical alpine skiing velocities of 10-30 m**·**s^{−1}, the duration of contact of the ski with the snow varies from 0.05 to 0.2 s, depending on ski length. Wu et al. (^{16}) performed a series of experiments with different types of snow using a specially designed porous-walled cylinder-piston apparatus. The piston was released and allowed to fall under its own weight on the snow sample in the porous cylinder to create a sudden compaction. The excess pore pressure of the transiently trapped air was captured and compared with a consolidation theory. Using previously determined data of the Darcy permeability of snow (^{8}), the theoretical results reasonably matched the experimental data. For a wind-packed snow layer, the excess pore pressure generation occurred in a period of 0.1 s and relaxed in 0.7 s. For a snowboarder gliding over wind-packed snow with a velocity of 10 m**·**s^{−1}, if the length of the snowboard was equal to 1.5 m, the approximate time that the snowboard remained in contact with the snow layer was 0.15 s. This inferred that sufficient time was available to adequately support the weight of the snowboard by the generated pore air pressure. The basic physics underlying the lift generation in soft porous media proposed by Feng and Weinbaum (^{7}) was qualitatively verified.

The fundamental insights gained from these studies on the lift generation in porous media provide a new perspective for understanding the phenomenon of skiing and have further led to the Wu-Weinbaum theory, the first realistic model for the lift mechanics of downhill skiing and snowboarding (^{15,18}). This theory incorporates lift contributions from both the transiently trapped air and the compressed ice crystals. It predicts that for fine-grained wind-packed snow, when the velocity of the skier is 20 m**·**s^{−1}, approximately 50% of the total lift force is generated by the trapped air for snowboarding and 40% for skiing. For highly permeable fresh powder snow, the lift contribution from the pore air pressure drops to <20%. It captures the key physics of stability and control during skiing or snowboarding and realistically predicts the performance of skiing or snowboarding as a function of the skier's velocity, the sliding friction between the planing surface and snow, various snow types, as well as the geometry of a ski or snowboard (*L*/*W*, where *L* is the length and *W* is the width of the board). However, as mentioned previously, the Wu-Weinbaum theory needs to be reexamined because of its limitations on the numerical simulations as well as its application being constrained to rectangular boards. The current article will be focused on the treatments of these limitations, which includes 1) providing corrections on the predictions of the Wu-Weinbaum theory by implementing improved numerical methods and 2) extending the Wu-Weinbaum theory to more complex shapes where the width of the planing surface changes with the axial location. In the following section, we shall develop a revised mathematic model for the lift mechanics of downhill skiing and snowboarding. This new model, taking into account the variation of width of the planing surface, will be solved numerically. A thorough results and discussion section will be provided on the basis of the numerical solutions. These results will be summarized, from which we shall conclude the article.

#### METHODS

For a skier/snowboarder gliding with velocity *U* (*Ux*, *UY*, *Uz*) over a snow layer as shown in Figure 1A, the sudden compaction of the snow leads to the generation of pore air pressure inside the compressed layer, *Na*, as well as the solid-phase lifting force from the ice crystals, *Ns*. The normal component of the weight *mg*, *FN*, is to be balanced by the summation of *Na* and *Ns*,

where *αh* is the angle of the inclined slope. Equation 1 can also be written in the form of an average pressure by dividing each term by the board area *As*. If one defines the total dimensionless average pressure arising from both the trapped air and the solid phase as *P*_{av}′ = ((*Na* + *Ns*) / *As*) / *P*_{0} = *P*_{ava}′ + *P*_{avs}′ and the total dimensionless loading pressure exerted by the skier or snowboarder as *P*_{avload}′ = *P**mg* / *P*_{0}, where *P**mg* = *mg* cos *αh* / *As*, equation 1 reduces to

Similarly, the sum of all torques about the center of mass (CM) must be zero too, that is,

where *Ff* is the snow friction force, *Ff* = *ηNs* where *η* is the coefficient of friction, *lc* is the normal distance of the CM from the ski surface, and *x*_{ccr}, *xa*, and *xs* are the *x* coordinates of CM, center of *Na*, and center of *Ns*, respectively. The setup of the coordinate system is shown in Figure 1B.

The lift force from the transiently trapped air is obtained by integrating the pore air pressure generated under the ski or snowboard,

where *w*(*x*) is the local width of the planing surface, as shown in Figure 1C, *P* is the pore pressure, and *P*_{0} is the atmospheric pressure.

In the Wu-Weinbaum theory (^{15,18}), a general equation for the pore pressure distribution beneath the planing surface was developed,

where *μ* is the viscosity of the air and *A* is the vertical velocity gradient, *A* = −*Uz*(*x*, *y*, *h*)/*h* where *h* is the local thickness of the snow layer. For the case when there is no lateral tilt, *h* = *h*(*x*), *K* = *K*(*x*), and *A* = (*U*/*h*)*dh*/*dx* where *U* = *Ux*, the pressure distribution in the *y* direction is parabolic (^{19}), and one obtains the centerline pressure distribution beneath a ski or snowboard,

where *Pc*(*x*) is the centerline pressure corresponding to the cross section at the location *x*, *P*_{0} is the pore pressure at the edges of the planing surface that is very close to the atmospheric pressure, and the local Darcy permeability changes as a function of compression, which was obtained by Wu et al. (^{18}) on the basis of Shimizu's (^{11}) empirical relationship,

where *L* is the length of the ski/snowboard; *d* is the mean diameter of the ice crystals; *k* = *h*_{2}/*h*_{1}, where *h*_{2} and *h*_{1} are the local thicknesses of the snow layer at the leading (*x* = *L*) and trailing (*x* = 0) edges, respectively (Fig. 1B); and *K*_{2} is the permeability beneath the leading edge, *K*_{2} = 0.077exp[(*h*_{0}/*h*_{2})ln(*K*_{0}/0.077*d*^{2})]*d*^{2}, where *h*_{0} and *K*_{0} are the undeformed thickness and permeability of the snow layer, respectively.

Wu et al. (^{15,18}) have examined the case when the width of the planing surface is a constant along the axial direction. However, commercially available skis/snowboards have complex geometries and variable width, as shown in Figure 1C. If the nose width is *W*, one defines the width factor of a snowboard as *f*(*x*) = *w*(*x*)/*W*. Introducing dimensionless variables,

we obtain a dimensionless equation for *pc*′(*x*′):

where

At the leading and trailing edges of a ski or snowboard, the pore pressure is close to the atmospheric pressure (^{15,18}), which provides the boundary condition for equation 9, *pc*′(0) = *pc*′(1) = 0. This equation can be solved numerically.

Once *pc*(*x*′) is obtained, the transverse pressure distribution is determined on the basis of its parabolic nature (^{12,15,16,18,19}),

The average dimensionless pressure generated by the trapped air inside the compressed snow layer is obtained by integrating the pore pressure over the entire surface,

The solid-phase (ice crystals) lifting force is obtained from quasistatic drained experiments performed by Wu (^{12}) and Wu et al. (^{16}), where static loads were applied incrementally on a snow layer, allowing air to escape freely from the porous media and all of the loading being supported by the solid phase. This force in its dimensionless form is given by

where *P*_{solid} (*x*′) is the local solid-phase pressure, *g*(*x*′) is the empirical relationship obtained in Wu (^{12}) and Wu et al. (^{16}), *φ*_{0} is the undeformed porosity of the snow layer, and *λ* = *h*_{2}/*h*_{0} is the compression ratio at the leading edge. The average pressure generated by the solid phase is then given by

The normal force balance on the skier, equation 2, is equivalent to

where *f*_{air} = *P*_{ava}′/*P*_{avload}′ and *f*_{solid} = *P*_{avs}′/*P*_{avload}′. Similarly, the moment balance, equation 3, can be written as

where *x*_{ccr}′ = *x*_{ccr}/*L*, *xa*′ = *xa*/*L*, *xs*′ = *xs*/*L*, *lc*′ = *lc*/*L*, and *Ff*′ = *Ff/mg* cos *αh*.

When a skier glides down a slope at velocity *U* over an undeformed snow layer of thickness *h*_{0} and Darcy permeability *K*_{0} without changing his or her location of CM, one has to adjust the tilt angle *γ* (or the compression ratio from the leading to trailing edge, *k* = *h*_{2}/*h*_{1}) as well as the compression ratio at the leading edge, *λ* = *h*_{2}/*h*_{0}, to satisfy the force and moment balance equations 15 and 16.

#### RESULTS AND DISCUSSION

The lift generation inside a compressible snow layer and its distribution between the trapped air and the ice crystals strongly depend on the geometry of the planing surface, *L*/*W*; the speed of the skier or snowboarder, *U*; the frictional coefficient *η*; the location of CM (*x*_{ccr}′, *lc*); and the properties of snow (*K*_{0}, *φ*_{0}, *d*), which have been extensively studied in Wu et al. (^{15,18}). In the current study, we shall focus on two major aspects of downhill skiing and snowboarding. First, we shall reexamine the force and moment balance equations and discuss their possible solutions, from which we shall provide a new set of parametric study as compared with the Wu-Weinbaum theory; second, we shall investigate the effect of width variation in the axial direction on the performance of skiing or snowboarding on the basis of our new model. The parameters chosen are as follows: *m* = 80 kg, *αh* = 15°, *η* = 0.04, *K*_{0} = 5.0 × 10^{−10} m^{2}, *φ*_{0} = 0.6, *d* = 0.42 mm, *h*_{0} = 10 cm.

For a given compression ratio, *λ*, at the leading edge (Fig. 1B), increase in the tilt angle (increasing *k*) will lead to the increase in both the lift force from the transiently trapped air and the ice crystals. Because the applied load is fixed, the compression ratio *k* from the leading to the trailing edge is uniquely determined on the basis of the force balance equation 2 or 15. One can adjust the compression ratio, *λ*, at the leading edge and obtain a series of (*λ*, *k*) that satisfy the force balance. Each (*λ*, *k*) combination generates a value of *xc*′ = *f*_{air}*xa*′ + *f*_{solid}*xs*′ − *Ff*′*l**c*′. When and only when (*xc*′ − *x*_{ccr}′) vanishes, the moment balance equation 3 or 16, is achieved.

##### Rectangular Snowboard or Ski (*f*(*x*) = 1)

We shall first look at the case of rectangular board as that discussed in Wu et al. (^{15,18}). For snowboarding, *L* = 1.16 m and *W* = 0.27 m, whereas for skiing, *L* = 1.7 m and *W* = 0.1 m. Unless specifically mentioned otherwise, the velocity of the skier is 20 m**·**s^{−1} and the location of the skier's CM is *x*_{ccr}′ = *x*_{cr}/*L* = 0.45, *lc* = 1.0 m for snowboarding and *x*_{ccr}′ = *x*_{cr}/*L* = 0.40, *lc* = 0.8 m for skiing (^{9}).

##### Multiequilibrium positions.

Figure 2A shows the change of (*xc*′ − *x*_{ccr}′) as a function of *λ* for snowboarding at 20 m**·**s^{−1} on a wind-packed snow layer that is 10 cm thick. It is clear from this figure that multiple solutions, corresponding to multiple zero values of (*xc*′ − *x*_{ccr}′), exist for this problem. In other words, there are several equilibrium positions for a given CM location. This observation reveals the nonlinear nature of the stability and control of skiing and snowboarding. It is a revision of the predictions of the Wu-Weinbaum theory (^{15,18}), where a unique (*λ*, *k*) was predicted. As a skier glides on a snow layer with an initial penetration depth at the leading edge, to achieve force and moment balance, he or she has to adjust the snow compression ratios both at the leading and trailing edges to the closest equilibrium location (*λ*, *k*) predicted in Figure 2A. One can shift from one equilibrium position to another by varying *λ*. This change is accompanied by a transfer of lift forces from the air to the solid phase or vice versa and a change in the angle of attack of the snowboard. Figure 2B shows the contribution of trapped air to the total lift, *f*_{air}, at different equilibrium positions. One notices from this figure that less compression at the leading edge (larger *λ*) leads to higher air lifting force, *f*_{air}.

It is evident from Figure 2B that the lift contribution from the trapped air, *f*_{air}, ranges from 31.1% to 58.4% for the different possible neutral equilibrium positions predicted in Figure 2A. The centerline pore pressure distributions and the associated solid-phase pressure beneath the surface of the snowboard for typical neutral equilibrium positions, (*λ*, *k*) = (0.715, 1.085), (0.685, 1.065), and (0.565, 1.015), are plotted in Figure 3. One finds from this figure that the general pore pressure distribution along the centerline of the snowboard is the same as that predicted by the Wu-Weinbaum theory, although multiple equilibrium positions exist for the same gliding condition. One notices that, with more compression at the leading edge, *λ* decreases, leading to the increase in the solid-phase pressure *ps*′ and thus decrease in the pore air pressure *p*′. Because less pore pressure is needed, less tilt angle is required and *k* decreases. Along with this decrease is the shift of peak pore pressure away from the trailing edge toward the middle of the snowboard, and the pore air pressure profile becomes flatter.

##### Parametric study.

*U* = 10, 20, and 30 m**·**s^{−1}.

As one increases his/her velocity, the contact time of the planing surface with the snow layer decreases, the air that is trapped inside snow has less time to escape before the pore pressure decays, and thus, one needs a smaller compression to generate the required lift force. This intuition is confirmed in Figure 4A where the lift contribution from the trapped air, *f*_{air}, for different possible neutral equilibrium positions are plotted as one snowboards on a wind-packed snow layer at velocities of 10, 20, and 30 m**·**s^{−1}. For *U* = 10 m**·**s^{−1}, *f*_{air} ranges from 24.4% to 42.8%, *λ* = *h*_{2}/*h*_{0} ranges from 0.525 to 0.625, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.065, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.517 to 0.587; for *U* = 20 m**·**s^{−1}, *f*_{air} ranges from 31.1% to 58.4%, *λ* = *h*_{2}/*h*_{0} ranges from 0.565 to 0.715, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.105, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.557 to 0.647; for *U* = 30 m**·**s^{−1}, *f*_{air} ranges from 38.2% to 68.5%, *λ* = *h*_{2}/*h*_{0} ranges from 0.585 to 0.775, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.095, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.576 to 0.708. It is clear that an increase in velocity leads to an increase in the trapped air's contribution to the total lift and a decrease in the compression of the snow layer.

The results shown in Figure 4A agree with the general predictions in the Wu-Weinbaum theory (^{15,18}) about the velocity effect on the lift generation; however, it provides multiple solutions for this nonlinear problem with multiple *f*_{air}, *λ*, and *k* combinations. This is in sharp contrast to the results provided in Wu et al. (^{15,18}), where a single *f*_{air}, *λ*, and *k* combination was predicted and summarized in Table 1 of Wu et al. (^{18}). The results presented herein are consistent with the experimental measurements in Wu et al. (^{13,16}) for the dynamic compression of wind-packed snow in a porous-walled cylinder-piston apparatus, which suggest that for snowboarding on fine-grained less permeable (wind-packed) snow at velocities greater than 10 m**·**s^{−1}, the lift forces generated by the trapped air in the compressed snow layer should provide a significant fraction of the total lift.

##### Friction (*U* = 20 m**·**s^{−1}, *η* = 0.02, 0.04, and 0.08).

Before the development of the Wu-Weinbaum theory (^{18}), most of the literature on ski or snowboard friction is focused on the presence of a micron-thick fluid film that forms beneath the running surface as a melt water lubrication layer (^{1-5}). Wu et al. (^{15,18}), for the first time, investigated the effects of surface smoothness on the lift generation underneath a ski or snowboard and obtained a single equilibrium solution for a certain value of friction coefficient. This result is reexamined in the current study.

In Figure 4B, we plot the lift contribution from the trapped air, *f*_{air}, for various values of friction coefficient, *η*, as one glides over a 10-cm wind-packed snow layer at a velocity, *U* = 20 m**·**s^{−1}. The values of *η*, 0.02-0.08, are chosen to span a broad range of thermal, snowboard bottom surface and ice crystal conditions (^{9}). For *η* = 0.02, *f*_{air} ranges from 43.1% to 64.5%, *λ* = *h*_{2}/*h*_{0} ranges from 0.625 to 0.755, *k* = *h*_{2}/*h*_{1} ranges from 1.035 to 1.115, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.604 to 0.677; for *η* = 0.04, *f*_{air} ranges from 31.1% to 58.4%, *λ* = *h*_{2}/*h*_{0} ranges from 0.565 to 0.715, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.105, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.57 to 0.647; for *η* = 0.08, *f*_{air} ranges from 32.1% to 47.1%, *λ* = *h*_{2}/*h*_{0} ranges from 0.525 to 0.615, *k* = *h*_{2}/*h*_{1} ranges from 1.010 to 1.035, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.520 to 0.590. It is clear from this figure that a decrease in the sliding friction leads to an increase in the trapped air's contribution to the total lift and decrease in the compression ratios at the leading and trailing edges, *λ* = *h*_{2}/*h*_{0} and *λ*/*k* = *h*_{1}/*h*_{0}. These results indicate that for downhill skiing and snowboarding, a decrease in the sliding friction not only reduces the snow frictional drag force that the skier encounters as he or she glides down the slope at a given speed but also increases the trapped air's contribution to the total lift. It provides a new insight into the role of friction in skiing and snowboarding performance. For a smooth air-cushioned glide, one wants a low value of *η*. This can be obtained by waxing the bottom surface of the snowboard or ski.

##### Shift of *xc*′.

Critical insights about the stability and control of skiing or snowboarding were obtained by Wu et al. (^{15,18}), who show that a snowboarder can keep in equilibrium positions by altering his/her CM. For snowboarding, this is achieved by shifting their weight from the front to the rear foot. Under most conditions, this shift in *x*_{cr}′ is small and varies between roughly 0.37 and 0.47. However, as shown in Figure 4C, it has a dramatic effect on the distribution of the lifting forces between the air and solid (ice crystal) phases. When one places more of their weight on the rear foot while snowboarding, *x*_{cr}′ decreases and approaches the lower limit of 0.373, whereas when one shifts their weight to the front foot, *x*_{cr}′ increases and approaches the upper limit of 0.473. When *x*_{cr}′ = 0.473, *f*_{air} ranges from 34.6% to 47.1%, *λ* = *h*_{2}/*h*_{0} ranges from 0.555 to 0.615, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.035, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.547 to 0.594. The contribution of the trapped air pressure is small; larger snow compression is required at both the leading and trailing edges to generate enough lift to support the skier. In contrast, when *x*_{cr}′ = 0.373, *f*_{air} ranges from 81.3% to 85.7%, λ = *h*_{2}/*h*_{0} ranges from 0.965 to 0.995, *k* = *h*_{2}/*h*_{1} ranges from 1.31 to 1.35, and *λ*/*k* = *h*_{1}/*h*_{0} is around 0.737. The contribution of the trapped air pressure is large; the tilt angle and the snow compression ratio at the leading and trailing edges are smaller.

Figure 4C shows the neutral stability condition in which both the force and moment balance equations are satisfied. If one shifts their weight (changes *x*_{cr}′) and does not change their angle of attack, the equilibrium condition is broken. To avoid falling and return to a new neutral stability condition, one must adjust the snow compression ratios at both the leading and trailing edges, corresponding to the changes in the *λ* and *λ*/*k* as prescribed by our revised model.

##### Shift of *lc*.

The vertical displacement of the CM will also affect the snowboarder's stability. Similar to the shift of *x*_{cr}′ in Figure 4C, if one raises or lowers his or her CM vertically through up-turn-down or down-turn-up body motion without changing their compression ratios at the leading and trailing edges, the initial neutral force and moment balance is broken. To maintain stability, one has to input a muscular moment or change the compression ratios of the snow layer to get back to a new neutral moment balance position. The latter requires no muscular input and is accompanied by a transfer of lift forces between the trapped air and the solid ice crystals as well as changes of snow compressions at the leading and trailing edges. We have plotted in Figure 4D the lift contribution from the trapped air, *f*_{air}, for different values of *lc*, *lc* = 0.8, 1.0, and 1.2 m, as one glides over a 10-cm wind-packed snow layer at a velocity, *U* = 20 m**·**s^{−1}. For *lc* = 1.2 m, *f*_{air} ranges from 31.2% to 51.1%, *λ* = *h*_{2}/*h*_{0} ranges from 0.565 to 0.665, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.055, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.557 to 0.630; for *lc* = 1.0 m, *f*_{air} ranges from 31.1% to 58.4%, *λ* = *h*_{2}/*h*_{0} ranges from 0.565 to 0.715, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.085, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.557 to 0.659; for *lc* = 0.8 m, *f*_{air} ranges from 31.2% to 64.5%, *λ* =*h*_{2}/*h*_{0} ranges from 0.565 to 0.755, *k* = *h*_{2}/*h*_{1} ranges from 1.015 to 1.115, and *λ*/*k* = *h*_{1}/*h*_{0} ranges from 0.557 to 0.677. It is clear that as one vertically lowers his or her CM from *lc* = 1.2 m to *lc* = 0.8 m, the potential trapped air's contribution to the total lift increases, and the solid-phase lift contribution decreases accordingly. Meanwhile, the required compression at the leading and trailing edges decreases. As shown in Figure 4D, for the same precompression at the leading edge, *λ*, one could obtain the same lifting force from the pore air pressure with different *lc*. However, smaller *lc* requires less compression at the trailing edge.

##### Skiing versus snowboarding.

As predicted by Feng and Weinbaum (^{7}) and Wu et al. (^{15,18}), one needs larger snow compression for skiing than for snowboarding to generate the required lifting forces to support the weight of the skier or snowboarder. This is due to the fact that the width of a ski is one-third that of a snowboard, and thus, larger pore pressure relaxation happens at the lateral edges. In Figure 5, we have compared the lift contribution from the trapped air, *f*_{air}, for skiing and snowboarding as one glides over a 10-cm wind-packed snow layer at a velocity, *U* = 20 m**·**s^{−1}. For snowboarding, *L* = 1.16 m, *W* = 0.27 m, *x*_{cr}′ = 0.45, *lc* = 1 m, whereas for skiing, *L* = 1.7 m, *W* = 0.1 m, *x*_{cr}′ = 0.4, *lc* = 0.8 m. As shown in this figure, multiple equilibrium positions are found in the case of snowboarding, with *f*_{air} ranging from 31.1% to 58.4%, *λ* = *h*_{2}/*h*_{0} ranging from 0.565 to 0.715, *k* = *h*_{2}/*h*_{1} ranging from 1.015 to 1.105, and *λ*/*k* = *h*_{1}/*h*_{0} ranging from 0.57 to 0.647. In contrast, limited equilibrium condition is found for the case of skiing. For the same precompression ratio at the leading edge, *λ* = 0.685, *f*_{air} is 53% for snowboarding and is 43.4% for skiing, *k* = *h*_{2}/*h*_{1} is 1.065 for snowboarding and is 1.285 for skiing, and *λ*/*k* = *h*_{1}/*h*_{0} is 0.643 for snowboarding and is 0.533 for skiing. It is clear that for the same required compression at the leading edge, the tilt angle *γ* of the planing surface (or the compression ratio from the leading to trailing edge, *k* = *h*_{2}/*h*_{1}) is much larger in skiing than in snowboarding, and thus, the snow layer at the trailing edge of the ski is compressed more, and one obtains more pore air lift for snowboarding than for skiing. The difference in *f*_{air} is not great considering the large difference in *L*/*W*. This is because the skier's CM is closer both to the trailing edge and to the board, and one needs to increase the tilt angle *γ* to satisfy the moment balance equation. By doing so, the center of pore air pressure shifts toward the trailing edge enhancing the pore pressure's contribution to the total lift and compensates in part for the pressure leakage at the lateral edges.

##### Real Snowboard or Ski (*f*(*x*) ≠ 1)

We choose a commercially available snowboard, A, as our studying case. The characteristic dimensions of the snowboard are as follows: total length = 1.38 m, effective edge length = 1.175 m, nose length = 0.308 m, and waist length = 0.264 m (Fig. 1C). For the same gliding condition, one obtains multiple solutions. In other words, there are several equilibrium positions for a given CM location as observed in the previous section for rectangular skis or snowboards. Figure 6A shows the contribution of trapped air to the total lift, *f*_{air}, at different equilibrium positions. It is evident that the lift contribution from the trapped air, *f*_{air}, ranges from 30% to 60% for the different possible neutral equilibrium positions. Less compression at the leading edge (larger *λ*) leads to higher air lifting force, *f*_{air}.

In Figure 6B, one plots the centerline pore pressure distributions beneath the surface of snowboard A for typical neutral equilibrium positions, (*λ*, *k*) = (0.585, 1.020), (0.655, 1.045), and (0.715, 1.080). It is observed from this figure that there are two pressure peaks along the length of the snowboard occurring near the nose section where the width of the snowboard is the maximum. This is in sharp contrast with the results predicted in Wu et al. (^{15,18}) as well as in the previous section for rectangular snowboards where a single pressure peak was observed. One attributes this behavior to the fact that the increased width of the snowboard at the nose prolongs the outflow of the trapped air in the lateral direction; hence, on the time scale that the snowboard is in contact with the snow, higher pore pressure is generated at these two sections. On the basis of similar reasoning, a narrower section of the snowboard near the waist section that is near the center of the snowboard entails a rapid pressure relaxation and hence a drop in the pore air pressure buildup. Evidently, the pore pressure reaches maximum near the trailing edge where maximum snow compression occurs. One also notices that as the compression ratio at the leading edge decreases, the compression ratio from the leading to the trailing edge, *k*, decreases. In other words, if one compresses the snow at the leading edge enough, he or she does not need much tilt angle to generate the required lifting forces.

#### CONCLUSIONS

In summary, we have developed a new realistic model for the lift mechanics of downhill skiing and snowboarding, which incorporates the shape variation effect in the lift force generation. We have performed a thorough discussion on the neutral stability of a skier or snowboarder and obtained multiple equilibrium solutions for the same gliding conditions. This is a significant modification to the Wu-Weinbaum skiing mechanics theory (^{15,18}). We have reexamined the results and conclusions in Wu et al. (^{18}) by performing a comprehensive parametric study in comparison with the results listed in Table 1 of Wu et al. (^{18}). Furthermore, the Wu-Weinbaum theory was extended to more realistic applications by taking into account the shape variation in the axial direction. A completely different pore pressure profile was observed underneath a real ski or snowboard.

Significantly, our revised model opens a new direction for the improvement of the performance of skiing or snowboarding by optimizing the shape of the board. For a smooth air-cushioned glide, one wants to have more lift from the pore air pressure; meanwhile, with more air trapped inside the snow layer, the frictional force that is proportional to the solid-phase lifting force decreases. Thus, the contribution of the transiently trapped air to the total lift, *f*_{air}, provides us a criterion for evaluating the performance of a snowboard. For example, we choose five commercial snowboards A**,** B**,** C**,** D, and E for comparison; their geometries are given as following: total length, effective edge length, nose length, waist length = 1.38, 1.175, 0.308, and 0.264 m for A; 1.42, 1.175, 0.308, and 0.264 m for B; 1.50, 1.170, 0.290, and 0.240 m for C; 1.56, 1.190, 0.294, and 0.250 m for D; and 1.62, 1.264, 0.315, and 0.268 m for E. Under the same gliding condition (*U* = 20 m**·**s^{−1}, *m* = 80 kg, *αh* = 15°, *η* = 0.04, *K*_{0} = 5.0 × 10^{−10} m^{2}, *φ*_{0} = 0.6, *d* = 0.42 mm, *h*_{0} = 10 cm, *x*_{ccr}′ = 0.45, *lc* = 1.0 m), the contribution of pore air pressure to the total lift, *f*_{air}, ranges from 31% to 60% for A, from 31% to 64% for B, from 29% to 63% for C, from 32% to 59% for D, and from 34% to 67% for E. Obviously, snowboard E performs better than the other four snowboards. On the other hand, by optimizing the shape of the board to achieve higher value of *f*_{air}, one could improve the performance of skiing and snowboarding (^{17}). The optimization involves extensive parametric studies and is another research project that is performed by our group at the Cellular Biomechanics and Sports Science Laboratory at Villanova University.

Lift generation inside highly compressible porous media under rapid compaction is a new concept for porous media flow. This concept was originally applied to explain the pop-out phenomena of red cells gliding over the endothelial glycocalyx in a tightly fitting capillary (^{7}), revealing the mystery of the longevity of red blood cells. Its application for downhill skiing and snowboarding (^{15,17,18}) reveals the mystery of how a 70-kg skier could glide over a soft snow layer without sinking to the base as would occur if the motion was arrested. An extreme application of this concept was recently developed by Wu et al. (^{13}) and was extended by Mirbod et al. (^{10}), where a futuristic train that runs on a soft porous track with minimum frictional drag was proposed. These applications were chosen for their novelty, yet the basic idea of using highly compressible porous materials to generate greatly enhanced lift forces has significant potential in the design of soft porous bearings with tremendously increased lubrication pressures and long life. Recently, a systematic approach has been developed by our group to examine the lift generation inside a soft porous medium with applications to soft porous bearings. A prototype of such a bearing has been constructed in the Cellular Biomechanics and Sports Science Laboratory at Villanova University. The model presented in this article, applied for the lift mechanics of downhill skiing and snowboarding, can be readily extended to a generalized lubrication theory for highly compressible porous media, which will provide theoretical foundations for the design and characterization of the new soft bearing systems.

This research is partly supported by Villanova University 2010 Summer Research Fellowship and Research Support Grants.

The results of the present study do not constitute endorsement by the American College of Sports Medicine.